Statically indeterminate structures when you observe a building under construction, you can see that there are beams supported on multiple columns. Indeterminate and statically determinate structures in this chapter we shall examine methods of analysis of different forms of statically 472 • chapter 16 / analysis of statically indeterminate structures figure 166 determination of the entire structure a g b b b e e e g g ff a. Chapter 2 statically indeterminate structures 2-1- introduction it is customary to divide structures into statically determinate and statically release the reaction at support 1 to reduce to a statically determinate system the deflection at point 1 will be ei ei wl d 324 8 4. Fe exam review for structural analysis prof v saouma oct 2013 deflection analysis of statically determinate beams, trusses and frames 3 stability analysis of beams, trusses and frames elementary statically indeterminate structures the only page in the “supplied-reference handbook” related to structural analysis. Code/name: ce1303 structural analysis-i year: iii sem:v unit-i deflection of determinate structures 1why is it necessary to compute deflections in structures computation of deflection of structures is necessary for the following reasons: a if the deflection of a structure is more than the permissible, the structure will not look.
102 types of statically indeterminate beams is called the degree of static indeterminacy 2 the excess reactions are called static redundants the structure that remains when the redundants are released is called released structure or the primary structure thus the slope and deflection equations are px. In statics, a structure is statically indeterminate (or hyperstatic) when the static equilibrium equations are insufficient for determining the internal forces and reactions on that structure based on newton's laws of motion, the equilibrium equations available for a two-dimensional body are ∑ → =: the vectorial sum of the forces acting on the body equals zero. Reactions and internal forces in statically indeterminate structure in the analysis since flexibility method requires deflection of statically determinate structure, a table of formulas for deflections for various load cases and boundary conditions is also given in this lesson for ready use the force method of analysis.
Determinate and indeterminate structures: structure is an assemblage of a number of components like slabs, beams, columns, walls, foundations and so on, which remains in equilibrium it has to satisfy the fundamental criteria of strength, stiffness, economy, durability and compatibility, for its existence. Structures of reinforced concrete for continuous frames and can be considered one of the greatest contributions to the analysis from indeterminate structures this method. Deflection of a truss the virtual work method can be used to determine the deflection of trusses we know from the principle of virtual work for trusses that the deflection can be calculated by the equation with n equal to the virtual force in the member and equal to the change in length of the member. It is generally classified into two categories as determinate and indeterminate structures or redundant structures any structure is designed for the stress resultants of bending moment, shear force, deflection, torsional stresses, and axial stresses.
Determinate and indeterminate structure december 12, 2016 shanmukha leave a comment structure is an assemblage of a number of components like slabs, beams, columns, walls, foundations and so on, which remains in equilibrium. Deflection of statically determinate beams and frames – other methods lecture 11 – beam and frame deflections a qualitative deflected shapes 1 qualitative deflection shapes • the elastic curve – is the curve that connects the centroids of all the cross-sectional areas along the length of a deflected member. Statically determinate of statically indeterminate if statically indeterminate, report the degrees of of determinacy the appropriate slope or deflection equations, yields expressions that supplement the structure involved 13 lecture 18 beams: statically indeterminate.
M9 truss deflections and statically indeterminate trusses truss deflection example calculate deflection of loading point e in pin-jointed truss shown below bars are at 90° or 45°to each other all bars have cross sectional area a, young's modulus e no temperature change occurs. Analysis of statically determinate structures the most common type of structure an engineer will analyze lies in a plane subject to a force system in the same plane. Deflections deflection diagrams and the elastic curve usually, before the slope and deflection are calculated, it is important to sketch the shape of the structure when. Unit-i deflection of determinate structures 1 why is it necessary to compute deflections in structures computation of deflection of structures is necessary for the following reasons: a if the deflection of a structure is more than the permissible, the structure will not look.
This course explores the analysis and design of engineering structures considering factors of deflection, buckling, combined loading, & failure theories ----- the copyright of all content and materials in this course are owned by either the georgia tech research corporation or dr wayne whiteman. The conjugate-beam method is an engineering method to derive the slope and displacement of a beam the conjugate-beam method was developed by h müller-breslau in 1865 the conjugate-beam method was developed by h müller-breslau in 1865. Bending deflection – statically indeterminate beams ae1108-ii: aerospace mechanics of materials aerospace structures & materials faculty of aerospace engineering dr calvin rans dr sofia teixeira de freitas recap split into two statically determinate problems 3) compatibility:. Deflection of determinate structures - answer (click here) moving loads and influence lines 1 deflection of determinate structures - answer (click here) 2 influence lines - answer (click here) 3 muller bresalu principle, begg's formeterde - answer (click here) 4.
This video demonstrates how to calculate the reactions and draw shear and moment diagrams of a statically indeterminate beam by using the method of superposition. However, certain methods could be more applicable depending on the characteristics of the structure or truss slope deflection method is more specific to indeterminate structures with low degrees of freedom contents 1 introduction solve the newly formed determinate structure. The traditional procedure for employing the virtual work equation to calculate structural deflections of statically indeterminate structures requires the solution of the internal action state of the structure to be available or to be calculable. Questions & answers on analysis of statically determinate structures and trusses the section contains questions and answers on idealized structure, determincy and stability, trusses types, coplanar trusses classification, joints and sections methods, space trusses and assumptions.
Introduction to statically indeterminate analysisindeterminate analysis s pport reactions and internalsupport reactions and internal forces of statically determinate structures can be determinedcan be determined using only the equations of equilibrium. Analysis of statically determinate trusses theory of structures asst prof dr cenk üstündağ common types of trusses a truss is one of the major types of engineering structures which provides a practical and economical solution for many engineering constructions, especially in the design of bridges and buildings that. Structure is generally classified into two categories as determinate and indeterminate structures or redundant structures for analysis of structures to find forces based on criteria discussed below structure is an assemblage of a number of components like slabs, beams, columns, walls, foundations and so on, which remains in equilibrium.


Mathematics can be a dreaded subject for many kids, but it doesn’t have to be that way. With the right tools and resources, math can be an enjoyable and fun experience. One such tool is cool math games. These activities are designed to make learning math fun and interactive for kids of all ages. In this blog, we will explore the benefits of cool math games for kids and how they can help your child excel in mathematics. We will also provide you with a list of offline and online math games that are free that you can play with your child at home or in the classroom.
Benefits of Cool Math Games for Kids
Cool Math Games offer a fun and engaging way for children to learn and practice math. These games provide an environment where kids can improve their math skills without the added stress of traditional classroom learning. Not only do they help kids develop problem-solving and critical thinking abilities, but they also allow parents to bond with their children while promoting learning. Teachers can use these games as a tool to supplement classroom teaching and reinforce math concepts. Cool Math Games offer a creative and interactive way for children to have fun while improving their math skills. If you’d like to know more about what kind of video games can make you or your child smarter, click here.
Offline Cool Math Games for Kids
Many offline math games that are free leverage various age-appropriate learning principles such as memory retention, critical thinking, and logic. These games can be used by parents and teachers to supplement traditional classroom learning.
Finding the right cool math game for your child is not difficult. The internet has several free resources that provide lists of popular offline cool math games, along with instructions on how to download them. Here is an example of such a list. With the help of these games, children can learn to solve mathematical problems creatively and effectively. Teachers can also use these games to make learning more interactive and fun.
Here are 5 examples of coolmath games for free that anyone can start playing:
- Math War: This game is played with a deck of cards. The players divide the deck equally and each player turns over two cards at a time. The player with the highest sum wins both cards. The player with the most cards at the end of the game wins.
- Math Bingo: This game is played like traditional bingo, but with math problems instead of numbers. The teacher calls out a math problem and the students mark their bingo card if they have the answer.
- Math Scavenger Hunt: This game is played by hiding math problems around the room or outside. The students must find the problems and solve them.
- Math Jeopardy: This game is played like the TV show Jeopardy. The teacher or parent creates categories and questions related to math. The students work in teams to answer the questions.
- Math Dice: This game is played with two dice. The teacher gives a math problem and the students roll the dice to get the answer. The first student to get five correct answers in a row wins.
DIY Math Games
When it comes to cool math games for kids, DIY options are a great way to engage children and help them develop their math skills. These games can be created using everyday materials such as dice, cards, and even Legos. DIY math games can help children learn important math concepts such as counting, addition, subtraction, multiplication and division. By creating their own games, children can also develop their creativity and problem-solving abilities while having fun at the same time. So next time you’re looking for a fun way to teach math to your kids, consider trying out some of these DIY math game ideas! Note however that these activities are most ideal for young children such as preschoolers.
Here are our top 3 picks for DIY cool math games:
- Number on the Floor Relay Race: If your little one is always bursting with energy and love to run around, then we’ve got a game for you that combines both their love for movement and fun! All you’ll need is some trusty painter’s tape, and your job is to create large numbers on the floor. Once that’s done, let the game begin! One player calls out numbers at random while the other dashes to the corresponding number on the floor. To really get the competitive juices flowing, why not time each other for some math-related excitement? So get ready to have a blast while learning, moving, and having loads of fun! Make everyone add and keep their own score to add some more maths to it.
- Flip the Pancake Equations: Instead of mixing up batter, you’ll be cutting out circles of cardboard to make your very own “pancakes”. You’ll be writing addition and subtraction equations on each “pancake”, with the corresponding answer on the flip side. Once that’s done, it’s time for the cooking showdown to begin! Your little chef will pick up a pancake and shout out the answer to the equation, before flipping it over with the spatula. If they get the answer right, they get to keep the pancake. If not, their opponent swoops in to claim it for themselves. Who will emerge as the ultimate pancake king or queen? The player with the most pancakes at the end, of course! So grab your spatula and get ready for some math-fueled cooking fun!
- Create a Pretend Grocery Store: With just a grocery store flier and some imagination, your child can transform into a customer and you into the friendly cashier. Cut out pictures of grocery items and their prices from the flier and pick out the items you have in your own home. Then, armed with real or fake money (depending on how generous or lazy you’re feeling), your little shopper can pick out an item from the flier, find the corresponding object in your house, and try to purchase it from you and figure how much money they need to pay or get back. The possibilities are endless as you role play different scenarios, with or without a register. With so much math involved, who said grocery shopping can’t be fun? So let’s get ready to shop ’til we drop!
Board Games that Teach Math
Fun games like Monopoly, Settlers of Catan, and Ticket to Ride all require strategic thinking and basic math skills like addition, subtraction, and multiplication. Other popular games like Math Dice or Sum Swamp are specifically designed to teach mathematical concepts in a more structured manner.
Here is a short description of the mentioned games:
- Math Dice – This game is designed for kids aged 8 and up. It helps children develop their math skills by using dice to solve problems. The game can be played by two or more players.
- Sum Swamp – This game is designed for kids aged 5 and up. It helps children learn basic math skills such as addition and subtraction. The game can be played by two to four players.
- Monopoly – This classic game is designed for kids aged 8 and up. It helps children learn basic math skills such as counting money and making change. The game can be played by two to eight players.
- Settlers of Catan – This strategy game is designed for kids aged 10 and up. It helps children learn basic math skills such as probability and resource management. The game can be played by three to four players.
- Ticket to Ride – This game is designed for kids aged 8 and up. It helps children learn basic math skills such as counting and probability. The game can be played by two to five players.
In addition to board games, offline cool math games also include puzzles and brain teasers, such as Sudoku or chess, that help develop critical thinking skills while reinforcing mathematical concepts. These types of games provide a fun and engaging way for kids to improve their math skills outside of the classroom environment.
Online Math Games That Are Free For Kids
Many of these games are easily accessible online for free, which makes them an affordable learning resource. Teachers can use these games as a teaching aid to make math more interactive and engaging in the classroom. Parents can also use these games to help their children improve their math skills outside of school hours.
There are many cool math games available online catering to different age groups and skill levels. These fun and interactive games not only improve mathematical skills but also critical thinking and problem-solving abilities in children. With so many options available online, it is easy to find a game that suits your child’s interests while making learning enjoyable at the same time. Here is a great resource with educational games to get you started.
Find Our Cool Math Games to Play by Category:
If you’re looking for cool math games for your kids to play online, you can easily find them organized by category. These categories can include addition, subtraction, multiplication, division, geometry, and more. This website is one such great resource. By selecting the appropriate category, you can help your child focus on a particular skill or concept that they need to work on. With a variety of options available, there are endless possibilities for making learning fun and engaging through these cool math games.
Parental Control Tools to Ensure Safe Gaming Experience.
Safety should be a top priority while allowing kids to play online games. Before letting your child play free math games online, it’s important to ensure that they have a safe gaming experience. Parental control tools can help you achieve this objective. Setting up age-appropriate content filters is vital to keeping your child away from inappropriate content. Monitoring the amount of time your child spends playing games and setting limits if necessary will ensure that their gaming experience remains healthy and balanced.
Educate your child on internet safety and the importance of not sharing any personal information while gaming. Also, encourage them to communicate with you about any concerns or issues related to their online gaming experience. With these measures, you can ensure that your child has an enjoyable, safe, and educational experience playing cool math games for free!
Cool math games are an excellent way to make learning math fun and engaging for kids. It helps them develop critical thinking skills, problem-solving strategies, and logical reasoning abilities. Offline games like DIY math games and board games that teach math can be played with family and friends. On the other hand, online cool math games provide access to a wide range of games by grade level or category. However, it is crucial to ensure that children have a safe gaming experience, and parental control tools can help with that. Click here, if you want to learn more about other benefits of gaming.


