Maths, Physics, ICT and EAP Teacher
Stephen holds an honours degree in Mathematics from Trinity College Dublin, an honours MSc in Mathematical-Physics from University College Dublin and a CELT from Dublin School of English. He has taught at DIFC for seven years, in the subject areas of Mathematics, Statistics, Physics, ICT and EAP. In addition to teaching, continuing to learn is important to him and he has taken a number of classes through Coursera, primarily in programming. Stephen has also been the Moodle administrator for DIFC for the past five years and is proficient in R, C++, Python and PHP. He lives beside the sea with his wife, where he enjoys taking part in cultural and town activities.


PPE and P3Ei plan to hold a Structural Equation Model (SEM) training on December 15, 2018 at the Faculty of Economics, UII. SEM is a statistical tool used to complete multilevel models simultaneously which cannot be solved by linear regression equations. SEM can also be considered as a combination of regression analysis and factor analysis. SEM can be used to solve equation models with more than one dependent variable and also recursive influence. SEM is based on covariance analysis so that it gives a more accurate covariance matrix than linear regression analysis. Statistical programs that can be used to complete SEMs are Analysis Moment Structure (AMOS) or LISREL.
The training will be opened for 30 participants, for students (S2 & S3), lecturers, researchers, private / public employees and the general public who have the desire to improve their understanding of quantitative research methods. Training will be held at 08.00-16.00 at the Faculty of Economics, UII. This training will later be shared by Mr. Dr. Zainal Mustafa EQ MM who has been very competent in their field.
For more detailed information, please see the leaflet below,


Dr. Damir D. Dzhafarov is Assistant Professor in the Department of Mathematics at UConn. Dr. Dzhafarov also serves as Associate Director of the University of Connecticut Logic Group, which The Institute has helped to support with funding. The Logic Group is an interdisciplinary research cluster with over 40 faculty and graduate student members from Mathematics, Philosophy, Linguistics, Psychology, Computer Science, Law, and other departments.
Logic is a multi-faceted subject that, at its core, concerns language, structure, computation, reasoning and problem-solving, and their limitations and relationships. As such, it is an important area of interest in many disciplines, particularly those represented in the Logic Group, which have always maintained a strong tradition of scholarship in logic at UConn.
Dr. Dzhafarov’s research is in mathematical logic, and specifically in the areas of computability theory and reverse mathematics.
In his research, he seeks to understand the logical strength of mathematical problems—to characterize how close they are to being algorithmically solvable, and to understand the techniques needed to prove various results concerning them.
In broad terms, this allows one to formalize the notions that one problem cannot be solved using the same methods as another, or that a particular technique used in a proof is indispensable. A well-known example, from a slightly different setting, is the P vs. NP problem, which asks if rapid (polynomial-time) solution-verification is essentially the same as rapid solution-finding. This work deepens our insights into the fundamental properties underlying different areas of mathematics, and reveals new connections between them.
Two of the most prominent frameworks for this analysis are effective mathematics (also called applied computability theory) and reverse mathematics. Roughly, the former seeks to understand the strength of a problem in terms of the complexity of describing its solutions, while the latter seeks to do this in terms of the axioms needed to prove that solutions exist. One is thus a computational approach, the other a syntactic (or proof-theoretic) one. It is a common intuition among mathematicians that theorems in very disparate areas of mathematics can be strongly evocative of each other, and that the same patterns of problem-solving crop up in solutions to apparently unrelated problems. These frameworks provide a confirmation for this intuition. Indeed, a striking fact demonstrated by this research is that a vast number of mathematical theorems can be broadly classified into just five main types. This suggests that the methods underlying much of mathematical practice, across different fields, are deeply interrelated.
Dr. Dzhafarov’s research has appeared in high tier mathematics journals, such as Advances in Mathematics, Transactions of the American Mathematical Society, and the Journal for Symbolic Logic. His work has been supported by a variety of national and international agencies, including the U.S. National Science Foundation and the Simons Foundation.
Dr. Dzhafarov obtained his Ph.D. at the University of Chicago, and prior to coming to UConn, was a Visiting Postdoctoral Fellow at the National University of Singapore, and then an NSF Postdoctoral Fellow at the University of Notre Dame and the University of California, Berkeley.


Sunday, 3.14 is Pi day, designated because the date matches the number Pi. Pi is the constant that equals the ratio of a circle's circumference to its diameter. Here at Linda's Bakery we have Pie, a delicious pastry in the shape of a circle!
This week through Sunday our Pie Tarts are on sale buy 1 get 1 free up to 6 free per customer. Sunday we will have our 6" Pies on sale for $3.14, so stop in and get one while they last!