Welcome Sophomore class!
This is my 3rd year teaching at The SAE and I am looking forward to teaching you math. You will find the topics we will cover this year below in the attached class Syllabus.
Trigonometry Test 11/1 - 11/2
Trig functions, inverse trig functions and modeling
Unit 1 Test - Monday/Tuesday 10/9-10/10
Similar Triangles, Special Right Triangles and using the Tangent Ratio.


Multiplication Posters with multiples 1-12
This resource features multiplication posters that display multiples 1-12 of large numbers ranging from 1 to 15. These posters are designed to be a helpful reference for students of all ages, providing a comprehensive overview of multiplication tables.
With a selection of 3 themes to choose from, including watercolor background, bright chevron background, pastel swirls, rainbow watercolor and farmhouse chic (shiplap), there is a poster to suit any classroom or learning environment. Additionally, there is a black and white option available for those seeking an ink-friendly alternative.
These posters are a fantastic way to capture your student's attention and encourage them to engage with multiplication concepts. Print them on brightly colored paper to make them even more eye-catching and engaging!
These posters are part of the math posters bundle HERECopyright © To The Square Inch LLC
All rights reserved by the author.
Permission to copy for single classroom use only.


Ketologger: What is the GKI?
Every once in a while I get a question asking: What is the GKI? What is g:k? Why is it green sometimes in the "Both" table.
This is the authoritative article on GKI:
"The glucose ketone index calculator: a simple tool to monitor therapeutic efficacy for metabolic management of brain cancer"
There's a lot of math in it but a lot of it deals with how to convert glucose values from mg/dL to mmol/L. (They use mM but as far as I can tell that is equivalent to mmol/L.) The GKI or g:k is simply glucose in mmol/L divided by ketones in mmol/L.
The image below is similar to the one displayed in the article except made from a Ketologger screen capture.
This is a glucose vs. ketone graph.
Glucose starts high and drops to a low level. Ketones start low and rise to a level someone might achieve when in nutritional ketosis.
Although I'm working on adding GKI graphing to Ketologger one of the reasons that I didn't from the start is that by looking at the glucose vs. ketone graph you can see a visual representation of the GKI. I'll explain below.
On the left of the graph in the screenshot above you see high/normal glucose and low ketones. This represents someone who is eating carbs and not in ketosis. Since glucose is high and ketones low you'll get a number higher than one for the GKI. (Eg. 6 divided by 1 gives you 6. Definitely larger than 1.)
As the graph moves to the right it represents someone who has lowered their carb intake enough so that ketones start to rise.
Near the center is where glucose and ketones meet. This is where the GKI is 1.0. When glucose in mmol/L equals ketones in mmol/L. Any number divided by itself is one. (Eg. 2.5 divided by 2.5 = 1.0)
Going rightward ketones have surpassed glucose. Most likely this person is in strict low carb or even maybe doing extended fasting. (Glucose has dipped quite a bit below what is considered normal fasting glucose for a non diabetic. For someone fasting or maybe even extermely low carb this may feel ok but it would probably be unpleasant to say the least for someone who is not fat adapted.) In this case glucose is a smaller number than ketones so when you divide glucose by ketones you get a number that is less than 1. That is a fraction. For eg. 2 divided by 4. That gives you on half which is 0.5.
To see that the GKI is less than one you can simply take look at the graph and see that ketones are above glucose. Or you can do the division of glucose by ketones and have the mathematical "picture" and know when the GKI is below 1.0.
This is the "Both" history view in Ketologger which pairs up glucose and ketone readings. This matches the graph in the screenshot above. The GKI/G:K is displayed on the right. A G:K of less than or equal to 1.0 is highlighted in green.
In the "Both" table Ketologger pairs up the closest readings of glucose and ketones and calculates the GKI (by dividing glucose by ketones in mmol/L). You'll see it in green when it's at or below 1.0.
As mentioned I'm working on adding the GKI graph on it's own which will appear in a future release.


Or, "How Do I Get To Become a Quiz General?"
So you just took one of our Civil War Trust quizzes and you managed to get ranked as a Quiz Corporal. How did that happen? How do you score these infernal quizzes?
Well, here's an inside look at the math behind our scoring process:
Each quiz question is initially worth 100 points.
- You get 10 seconds to answer the question. If you get it right in ten seconds or less you get 100 points for that question. The timed portion of the quiz does not apply to the answer pages.
- After 10 seconds have elapsed, you begin to lose 2 points per each second. Uh-oh!
- After 40 seconds have elapsed, the quiz stops deducting points for time. The maximum score that you can get for a right answer is then 30 points.
- The quiz ignores fractional seconds. Don't worry -- this is not a 100m Olympic sprint.
- You get 0 points for each incorrect answer. Sorry, no partial credit.
After you have answered all the questions the scores from each question are added together to produce a total score. We then compare that score against the maximum total score to produce a range of quiz grades.
Those grades are:
- 85% or better = Quiz General (time to start bragging)
- 75% or better = Quiz Colonel (pretty darn good if you ask us)
- 50% or better = Quiz Captain (still an officer)
- 49% or lower = Quiz Corporal (I hope you don't mind sleeping on the ground?)
At the end of each timed quiz, you should be able to see your score and the quiz grade. From that page, you can let all your Facebook fans know how you did on the quiz.
Any questions or suggestions regarding our quizzes? Write to us at firstname.lastname@example.org.


Printable Multiplication Worksheet 4Th Grade. It's time for them to practice repeated addition and equal groups to understand multiplication better. Finding the total with arrays.
Finding the total with arrays. Students also learn how to multiply difficult numbers in. Grade 4 multiplication recommended topics for you multiplication facts
As Multiplication Is A Complex Yet Crucial Concept, These Worksheets Simplify The Studying Process Through Fun Exercises And Problems That.
Here you will find our selection of printable 4th grade multiplication worksheets which will help your child learn to multiply a range of numbers up to 4 digits by a single digit. A comprehensive lesson plan this lesson contains everything that you need to introduce division to your students. Web the pdf based printable multiplication worksheets for 4th grade can be solved offline once downloaded from the website as per the student’s convenience.
Web 4Th Grade Math Worksheets:
Web grab your free 4th grade multiplication worksheets. Web free printable multiplication worksheets for grade 4 multiplication mastery awaits! Parking lot multiplication word problems #1.
Web These Free Printable Multiplication Worksheets Are A Great Practice Or Refresher For Homeschool Math.
Splashlearn's multiplication worksheets for 4th graders can help them enhance their multiplication skills with learning retention. We have a separate page for our grade 4 multiplication in columns worksheets. All you need to do is download,… subjects:
Basic 4Th Grade Math Worksheets To Help Your Student.
Finding the total with arrays. Web grade 4 multiplication worksheets you are here: It includes several types of activities for your grade 4 students such as coloring pages, mystery pictures, classic worksheets, and other printables.
Array Multiplication (Part One) Worksheet.
Web our grade 4 multiplication in columns worksheets range in difficulty from 2 x 1 digits to 3 x 3 digits. First, you’ll receive four fun multiplication worksheets. Web 4th grade multiplication worksheets (154) results found sort by:


Class Information for MGT 3: Quantitative Methods in Business - Willoughby
Session: MGT 3: Quantitative Methods in Business - Willoughby
Class Size: 0
# Openings: 0
Class Session Date(s): 09/28/17 - 12/15/17
Registration Dates: 09/15/2017 03:00 PM - 10/23/2017 08:00 AM
Current Price: $0.00
MGT 3: Quantitative Methods in Business Prerequisites: none Quantitative Methods in Business Introduces techniques for developing and analyzing data to make informed management decisions: statistical inference, probability, regression analysis, and optimization. Using these analytics approaches, theory based formulas and spreadsheets; students explore managerial applications across all areas of business activity.
Additional InformationNot sponsored. You will need to wait list for this class.
This session is full. Join the Waiting List


The hyperreal line is an extension of the real line. It contains all real numbers, and some infinite and infinitesimal numbers.
We say that an element of the hyperreal line x is finite iff |x| < n for some integer n. x is infinite iff it is not finite. x is infinitesimal iff |x| < 1/n for all integers n. (Note that 0 is trivially infinitesimal.)
In the normal real scale without magnification, the hyperreal line (nonstandard analysis' version of the real line), looks like the real line, thus:
/ -1 0 1 2 \ / ........... __________|___|___|___|_________ ..........\ \ negative finite positive / \ infinite infinite/
What makes the hyperreal line different from the real line is that it contains an infinite number H, and an infinitesimal 1/H = d.
If we look at 1 through an "infinitesimal microscope", and magnify by the infinite amount H, then we can distinguish points which are an infinitestimal distance d apart, thus:
1-2d 1-d 1 1+d 1+2d 1+3d ____________|______|______|______|______|______|______________
If we look at H through an "infinite telescope", then we can see infinite numbers on the ordinary real scale, thus:
H-2 H-1 H H+1 H+2 H+3 _____________|______|______|______|______|______|______________The ideas of the infinitesimal microscope and infinite telescope are due to Keisler. A more detailed version of this page can be found in his paper, "The hyperreal line", in Real numbers, generalizations of the reals, and theories of continua (ed. Ehrlich P) Kluwer Academic Publishers, Netherlands
Back to NSA page


Sunday, April 16, 2006
Thanks to Suzanne for sending a link to this fascinating project. The Institute For Figuring is crocheting a coral reef--in their words, it's "a woolly celebration of the intersection of higher geometry and feminine handicraft, and a testimony to the disappearing wonders of the marine world. " They're accepting samples from crocheters. Here's the complete article.
On a related topic, NPR just did a piece on hyperbolic geometry and crochet. A lot of the pictures look fairly similar to the coral reef representations.


Maths TeacherMPS-UPS Permanent, full time contractRequired September 2023 (or earlier if available)
We are looking to recruit an enthusiastic and innovative teacher of Mathematics to join our strong team of subject specialists in September 2023 or earlier if available. Since the introduction of the new specification, 85% of Year 11 students at Norton Hill have achieved a standard pass and 65% have achieved a strong pass in GCSE Mathematics. In A-Level Mathematics 70% of students achieved A*-B passes and in Core Maths 80% of students achieved A or B passes.
Closing date: Friday 10th March Interviews: To be confirmed


What is the volume of one mol of iron where iron is 558 g/mol and the density of iron is 786 x 10^3 kg/m^3 2 relevant equations p = m/v 3 the attempt at a solution i can get the numerical answer (checked back of book), but i'm not sure how my units worked out v = m/p = (558 g/mol) / (786 x 10^3. Of 24 dm 3 or 24,000 cm 3 at rtp (room temperature and pressure) this volume is called the molar volume of a gas this equation shows how the volume of gas in dm 3 at rtp is related to the number of moles: volume of gas at rtp = number of moles × 24 qquestion calculate the volume of 05 mol of carbon dioxide at rtp a. Density of water is 1000 g/l the gram molecular mass of water is 1802 the number of moles in 1 litre = 1000 / 1802 = 555 moles 1 mole of water = 6022 x 10^23 molecules of water so 555 moles of water = 555 x 6022 x 10^23 = 334 x 10^25 molecules of water 334 x 10^25 molecules of water = 1 litre 1 molecules. Many packages mentioned above will be able to give more accurate results, but the use of the added accuracy is questionable very small changes in conformation of a molecule or changes in the hydrogen bonds around the molecule, can have surprisingly large effects on the very well defined voronoi volumes. Calculations based on formulae 76 - molar volume of a gas when the molar mass of any gas is divided by the density of gas at stp, the value 224 is obtained here are some examples: gas, molar mass (g/mole), density at stp (g/ l), molar mass/density (l/mole) n2, nitrogen gas 28014 125 224.
The molar volume of an ideal gas at stp, which we define to be 0∘c and 1 atm arbitrarily (because we're old-fashioned and stuck in 1982) is 22411 l/mol to calculate this we can use the ideal gas law of pv=nrt at stp (standard temperature and pressure), we chose: p=1 atm v= n=1 mol. A given volume v of any ideal gas will have the same number of molecules the mass of the gas will then be proportional to the molecular mass a convenient standard quantity is the mole, the mass of gas in grams equal to the molecular mass in amu avogadro's number is the number of molecules in a mole of any. Answer to what is the volume of one mole of air, at room temperature and 1 atm pressure.
Calculate the molar weight of a gas a sample of which (0638 g) occupies a volume of 223 ml at 300 k and 750 mmhg 5 a mixture of ar (328 x 10-3 mole) and n2 (192 x 10-2 mole) was collected in a vessel whose capacity was 625 ml what would be the total gas pressure at 298 k 6 a spark is passed through a. If you have a solution, you multiply the molarity by the volume in litres moles from volume of pure liquid or solid there are two steps: multiply the volume by the density to get the mass divide the mass by the molar mass to get the number of moles example how many moles are in 1000 ml. Our table of molar volumes has over 130 values covering 97 elements each value has a full citation identifying its source the integrated unit conversion calculator can quickly convert a value to the units that molar volume ( for solid at 298 k, 1 atm unless specified ) click button to see citations actinium 22554 cm3/mol.
Use molar mass to make conversions between mass and moles of a substance explain avogadro's hypothesis and how it relates to the volume of a gas at standard temperature and pressure convert between moles and volume of a gas at stp calculate the density of gases at stp use the mole road. Weight of water = 20158 g + 159994 g weight of water = 180152 g one mole of water weighs 180152 grams unless you have a good sense of mass, this value probably doesn't have much meaning to you it's easier to grasp how much water is in a mole if you find the volume of this amount of mass. Molar volume of ideal gas, vm, is defined as the volume of 1 mole of the ideal gas at a specified temperature and pressure molar volume of gas, vm, is therefore the gas volume per mole of gas, so the units of molar volume of gas are: unit volume/mol or unit volume ÷ mol the metric unit (4) for volume is the litre, l, so the.
Molar volume, or volume of one mole of gas , depends on pressure and temperature, and is 224 liters - at 0 °c (27315 k) and 1 atm (101325 pa), or stp (standard temperature and pressure), for every gas which behaves similarly to an ideal gas the ideal gas molar volume increases to 240 liters as the.
Molar volume avogadro's law states that: 1 mole of every gas occupies the same volume, at the same temperature and pressure at stp (standard temperature and pressure), this volume is 224 liters at rtp (room temperature and pressure), this volume is 24 dm3 (liters) we can also say: the molar volume of a gas is. One mole of any gas occupies the same volume when measured under the same conditions of temperature and pressure in this experiment, the volume of one mole of hydrogen is calculated at room temperature and pressure. The molar volume, symbol vm, is the volume occupied by one mole of a substance at a given temperature and pressure it is equal to the molar mass (m) divided by the mass density (ρ) it has the si unit cubic metres per mole (m3/mol), although it is more practical to use the units cubic decimetres per mole (dm3/mol) for.
1 mole of any gas at stp occupies 224 liters of volume using this information, the volume occupied by any number of moles (or grams) can be determined conversely, the number of moles present in any volume of gas at stp can also be determined, and if the identity of the gas is known, this number of moles can be. Molar volume is another way to view number density the molar volume is the volume of 1 mole of substance it can be simply found by dividing the volume of a sample by the number of moles of that sample v m = v n this is simply the inverse of the number density it is a useful quantity to think about things from a. Using the ideal gas law, you would find the volume of 1 mole of a gas at standard temperature and pressure (stp) stp = 1 atm of pressure and 273 k for temperature p = 1 atm v = n = 1 mole r = 00821 atm l/mol k k = 273 k pv=nrt solves to v=nrtp v=1mol 00821(atm l)mol k 273k 1atm. A bbc bitesize secondary school revision resource for higher chemistry on calculations: moles, avogadro constant, calculating molar volume, reactant gas.


CloudWatch is a great concept — super-easy to configure and inexpensive. And at first glance, it actually looks pretty nice. But after I spent about 30 minutes with it, I realized it wasn’t easy to use. The units used are especially hard to interpret. This is my best attempt to explain what the network values mean.
Start by looking at per-instance network values. In the AWS console, go to CloudWatch. Type “ec2” in the metric search box, and click on “EC2 > Per-Instance Metrics”. In the Search box, paste in the instance ID of one of your EC2 instances, and then select “NetworkOut”.
Values that are recorded by CloudWatch for
NetworkOut are just the number of bytes output since the last reading. Looking at network traffic in units of “Bytes” (as opposed to “Bits per second” or even “Bytes per second”) is a little like saying “the car went 1 mile” and expecting the listener to understand how fast that was.
First, you have to know what the interval is between readings. By default, CloudWatch metrics are recorded once every 5 minutes unless you pay for detailed metrics (this is true for EC2, but you need to consult the documentation for other services — EFS records every minute, for example, but S3 reports basic metrics once per day by default — you really need to do a deep dive into each metric to understand its frequency, which is another reason CloudWatch is hard to use).
So if you see value of 300,000,000 Bytes for
NetworkOut (without detailed metrics enabled), you have to divide by 300 (300 seconds in 5 minutes) to get the Bytes per second value of 1,000,000. If you want a nice familiar metric like Mbps, you need to multiply by 8 and then divide by (1024 * 1024), for a value of 7.6 Mbps.
If you enable detailed metrics, you’ll get one value each minute, so the math changes a little bit. Instead of dividing by 300, you divide by 60. But you should note that the CloudWatch graph by default plots one point for every 5 minutes, unless you change that to 1 minute under the “Graphed Metrics” tab.
So what do the different “Statistic” types do? If your graph is set up so that you have one underlying data point for every point on the plot, then they don’t really do much of anything. They really only matter when you have more than one data point factored into each point on the plot. For example, if you have detailed metrics enabled, but your graph is configured with a period of 5 minutes, you have 5 data points being factored into each point in the plot. If you choose “Average”, the plotted point will be the average of the 5 data points. If you choose “Maximum”, the plotted point will be the highest value of the 5 data points. “Sum” will be the sum of the 5 data points (which would effectively be the same as if you had basic metrics measuring every 5 minutes and plotting with a period of 5 minutes).
Things get more interesting if you’re looking at metrics across a group of instances. If you have an auto-scaling group with more than one EC2 instance, you can aggregate metrics across those instances. In the AWS console, go to CloudWatch. Type “ec2” in the metric search box, and click on “EC2 > By Auto Scaling Group”. In the Search box, paste in the name of the auto-scaling group, and then select “NetworkOut”.
Now that you’re looking at numbers for more than one instance, the statistic selected makes a big difference.
If you have basic metrics, and you’re plotting one value every 5 minutes, the “Average” statistic will average the data points across the instances in your auto-scaling group. The “Maximum” statistic will give you the network volume for the instance that moved the most bytes. “Sum” will total up all the traffic across the instances, giving you a measure of the total volume of traffic handled by the auto-scaling group.
If you have detailed metrics, things can get a little complicated. “Average” is the average across the 5 data points across the instances of the auto-scaling group. “Maximum” is the largest data point recorded during a single minute in the 5-minute period on one of the instances. “Sum” is the sum of all 5 data points on all of the instances in the auto-scaling group.
Imagine you have 4 EC2 instances in your auto-scaling group. Depending on the metrics you’re using and the period of your graph, the number of data points taken into account for each point on the graph will vary.
|Metrics||Period||Num data points|
|Basic (5-min)||5 min||4|
|Detailed (1-min)||5 min||20|
|Detailed (1-min)||1 min||4|
While this isn’t too bad once you get the hang of it, the fact remains that the network unit used by CloudWatch is the Byte. I’m not sure I’ll ever be able to look at a glance at a bytecount (with its implicit measurement interval) and determine whether an instance is moving “a lot” of traffic. I think in terms of Mbps. I know that an instance moving 800Mbps is doing a pretty nice amount of traffic. But converting a value of “Bytes” to Mbps is too much, especially when one instance might be using Basic Monitoring and another might be using Detailed Monitoring.
You might want to look to third-party solutions to really maximize your CloudWatch visualizations. I have looked at a number of different solutions, and I have settled on running our own InfluxDB database with Grafana for the front end. We use custom code to query CloudWatch and feed the values into InfluxDB, converting the “units-since-last-measured” metrics into sensible “units-per-second”. With the InfluxDB/Grafana combination, it’s vital that you make this conversion so that you can do things like take the mean of several values for rolling up your data into long-term retention policies.


So, through my adventures on Twitter, I came across a Facebook group called TEACHERS LOVE SMARTBOARDS. I became a fan and was given a free download for a smartboard application called “combination safe”. This is a notebook application that has a combination lock and a math question. You answer the math question using the combination lock. If you get the correct answer, the safe will open! You can set the question and the answer. Fom this page, I was also referred to a podcast called GET SMART WITH SMARTBOARDS. This is a free download from iTunes. So far, there are 8 episodes. I downloaded all 8, but have only listened to the first. It is full of neat information. There is also a website to go with it that has all the links they talk about in the podcast. Also from the Facebook group, I found out about some teacher training online. It has many lessons for use in the classroom as well as accompanying videos to show how to use them!
I am just so grateful that I applied for this program and that I was accepted.


An Axiom of Fairness
Consider this scenario: Two people are given 100 rupees. Now we ask the following question: How will these two people divide this 100 rupees among each other, given that we stipulate a condition that none of them will get anything if negotiation between them fails? That is, we take back the Rs. 100 away from them if they fail to reach an agreement. Common sense dictates that these two people divide the sum according to 50-50 rule. In other words, each person gets half of the 100 rupees, which is 50 rupees. This solution is attractive because it fits our intuitions of fairness. Thus we can even claim that the proposed solution is a "fair" solution. However, it is worthwhile to consider why this particular 50-50 division should be considered as a fair division. One possible answer to this question is to say that each person is in some way "equal" to each other. Hence, they should get equal amount during the division of the money. But this begs a further question: in what way are they "equal" to each other? To answer this question let us call these two people Alice and BoB. We can make this idea of being "equal" more precise by considering symmetric situations. Let us first consider Alice alone. If Alice stays alone, she gets nothing; so her self worth is zero. Now if Bob comes along to join her, then the value of their partnership becomes 100 rupees. That is, they get a chance to retain the 100 rupees. Thus we say that Bob's marginal (or incremental) contribution to this partnership is 100 rupees. Then again, we can switch the order of consideration. In the second situation, We can first consider Bob alone, due to which his self worth is zero. If Alice comes and partners with him, then their group value becomes 100 rupees. Thus, in this second situation, we say that Alice's marginal contribution to the partnership is 100 rupees. But we have no reason to prefer any one of these two order of consideration -- whether Alice comes first and is joined later by Bob, or Bob comes first and is later joined by Alice. Since both cases are equally likely, if we average the marginal contributions of Alice and Bob in these two cases, we find that both Alice and Bob makes an average marginal contribution of 50 rupees. This is their worth or value that they each individually on average contributes to the partnership. Thus, it is fair to divide the 100 rupees using 50-50 rule.


Item #113- Fill an entire water bottle with change and use the money to complete another goal on the list
Step number one...where to find a giant water bottle?
Answer: Hijack it.
I came across an empty bottle. Not that anyone is going to miss an empty water bottle and besides it's for a good cause.
Now my poor lonely, empty, 5 gallon bottle (damn that's gonna be a lot of change) is drying as we speak.
Ah, water bottle...we've only just begun.
So exactly how much change can you fit in a 5 gallon bottle?
Good question. The answer is, I don't know, hence why the goal is on the list in the first place. I'm imagining that depending on what denomination of change goes in it could potentially be several hundred dollars. I think a low estimate would be, say, if it were all pennies, somewhere in the $500 range. I'm not a math person so I will not even attempt to figure the equation out in my head. I don't even know how long it will take to fill the whole thing. I figure if I stick with it and don't break down in a moment of desperation and cash it in, it'll take roughly a year to fill it completely to the top.
I will post pictures periodically updating the progress of filling the bottle. Until then...


On August 10th, the launching ceremony of the Jilin National Center for Applied Mathematics was held at the lecture hall of the Mathematics Building of Jilin University. Dr. Jiang Zhiying, Chair of Jilin University Council, unveiled the plaque and Prof. Dr. Zhang Xi, President of Jilin University gave the inaugural speech.
In his speech, Prof. Zhang, on behalf of Jilin University and Dr. Jiang greeted the guests and expressed his gratitude to all the units and individuals who have contributed to the preparation and construction of the center. He hoped that the center, one of the first batch of national applied mathematics centers, would seize the historic opportunity and accelerate the construction of a "mathematics +" innovation platform integrating production, education, research, and application. He expressed his hope to see more young students devote themselves to basic research and become a strong force in building China into a scientific and technological power in the future.


An abacus is a calculation tool. It is used to perform a mathematical function. The basic function of a calculation is Addition, Subtraction, Multiplication and Division. The full form of ABACUS is Abundant Beads, Addition and Calculation Utility Systems. It is also called a counting frame. We have learned abacus in LearnClue website so go through it for better understanding.
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root. Spreadsheet. January 30th , 2021.
Check out the web's best free mortgage calculator to save money on your home loan today. This spreadsheet was originally posted on the mortgage forums of moneysavingexpert.com here.
Mortgage calculator spreadsheet uk. Your payments should match the total cost of the loan from the mortgage calculator. Regular version for excel 2007 or later. Use a mortgage calculator if you're using help to buy.
Also offers loan performance graphs, biweekly savings comparisons and easy to print amortization schedules. The total amount you repay is a combination of the mortgage debt and the interest charges accrued over the lifetime of the mortgage. If you have a flexible mortgage you can make huge savings by making regular overpayments or making lump sum overpayments.
Many thanks, this place is a real encouragement, even though i lurk mostly! The home mortgage tracker spreadsheet allows you to track and analyze your current loan agreement to easily consider the total cost of the mortgage. Our flexible mortgage calculator allows you to calculate the savings you can make.
Your principal should match up exactly with the original loan amount. The cumipmt function requires the analysis toolpak, which comes with excel. Free online mortgage calculator specifically customized for use in the uk including amortization tables and the respective graphs.
Below are free loan and calculator spreadsheets. I know there are loads of mortgage repayment and overpayment calculators out there, just wondering if anyone had a skeleton spreadsheet (simple layout) i could make use of! It combines information like your interest rate, number of periods, and principal to arrive at an amount for each monthly payment.
We have offered a downloadable windows application for calculating mortgages for many years, but we have recently had a number of people request an excel spreadsheet which shows loan amortization tables. Bespoke microsoft excel spreadsheet development & vba programming home | about roger | downloads | engaging roger | contact free excel mortgage calculator spreadsheet Your interest should match the interest cost from the mortgage calculator.
Mortgage payments can be easily found using your chosen spreadsheet program. Since creating this spreadsheet, i've created many other calculators that let you include extra mortgage payments.the most advanced and flexible one is my home mortgage calculator. The links below are the latest versions of the spreadsheet, version 1.14.
Mortgage calculators help you understand how much you can borrow, the stamp duty tax, and what your mortgage repayments will be in the uk. Note that this mortgage calculator assumes your interest rate will remain the same for the full term of the mortgage, and you are on a capital repayment, rather than an interest only, mortgage. And our miles per gallon calculator will help you determine the average mpg your vehicle gets.
Ensure that any overpayment you make goes to reduce the debt (so shortening the term) rather than reducing your monthly payments. Get a quick and clear picture of what it will take to pay off your mortgage with this accessible mortgage loan calculator template. Shows the cost per month and the total cost over the life of the mortgage, including fees & interest.
Get a quick and clear picture of what it will take to pay off your mortgage with this accessible mortgage loan calculator template. In addition, explore various financial calculators, as well as hundreds of other calculators addressing math, health, fitness, and more. Use the interactive dashboard to understand when your loan will be paid off, how much interest you are paying the bank, and what happens if you increas
Estimate your monthly payments with pmi, taxes, homeowner's insurance, hoa fees, current loan rates & more. If they match, you've done the formulas correctly. 11 extra payment mortgage spreadsheet.
Calculate the difference in total interest paid on a mortgage loan when making additional monthly payments. Download edit in browser share. It's important you always get a specific quote from.
The spreadsheet is easy to use and navigate, and there are plenty of comments explaining terminology in a simple way. Our loan amortization spreadsheet can help you determine your next mortgage payment or car payment. You can use our lease mileage spreadsheet to make certain you don't exceed your allowed lease miles.
A £180,000 loan structured over 25 years will see the borrower pay £56,581.78 in interest over the life of the loan. This function, in all major spreadsheet programs (microsoft excel, google spreadsheet, and apple numbers), is known as pmt, or the payment function. It has only been designed to give a useful general indication of costs.
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Use our helpful form below to calculate how much mulch you’ll need for your upcoming project.
How Much Mulch Do I Need?
If you’re just adding new mulch to a new garden, mulch should be 3-4 inches thick.
If you’re re-mulching an existing area, mulch should be 1-2 inches thick.
Square Foot Calculations:
Rectangle: length x width
Circle: 3.1416 x the radius squared
Triangle: 1/2 x length x width


This document is intended as a study help guide for students of general mathematics principles and processes. It is also a set of useful reminder sheets. Started as tutoring notes for a Reconstructing Maths course, they are often added to.
The value of each number changes with its position in relation to the dot marking the difference between whole
units and part units, because the face value of the number is multiplied by the value of its position. The value
of the Units position is always its face value, and the other positions are calculated as a multiple of the base value.
In the decimal system (base 10), each number position is worth 10 times that of the one on its right. The dot between the whole units and fractions is known as the decimal dot. The most often used values are:
million.......ten thousand.......hundred..........unit..........tenth.....thousandth....hundred thousandth
___ ___ ___ ___ ___ ___ ___ . ___ ___ ___ ___ ___ ___
hundred thousand...thousand...........ten.....decimal dot...hundredth....ten thousandth....millionth
To make large whole numbers easier to read, they are shown in divisions of 3 positions that are divided by a comma (,), thus a million is shown as 1,000,000 - not as 1000000.
|Numbers||The only numbers or numerals used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.|
|Base||Unless specifically informed otherwise (eg 5 8) you will use base 10.|
|Extra Zeroes||Zeroes before the decimal dot, or first number, are assumed.|
|Zeroes after the decimal dot, or last decimal place, are assumed.|
In all mathematical equations the basic principle is that the final result of all the actions and values
on one side of the equals symbol, =, is exactly the same as on the other side. Thus what is on one
side will always exactly balance what is on the other.
In most mathematical situations we are seeking to calculate the final result, but sometimes we know some values on both sides and this principle can be used to help find the missing values.
The main function processes used are:
|Addition||Increasing the value of one number by one or more numbers.
.....Example:....1 + 2 = 3....one plus two equals three.
|Subtraction||Decreasing the value of one number by one or more numbers.
.....Example:....3 – 2 = 1....three minus 2 equals one.
|Multiplication||Increasing the value of one number by another as if you had added
the first number to itself as often as the second number is. The term times is
often used to express the multiplication function.
.....Example:.....2 x 3 = 6.....two times three equals six.
|Division||Finding out how often one number will go into another number.
.....Example:....6 ÷ 3 = 2.....six divided by three equals two.
|Square||The square of a number is the value found when you multiply that
number by itself. This is expressed by a little 2 after and above it.
.....Example:.....32 = 9.....three squared equals nine, or 3 x 3 = 9.
|Square Root||The square root of a number is the value that, when multiplied by
itself will give you the number you started with. This is expressed by the root
symbol √, this is usually shown with a value as 2√
(square root) or 3√ (cube root) etc; if no value it is assumed
.....Example:.....2√ 9 = 3.....square root of nine equals three.
|Cube||The cube of a number is the value found when you multiply that number
by itself and then multiply the answer by the original number again. This is express as
a little 3 after and above it.
.....Example:.....23 = 8.....two cubed equals eight, or 2 x 2 x 2 = 8.
|Cube Root||The cube root of a number is the value that, when multiplied by itself
and then that answer is again multiplied by the value, will give you the number you
started with. This is expressed by the root symbol with the value of three, 3√.
.....Example:.....3√ 27 =3.....cube root of twenty-seven equals three
...........................................27 = 3 x 3 x 3, 27 = 9 x 3, or 3 x 3 x 3.
|Powers||The process used in squares and cubes can be extended infinitely and the small raised number is known as a power. Thus 22 is also known as two to the power of two. The number 26 is expressed as two to the power six, or 2 x 2 x 2 x 2 x 2 x 2 = 64. Often very large large numbers are written as 1.325 x 1013, this is called scientific notation as it is shorter than 13,250,000,000,000.000|
|NB:||This same power numbering process can also be applied to roots and thus you could have a number 9√ 512. Luckily all you are likely to come across in daily usage are square roots and cube roots|
Although 2 x 6 is easy to do in the mind, 453 x 278 is not so easy, thus breaking it down into smaller
steps makes it faster and easier. Long Multiplication breaks it into a series of single number times single
number calculations that you can then add up, remember to carry forward higher position values in the
single digit multiplications. This process also makes the workings easier to see and check. Two main methods:
|3,624||multiply each number in the top line by 8|
|31,710||write a zero then multiply each number in the top line by 7|
|90,600||write 2 zeroes then multiply each number in the top line by 2|
|125,934||add up the multiplication lines|
|24||multiply 3 x 8|
|400||write a zero then multiply 5 x 8 (equal to 50 x 8)|
|3,200||write 2 zeroes and multiply 4 x 8 (equal to 400 x 8)|
|210||write a zero and multiply 3 x 7 (equal to 3 x 70)|
|3,500||write 2 zeroes and multiply 5 x 7 (equal to 50 x 70)|
|28,000||write 3 zeroes and multiply 4 x 7 (equal to 400 x 70)|
|600||write 2 zeroes and multiply 3 x 2 (equal to 3 x 200)|
|10,000||write 3 zeroes and multiply 5 x 2 (equal to 50 x 200)|
|80,000||write 4 zeroes and multiply 4 x 2 (equal to 400 x 200)|
|125,934||add up the multiplication lines|
Although 6 ÷ 3 is easy to do in the mind but 46,480 ÷ 35 is not so easy thus
a process to handle smaller steps is easier and faster. Long Division breaks it into a
series of smaller calculations. Remember to keep position values aligned properly. It
also makes the workings easier to see and check. If the answer goes into decimal places
this is easy to calculate as you can just keep adding zeroes onto the end.
The number being divided into the other number is called the divisor. (A) Write the equation down as 35)46,480 and draw a line above the number. (B) Then see if the divisor will go into the first digit of the number, if not try the first two digits of the number; this will work as 46 is larger than 35. (C.) Work out how many times it will go and write that number above the line. (D.) Then multiply that number by the divisor and write it below the number, (E.) draw a line and subtract it from the digits of the number being used, writing the left over under that. (F.) Then bring the next digit of the number down and write it beside the remainder you have. Repeat steps B to F until you have a remainder of zero or you have enough decimals to suit you. If needed when getting into decimals you can just add a zero onto the remainder and continue going through the steps. Make sure you have your decimal dot right.
|35||)46480||First pass 46 divided by 35|
|114....||subtract the 35 from 46 and bring down the next digit|
|35||)46480||First pass 46 divided by 35|
|35......||subtract the 35 from 46|
|114....||bring down the next digit and second pass 114 divided by 35|
|105....||subtract the 105 from 114|
|98...||bring down the next digit and third pass 98 divided by 35|
|70...||subtract the 70 from 114|
|280.||bring down the next digit and fourth pass 280 divided by 35|
|280.||subtract the 280 from 280|
|nil||finished no more to do.|
Sometimes it is also possible to make the calculation easier by simplifying it by dividing both the number and the divisor by a number which goes evenly into both. In this example both are multiples of 5 and can be divided by 5. This would then make the equation 9,296 divided by 7, which will give the same result as both sides of the calculations have been altered to the same extent.
A prime number is any number that will give a whole number answer when
divided by itself or one (1); that is, it is only divisible by itself or one. The
only prime numbers under 10 are 1, 2, 3, 5, and 7. The number 2 is the only
even prime number as all other even numbers can be divided by two; in a
similar fashion any number ending with the digit 0 or 5 (except 5) is not a
prime number as it can be divided by 5.
Most numbers can be broken down into their primes and expressed as a function of their primes, the number 30 can be expressed by its prime factors as 2 x 3 x 5. The convention in listing primes is to list the lowest number first and the rest in increasing value. Many numbers will be expressed as prime factors by the use of powers (see Functions), thus the prime factors expression of 60 is 22 x 3 x 5 which is the same as 2 x 2 x 3 x 5, and 120 is 23 x 3 x 5 and 180 is 22 x 32 x 5.
When calculating the prime factors of a number you should try dividing the number by the lowest number possible that gives an answer in whole numbers, then repeat the process until you cannot divide it by anything else. The easiest way to work out the prime factors is to start at the bottom left of a piece of paper and then do a series of long division calculations working your way up the page, the prime factors are then shown as the division numbers on the left and the final number on top. If a number is not suitable at any step it is not suitable for any later step as it would have worked in the earlier step if it could. Start with the lowest number possible and work upwards, this simplifies the task as you go, use 2 if the number is even, 3 if odd. For example establishing the primes in the number 63,063. Go to the bottom left corner, step 1:
|...143||uneven try 7 again, no; try 11|
|1001||uneven number try 7 again|
|Step 4||7)7007||(NB: 3 was no good last step so don’t try it)|
|..7007||uneven number try 3 again, no; try 7|
|Step 3||3)21021||(NB Skipped 5 as it did not end in 5 or 0)|
|21021||uneven number try 3 again|
|Step 1||63063||uneven number try 3 first.|
|In real life it will show like this, primes are in bold numbers.
The answer is 3 x 3 x 7 x 7 x 11 x13 or 32 x 7 2 x 11 x 13.
The number below the line is known as the denominator and in many cases it is necessary to bring the fractions
into a common denominator in order to complete the calculation. This makes the calculation easier as you can
then handle the above the line numbers as whole numbers. Adding 1/2
and 1/4 is not readily possible, but changing the
1/2 to 2/4 makes
it easier to add as they are both now / 4 and it becomes a case of
(2 + 1)/4 = 3/4. The same
applies to any calculation with fractions.
In some cases the only way to bring fractions into a common denominator is to multiply both halves of each side of the calculation by the denominator of the other side. Thus the common denominator for 4/9 + 2/5 would be converted by multiplying the 4 x 5 and the 9 x 5 to become 20/45 then multiplying the 2 x 9 and the 5 x 9 to be 18/45 making the calculation into 20/45 + 18/45 or 38/45.
It is standard practice to convert fraction answers into whole numbers and fractions where relevant and to simplify the fraction to its lowest possible number. Thus an answer of 48/45 changes to 1 3/45 and would end up as 1 1/15.
When the calculation includes a number that is a whole and a fraction like 1 2/5 then it is best to convert the whole number into fraction and this would become 6/ 5 prior to doing the calculation.
Fraction divisions can be simplified by turning the divisor upside down and changing the division to multiplication. Thus 4/5 ÷ 2/10 becomes 4/5 x 10/2 or (4 x 10)/(5 x 2) which equals 40/ 10 or 4.
Fraction multiplications can be simplified by drawing a long line between the top and bottom, you can also make the same changes to both halves. 9/10 x 10/ 18 is the same as (9 x 10)/(10 x 18) by taking away the 10 in both halves you get 9/18 or 1/ 2.
Decimal calculations are done the same as fractions once you have converted the decimal into a fraction by adding a zero to the denominator for each position after the decimal dot. Thus 1 = 1/1 and 1.5 = 15/10 and 1.55 = 155/100 etc.
Percentages, usually written as 95%, are a specialised form of showing values as hundredths; 95% is really
95/100. If the percentage has a decimal like 12.5% the decimal
is removed by adding a zero to the denominator, 125/1000. A
number is converted to a percentage by multiplying it by 1/100.
A number can be shown as a percentage of another number by dividing it by the second and multiplying by 100 For
example 5 as a percentage of 25 is calculated 5 ÷ 25 x 100; is 5/25
x 100 or (5 x 100)/25 which is 20, thus 5 is 20% of 25.
The difficulty with percentages comes in calculating things with a percentage, such as commissions, taxes, interest. Especially with reverse and multiple level calculations. The confusion is due to not clearly identifying which is the whole or 100% value.
The three main types of percentage calculations are simple, reverse and accumulative.
This is the easiest and most common and used in calculating simple interest.
To calculate the percentage of a number you multiply it by the percentage as a fraction. Thus 15% of 450 is 450 x 15/100 or (450 x 15)/100 = 67.5. This gives you the value that the percentage is of the original figure.
To calculate the value with 15% added on, you add 100 to the percentage fraction; 450 with 15% (that is 450 plus 15%of 450 added on) is 450 x 115/100 = 517.5, which is the value of the original figure plus the percentage added to it.
This is used to take a figure back to its original value before a percentage was added. Often used to
calculate pre-tax values or pre-commission values.
To calculate the original value from a number that includes a percentage you divide the number by 100 + the percentage and multiply by 100. The number 600 includes 20% so the calculation is (600 ÷ 120) x 100 or 600/120 x 100 = 500. Most people try to take 20% off the 600 giving 480, having forgotten that the 600 is 120%.
This is where you have more than one percentage being applied, often used to calculate tax added
retail prices. You start with purchase price add profit margin, commission, then tax; all percentages of
the figure before it, not the original value. You start with one value add a percentage, then add a percentage
to the new value. The most common fault is thinking you can add the percentages together and apply them
to the original value, when it is really a series of separate calculations of price + profit, then that + commission,
and then that + tax. .
Example:..........800 with 25% profit, 15% commission, 10% tax is
.........................800 x 125/100 = 1,000 x 115/100 =1,115 x 110/100 = 1,226.50;
.........................25% + 15% + 10% = 50% then 800 x 150/100 = 1,200 is wrong
The perimeter is the outside edge of a shape. Most can be calculated by a simple formula. A parallelogram, square or rectangle perimeter is calculated by adding the length of the four sides together but easiest as (b + h) x 2 or (breadth + height) x 2.
The area is the space inside a 2 dimensional (flat) shape like a square or triangle.
The area of a parallelogram, square or rectangle (these are special parallelograms) is calculated by multiplying the breadth x height (b x h).
The area of a triangle is 1/2 b x h as any triangle when mirror imaged and added to itself will create a parallelogram.
Volume is the space inside a 3dimensional shape like a cube. This is calculated by using the area equations above to calculate the area of the base and then multiply it by the depth. Cones and pyramids are slightly different and not covered in this document.
To work out how much of a whole any one component is of a mix you add up the ratios and
divide the total mix by that value and then multiply that answer by the value of the relevant component.
Sometimes it is easier to calculate what fraction of the mix that component represents and then calculate
what that fraction is of the mix.
Example:..........sand / gravel / cement mixes in a ratio of 1/3/4 for 64 kilo mix
.........................1 + 3 + 4 = 8; 64 ÷ 8 = 8, values are 1 x 8 / 3 x 8 / 4 x 8, or 8 / 24 / 32;
.........................Same mix but you only want the amount of cement for 60 kilos
.........................1 + 3 + 4 = 8; 4 ÷ 8 = 1/2; 60 x 1/2 = 30
The probability, or chance, that a particular event will occur is calculated as the number of times that
event is possible, divided by the total number of event options available, and then expressed as a fraction.
The most common demonstrations involve a six-sided die. Each number has one chance of being rolled,
and the total number of options is 6, thus a probability of 1/6
for each number.
Probabilities are more complex when you have two or more probability events occurring together, as in throwing two or more dice together. Then the total number of options is calculated by multiplying the total number available on the first event by the total number available on the second event, etc. The chances of a specific result are calculated by working out all the possible answers; then counting the number of times that event can occur. With two six-sided dice the total options are 6 x 6 = 36 (with three dice the probability is 6 x 6 x 6 = 216) whilst the chances of rolling 12 (6 + 6) is 1 (1/36 ), the chances of rolling 11 (5 + 6 & 6 + 5) is 2 ( 2/36 , 1/18).
To calculate the various event values you draw a matrix, or grid with the number of events for one factor on the top and the other factor on the side. Then calculate the different event option in the matrix boxes. For two six-sided dice it would be:
Missing digit calculations are where you have all but one number in a figure and
know another aspect of the figure (usually that it is evenly divided by a certain value)
and you have to work out what the value of the missing digit is.
This is calculated by assuming that the missing digit is 0 (zero), then dividing the number that this makes by the divisor supplied. You then see what is left over and then check which of the available values will give you an even multiple of the divisor if the left over is added to the various values given. The trick here is to remember the positional value of the missing digit. If the missing digit is in the 10 value position it is a multiple of 10 plus the left over, etc.
Example:..........4568x is a number that can be evenly divided by 3.
Divide 3 into 45680 this gives you 15226 with two left over, thus x is a number that is a multiple of 3 plus 1. The only possible options are 1, 4, or 7.
Example:..........456x1 is a number that can be evenly divided by 3.
Divide 3 into 45601 this gives you 15200 with one left over, thus x is a number that is a multiple 10 and a multiple of 3 plus 2. In this case it is easier to look at values with the left over added eg 11, 21, 31, 41, 51, 61, 71, 81, 91 the only ones divisible by 3 are 21, 51 and 81; so x must be 2, 5, or 8. NB different values to the solution above.
Today the majority of countries use metric measuring scales. These all have a
decimal base and a common set of words to express the value relevance of the scale
in use. The whole unit is expressed by the relevant scale point without a value reference,
eg metre, gram, litre. Also some scales are multiple unit values, eg in weights you have
grams and tonnes; and some possible mixes are not used, eg centigram, decimetre, or
decametre. The main words and their values (with some common examples) are below;
some other examples are megahertz and, kilowatts.
Triangles are enclosed shapes that have three straight sides. There are three main types of triangles.
The value of the three angles in any triangle will always add up to 180o, the value of a straight line.
This can be proven by cutting a triangle out of paper, and tearing it into three parts; then placing the
angle points together with the sides touching but not overlapping. Knowing this we can always work
out the value of an angle in a triangle when we know the value of the other two angles. Main types are:
Equilateral..........all three sides are the same length, and so are the angles.....
Isosceles............has two sides of equal length............................................. .....
Scalene..............no two sides are the same length......................................... .....
Right angled.......it has a 900 angle which is just ‘right’ for construction use..... .....
Triangles can be duplicated and then put together to make up other shapes, like a parallelogram, square, rectangle, rhombus, octagon, hexagon, pentagon. With these last three they would make a ‘regular’ shape; that is one where all the angles are the same, the sides usually are too.
There is a special triangle called a ‘right angle’ triangle because it has an angle that is 900,
just right for building and ensuring things are square. Right angle triangles can be scalene or isosceles
triangles but not equilateral triangles.
In a right angle triangle the side opposite the 900 angle is called the hypotenuse and its length can be calculated from the other two sides as the ‘square of the hypotenuse will always equal the sum of the squares of the other two sides’. Thus you calculate the square for the other two sides, add them together and get the square root of that figure
One of the most common is called a ‘3/4/5’ triangle as the side lengths are kept to those ratios. 3 squared is 9, 4 squared is 16, that adds to 25 the square of 5. This is a scalene right angle triangle. The other angles are usually 300 and 600.
An isosceles right angle triangle would have the two sides from the 900 the same length and the other two angles are 450 each, as the angles is the same for each. In this case the value of the other two angles will always be 450
With an equilateral triangle the value of the angles will always be 600 as that is
the only way you can get the length of all sides to be the same.
NB: All these relationships above can be used to help calculate the value of angles and areas in other shapes.
From what we know about triangles we can use them to calculate the angles of multi-sided
shapes by simply changing the shape into a group of triangles. In all cases we can calculate
the total value of the angles in the shape, and in a regular shape we can calculate the
value of each angle. A regular shape is one where all the angles are the same, thus the sides
are the same and it has a uniform appearance.
Regardless of what the shape is you need only place a dot in the centre of the shape and then draw a line to the middle of each angle, thus dividing the shape up into a number of triangles; as per below. And then imagine a circle around the joining point.
hexagon... ........octagon... ........regular pentagon...
........pentagon... ........trapezium... ........parallelogram...
In these examples we can see that each shape is now a number of triangles with a circle in the middle. The sum of all the angles in any triangle is 1800 and all the triangles join to make a circle of 3600 in the centre. Thus we need only add up the number of triangles, multiply that by 1800 and subtract 3600 to have the sum of the outside angles; this will work in all the shapes.
Hexagon is 6 x 180 = 1080 – 360 = 7200 .........Octagon is 8 x 180 = 1440 – 360 = 10800
Pentagon is 5 x 180 = 900 – 360 = 5400 .........Parallelogram / Trapezium is 4 x 180 = 720 – 360 = 3600
The bolded answers above are the sum of the angles in each shape. Both pentagon calculations are the same, the sum of all the angles will remain the same but the individual angle values will be different between the regular and irregular pentagons. If the shape is a regular one then the value for each shape will be the bolded answer divided by the number of angles. Thus the octagon angles are 1080 ÷ 8 = 1350, the hexagon angles are 720 ÷ 6 = 1200, the regular pentagon angles are 540 ÷ 5 = 1080.
Tessellation is a fancy word that means patterned or patterning (depending upon usage). Some
shapes can tessellate, some cannot. To be able to tessellate the shape MUST have angles that can
divide evenly into 3600, or be able to be grouped into a set of 3600.
In the shapes above the regular octagon and regular pentagon cannot tessellate as their angles will
not go evenly into 3600, whereas the regular hexagon and irregular pentagon will;
as will all squares and rectangles. Some tessellation examples are:
.... .... ....
Most mathematical processes are simple calculations using only one function as either a
single process or a repeated process; such as adding up a column of figures. When the calculation
used two or more processes it becomes a complex calculation, most complex calculations involve
only a process and its opposite, such as addition and subtraction in a column of figures. When the
calculation involves mixed processes you must process them in a set order to ensure a correct answer.
The order of processing functions is remembered by the term BoDMAS. This is made
up from the letters of the processes in their processing order, Brackets opened, Division,
Multiplication, Addition, Subtraction. By having everybody following the same
order you all get the same results.
You perform one set of calculations at a time and start with the one in the deepest set of brackets and then repeat for each set of brackets until there are no brackets, then complete the other calculations. When processing within each set of brackets you process in the DMAS order. The example below shows the value of correct order.
Example:.......... 4 x 5 + 2 = ................Processed as if it had brackets:
A.......................(4 x 5) + 2 = 22...........20 + 2 = 22
B........................4 x (5 + 2) = 28............4 x 7 = 28
BODMAS tells us to process the calculations as process method A.
A formula is a way of expressing a constant mathematical relationship without using specific
numbers or values. Letters or symbols are used to express certain values and concepts. In
calculating the area of a rectangle you always multiply the length of one side by the length of the
other side that joins it at a corner; these are referred to as the height and breadth. Thus H x B = A
(where H is height, B is breadth, A is area). To calculate the actual area you just replace the H and B
with the relevant real values. This formula, like all formulas, will always work.
This is useful in helping to understanding mathematical relationships, and in finding mathematical relationships where we don’t know all the values. The formula allows us to play with the equation components and view various mathematical relationships without being distracted by the values. Change H x B = A into A ÷ B = H; and you create a formula to find the height where the total area and breadth are known. This process of manipulating the formula is known as Algebra.
When you can get a component of the formula or equation by itself on one side of the equals sign it is said to be isolated. By isolating a component we can then use the values of the other components to calculate the value of the isolated component. Thus algebra is used to isolate unknown component from the known components to permit its calculation.
In using algebra to manipulate formulas and equations we must always remember the Basic
Principle to keep both sides of the equals sign (=) balanced and BODMAS. In order to keep
both sides of the equation balanced, what is done to one side must be applied to the other side
in the exact same manner. Divide one side by a value you must divide the other side by the same
value, the same with all other functions. To manipulate the equation you don’t need to know the
values or what they relate to.
S = T x D........we know S and D and must find T so we divide both sides by D
.......................S ÷ D = T x D ÷ D.........Simplifies to.........T = S ÷ D
D ÷ C = L.......we know C and L and must find D so we multiply both sides by C
.......................D ÷ C x C = L x C.........Simplifies to.........D = L x C
A + B = C.......we know B and C and must find A so we subtract B from both sides
.......................A + B - B = C - B.........Simplifies to.........A = C - B
A - B = C.......we know B and C and must find A so we add B to both sides
.......................A - B + B = C + B.........Simplifies to.........A = C + B
With very complex formulas we can do this a number of times to isolate a variable.
D x G x M + L - C = H x K........we know all except M and want to know its value
.......................We can divide both sides by D, then G, then add C and minus L
.......................G x M + L - C = H x K ÷ D.........M + L - C = H x K ÷ D ÷ G
.......................M + L = H x K ÷ D ÷ G + C........M = H x K ÷ D ÷ G + C - L
In algebra or a formula multiplication and division signs are usually omitted. You show multiplication by putting the symbols together without any function, A x B is shown AB; and division is done with a divisor, D ÷ G is shown as D/G. The equations above with division and multiplication would normally be written as:
S = TD..............................................converts to.........T = S/D
D/C = L..............................................converts to.........D = LC
DGM + L - C = HK..........................converts to.........M = HK/DG + C - L
NB: The two division values can be placed together as a multiplying divisor.
A grid has a lot of lines running up and down and left to right, the darker line in each direction is the centreline for each axis. The horizontal or ‘x’ axis is positive above the line and negative below. The vertical or ‘y’ axis is negative on the left and positive on the right. A set of co-ordinates will have a number for each axis, eg 3,4 means 3 positive lines horizontally and then up, whilst 3, -4 means 3 across and 4 down. Always remember to walk along the aisle before climbing the ladder. A common use for grids is in locating things on maps and street directories.
Copyright by Ernest E Bywater


optional mathematics by A. A. Reinecke Posted on July 10, 2017 by reubenwoolley Bio: A. A. Reinecke is a writer, playwright, and poet. Her likes include caffeine and the correct use of Oxford commas. Share this:TwitterFacebookLike this:Like Loading... Related
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The orbits followed by two point masses obeying Newtonian mechanics are conic sections, such as ellipses, hyperbolas, parabolas, and lines. Understanding them is important to understanding orbital mechanics. Conceptually, conic sections are formed from the intersection between a plane and a cone. More formally, it is the loci of points in a plane where the ratio of the distance from a fixed point (the focus) to a fixed line (the directix) is a constant.
Stable orbits have the shape of ellipses. Ellipses can be described by the length of the major and minor axes can also be used. These aren't useful for orbits.


A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a space that maps the possible states of the system into the complex numbers. The laws of quantum mechanics (i.e. the Schrödinger equation) describe how the wave function evolves over time. The values of the wave function are probability amplitudes — complex numbers — the squares of the absolute values of which, give the probability distribution that the system will be in any of the possible states.
For example, in an atom with a single electron, such as hydrogen or ionized helium, the wave function of the electron provides a complete description of how the electron behaves. It can be decomposed into a series of atomic orbitals which form a basis for the possible wave functions. For atoms with more than one electron (or any system with multiple particles), the underlying space is the possible configurations of all the electrons and the wave function describes the probabilities of those configurations.
The modern usage of the term wave function refers to a complex vector or function, i.e. an element in a complex Hilbert space. Typically, a wave function is either:
- a complex vector with finitely many components
- \vec \psi = \begin c_1 \\ \vdots \\ c_n \end,
- a complex vector with infinitely many components
- \vec \psi = \begin c_1 \\ \vdots \\ c_n \\ \vdots \end,
where each component may be complex function of one or more real variables (a "continuously indexed" complex vector)
- \psi(x_1, \, \ldots \, x_n).
In all cases, the wave function provides a complete description of the associated physical system. An element of a vector space can be expressed in different bases; and so the same applies to wave functions. The components of a wave function describing the same physical state take different complex values depending on the basis being used; however the wave function itself is not dependent on the basis chosen; in this respect they are like spatial vectors in ordinary space: choosing a new set of cartesian axes by rotation of the coordinate frame does not alter the vector itself, only the representation of the vector with respect to the coordinate frame. A basis in quantum mechanics is analogous to the coordinate frame: choosing a new basis does not alter the wavefunction, only its representation, which is expressed as the values of the components above.
Because the probabilities that the system is in each possible state should add up to 1, the norm of the wave function must be 1.
Spatial interpretationThe physical interpretation of the wave function is context dependent. Several examples are provided below, followed by a detailed discussion of the three cases described above.
One particle in one spatial dimension
The spatial wave function associated with a particle in one dimension is a complex function \psi(x)\, defined over the real line. The positive function |\psi|^2\, is interpreted as the probability density associated with the particle's position. That is, the probability of a measurement of the particle's position yielding a value in the interval [a, b] is given by
- \mathbf_ = \int_^ |\psi(x)|^2\, dx .
This leads to the normalization condition
- \int_^ |\psi(x)|^2\, dx = 1 \quad .
since the probability of a measurement of the particle's position yielding a value in the range (-\infty, \infty) is unity.
One particle in three spatial dimensions
The three dimensional case is analogous to the one dimensional case; the wave function is a complex function \psi(x, y, z)\, defined over three dimensional space, and its complex square is interpreted as a three dimensional probability density function:
- \mathbf_R = \int_R |\psi(x, y, z)|^2 \, dV
The normalization condition is likewise
- \int_R |\psi(x, y, z)|^2\, dV = 1
where the preceding integral is taken over all space.
Two distinguishable particles in three spatial dimensions
In this case, the wave function is a complex function of six spatial variables, \psi(x_1, y_1, z_1, x_2, y_2, z_2) \ , and |\psi|^2\, is the joint probability density associated with the positions of both particles. Thus the probability that a measurement of the positions of both particles indicates particle one is in region R and particle two is in region S is
- \mathbf_ = \int_R \int_S |\psi|^2 \, dV_2 \, dV_1
where dV_1 = dx_1 dy_1 dz_1, and similarly for dV_2.
The normalization condition is then:
- \int \int |\psi(x, y, z)|^2 \, dV_2 \, dV_1 = 1
in which the preceding integral is taken over the full range of all six variables.
Given a wave function ψ of a system consisting of two (or more) particles, it is in general not possible to assign a definite wave function to a single-particle subsystem. In other words, the particles in the system can be entangled.
One particle in one dimensional momentum space
The wave function for a one dimensional particle in momentum space is a complex function \psi(p)\, defined over the real line. The quantity |\psi|^2\, is interpreted as a probability density function in momentum space:
- \mathbf_ = \int_^ |\psi(p)|^2\, dp
As in the position space case, this leads to the normalization condition:
- \int_^ |\psi(p)|^2\, dp = 1 .
The wave function for a spin-½ particle (ignoring its spatial degrees of freedom) is a column vector
- \vec \psi = \begin c_1 \\ c_2 \end.
The meaning of the vector's components depends on the basis, but typically c_1 and c_2 are respectively the coefficients of spin up and spin down in the z direction. In Dirac notation this is:
- | \psi \rangle = c_1 | \uparrow_z \rangle + c_2 | \downarrow_z \rangle
The values |c_1|^2 \, and |c_2|^2 \, are then respectively interpreted as the probability of obtaining spin up or spin down in the z direction when a measurement of the particle's spin is performed. This leads to the normalization condition
- |c_1|^2 + |c_2|^2 = 1\,.
A wave function describes the state of a physical system, | \psi \rangle\,, by expanding it in terms of other possible states of the same system, | \phi_i \rangle. Collectively the latter are referred to as a basis or representation. In what follows, all wave functions are assumed to be normalized.
Finite dimensional basis vectors
A wave function which is a vector \vec \psi with n components describes how to express the state of the physical system | \psi \rangle as the linear combination of finitely many basis elements | \phi_i \rangle, where i runs from 1 to n. In particular the equation
- \vec \psi = \begin c_1 \\ \vdots \\ c_n \end,
which is a relation between column vectors, is equivalent to
- |\psi \rangle = \sum_^n c_i | \phi_i \rangle,
which is a relation between the states of a physical system. Note that to pass between these expressions one must know the basis in use, and hence, two column vectors with the same components can represent two different states of a system if their associated basis states are different. An example of a wave function which is a finite vector is furnished by the spin state of a spin-1/2 particle, as described above.
The physical meaning of the components of \vec \psi is given by the wave function collapse postulate:
- If the states | \phi_i \rangle have distinct, definite values, \lambda_i, of some dynamical variable (e.g. momentum, position, etc) and a measurement of that variable is performed on a system in the state
- |\psi \rangle = \sum_i c_i | \phi_i \rangle
- then the probability of measuring \lambda_i is |c_i|^2, and if the measurement yields \lambda_i, the system is left in the state | \phi_i \rangle.
Infinite dimensional basis vectors
The case of an infinite vector with a discrete index is treated in the same manner a finite vector, except the sum is extended over all the basis elements. Hence
- \vec \psi = \begin c_1 \\ \vdots \\ c_n \\ \vdots \end
is equivalent to
- |\psi \rangle = \sum_ c_i | \psi_i \rangle,
where it is understood that the above sum includes all the components of \vec \psi. The interpretation of the components is the same as the finite case (apply the collapse postulate).
Continuously indexed vectors (functions)
In the case of a continuous index, the sum is replaced by an integral; an example of this is the spatial wave function of a particle in one dimension, which expands the physical state of the particle, | \psi \rangle, in terms of states with definite position, | x \rangle. Thus
- | \psi \rangle = \int_^ \psi(x) | x \rangle\,dx.
Note that | \psi \rangle is not the same as \psi(x)\,. The former is the actual state of the particle, whereas the latter is simply a wave function describing how to express the former as a superposition of states with definite position. In this case the base states themselves can be expressed as
- | x_0 \rangle = \int_^ \delta(x - x_0) | x \rangle\,dx
and hence the spatial wave function associated with | x_0 \rangle is \delta(x - x_0)\, (where \delta(x - x_0)\, is the Dirac delta function).
Given an isolated physical system, the allowed states of this system (i.e. the states the system could occupy without violating the laws of physics) are part of a Hilbert space H. Some properties of such a space are
- 1. If | \psi \rangle and | \phi \rangle are two allowed states, then
- a | \psi \rangle + b | \phi \rangle
- is also an allowed state, provided |a|^2+|b|^2=1. (This condition is due to normalisation.)
- 2. There is always an orthonormal basis of allowed states of the vector space H.
The wave function associated with a particular state may be seen as an expansion of the state in a basis of H. For example,
is a basis for the space associated with the spin of a spin-1/2 particle and consequently the spin state of any such particle can be written uniquely as
- a|\uparrow_z \rangle + b|\downarrow_z \rangle.
Sometimes it is useful to expand the state of a physical system in terms of states which are not allowed, and hence, not in H. An example of this is the spacial wave function associated with a particle in one dimension which expands the state of the particle in terms of states with definite position.
Every Hilbert space H is equipped with an inner product. Physically, the nature of the inner product is contingent upon the kind of basis in use. When the basis is a countable set \\,, and orthonormal, i.e.
- \langle \phi_i | \phi_j \rangle = \delta_.
Then an arbitrary vector | \psi \rangle can be expressed as
- | \psi \rangle = \sum_i c_i | \phi_i \rangle
where c_i = \langle \phi_i | \psi \rangle.
If one chooses a "continuous" basis as, for example, the position or coordinate basis consisting of all states of definite position \, the orthonormality condition holds similarly:
- \langle x | x' \rangle = \delta(x - x').
We have the analogous identity
- \langle x | \int \psi(x') | x' \rangle \,dx' = \int \psi(x') \delta(x - x')\,dx' = \psi(x).
Whether the wave function is real, and what it represents, are major questions in the interpretation of quantum mechanics. Many famous physicists have puzzled over this problem, such as Erwin Schrödinger, Albert Einstein and Niels Bohr. Some approaches regard it as merely representing information in the mind of the observer. Others argue that it must be objective.
- Introduction to Quantum Mechanics (2nd ed.)
wavefunction in Bosnian: Talasna funkcija
wavefunction in Czech: Vlnová funkce
wavefunction in German: Wellenfunktion
wavefunction in Modern Greek (1453-): Κυματοσυνάρτηση
wavefunction in Spanish: Función de ondas
wavefunction in Persian: تابع موج
wavefunction in Finnish: Aaltofunktio
wavefunction in French: Fonction d'onde
wavefunction in Galician: Función de onda
wavefunction in Hebrew: פונקציית גל
wavefunction in Hungarian: Hullámfüggvény
wavefunction in Italian: Funzione d'onda
wavefunction in Japanese: 波動関数
wavefunction in Korean: 파동함수
wavefunction in Lithuanian: Banginė funkcija
wavefunction in Dutch: Golffunctie
wavefunction in Norwegian: Bølgefunksjon
wavefunction in Polish: Funkcja falowa
wavefunction in Portuguese: Função de onda
wavefunction in Romanian: Funcţie de undă
wavefunction in Russian: Волновая функция
wavefunction in Slovak: Vlnová funkcia
wavefunction in Slovenian: Valovna funkcija
wavefunction in Swedish: Vågfunktion
wavefunction in Turkish: Dalga fonksiyonu
wavefunction in Ukrainian: Хвильова функція
wavefunction in Vietnamese: Hàm sóng
wavefunction in Chinese: 波函数