Download GeoGebra 6 free most current complete version offline set up for Mac OS X. GeoGebra is program for undertaking mathematical calculations at all academic levels.
GeoGebra 6 for Mac Critique
GeoGebra is computer software for executing mathematical calculations at all academic ranges. This app brings together matters of geometry, algebra, spreadsheets, charts and graphs, figures and calculus in one particular simple and compact package deal. This software package is one particular of the main merchandise in the industry of dynamic computing in many disciplines of arithmetic. The purposes of this products can possibly be really standard and be used by pupils in arithmetic for day-to-day computing. Or to be comprehensive adequate to be applied to other science and technologies researchers in other technological engineering disciplines.
This computational software is one of the ideal computing methods to accelerate learning, especially in pure arithmetic. Teachers and college teachers can use the applications talents to enhance the educating course of action for their students at whatsoever degree they are at. One of the attention-grabbing capabilities of this software package is the dynamic and dwell conversation between information tables, algebraic calculations and graphs and graphs. In point, alterations are produced to each part at any time, which will conserve you a ton of time. In this case, it is easier to exhibit the connection among variations and outputs.
The GeoGebra interface is incredibly simple and person-friendly. Of training course, this simplicity is not a detriment to its capabilities. If you have enough theoretical expertise of the topic you may well be utilizing, it might choose a handful of hrs or fewer. Another appealing feature of the application is the potential to build instructional world-wide-web internet pages. Dependent on distinct calculations and graphs, world-wide-web web pages can be made and delivered to other men and women. This application is free and for non-professional purposes there is no restriction on its use.
Characteristics GeoGebra 6 for Mac
- Powerful software for mathematical computations
- Collects algebra, geometry, spreadsheets, charts, and diagrams
- One particular of the ideal options for dynamic computing
- Work with diverse mathematical branches
- A precise atmosphere for the researchers
- Accomplish correct algebraic calculations
- Make customizations to analyze the output
- Mathematical calculations
- Work with different science and know-how jobs
- Dynamic and vivid communication between the information, calculations, and charts
- Delivers precise effects and complete dependable calculations
- Quite a few other options
Technical Specifics of GeoGebra 6 for Mac
- Mac Program Comprehensive Name: GeoGebra, GeoGebra Typical 6..729
- Model: 6..729
- Set up App File Identify: GeoGebra_Classic_6_MacOS_6..729..rar
- Entire Application Dimensions: 102 MB
- Developers Homepage: GeoGebra
Method Necessities for GeoGebra 6 for Mac
- Working Process: Mac OS X 10.13 or afterwards.
- Memory (RAM): 2 GB of RAM demanded.
- Tough Disk Place: 2 GB of free area necessary.
- Processor: Intel Main 2 Duo or afterwards.
GeoGebra Vintage 6..729 for Mac Free Download
Click on on the under button to download the entire full model set up of GeoGebra 6..652. for Mac for macOS Huge Sur with Apple M1 Сhip guidance.
GeoGebra Traditional 6..729
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Math is one of the most important and also challenging subjects to learn. Although many people admit that they hate math; they still use it n the daily lives. As an example, they will need to measure the proper amount of detergent for laundry and to balance their checkbook. For this reason, it can be quite difficult to define what parents want for their children at an early age. Kindergarten is a place where children can have subtle instructions on basic concepts of math. In general, it should be quite easy for us to group mathematical concepts, so children can understand them better. As an example, children can know the difference between money basics, geometry and others. By their 5th birthday, children should be able to count to twenty and in some cases, higher than that. It is very important for children to know about numbers. Flash cards and toys can be used to provide excitement to children. It shouldn’t be long before children have the ability to make simple additions and subtractions.
Grouping, ordering and patterns are essential math skills. With grouping, children can recognize numerical numbers very easily. Then, they will need to sort these numbers based specific characteristics. As an example, if they are given the numbers; 17, 13 and 20; they will need to sort them. The teaching process can be provided when playing games and reading stories. Patterns are quite similar to earlier math skills; grouping and ordering. In this case, teachers should know how they need to figure out patterns and this requires strong critical thinking. It is advisable to use different kinds of teaching methods. In this case, it would be much easier for us to teach children on how to do things. With proper grouping, ordering and patterns skills; children will be able to know what will come next. In any case, it is important for us to start off simple and then gradually provide a more difficult pattern for children. At the same time; children should be given the less and more concept. This is a concept that any five year old children should know. Both are obvious mathematical ideas and it should be quite easy to instruct children to choose which one is less and which one is more.
Money and coins are good tools for providing basic financial skills. Preschoolers already know they need money to buy something. So, parents should provide an instruction on the value of money and coins. Children will be happy that they finally know about the finance world. In this case, children will have a lot of fun, especially parents provide cash register toy for them. Siblings and children may establish a pretend shop and they can exchange items for money. Parents should provide children with limited amount of coins and money. In this case, children will know that they can’t buy everything and it is important for them to figure out things that they can buy.


Introduction to measurements and statistics for shooters
Our world is random. When anything is measured with sufficient level of precision several times, the results are usually slightly different. Why?
Every physical object consists of a huge number of tiny particles - atoms, molecules - that are moving (flying in gas or in liquid, or vibrating in solid bodies)
under the influence of heat. Heat itself is but a measure of this internal movement. The physical parameters of the body (length, temperature, weight)
are determined in part by this microscopic dance of its molecules and the similar movement of the the surroundings. The effects of this dance vary in time, which leads
to the slight jitter of the body's parameters.
Furthermore, the measuring devices are physical bodies which are susceptible to similar jitter, which adds another degree of randomness to the measurements. Because
precise instruments amplify the value to measure it, the random jitter within the instruments themselves is also amplified, often disproportionately.
This randomness is not always perceptible because our measurements are typically very coarse. Occasionally, however, it gets they get magnified and can be easily observable.
Shooting is one activity especially amenable to such amplification of randomness, because even the slightest variability leads to large differences over the long distance.
In probability and statistics, a random variable is a variable whose value is subject to variations due to chance. Every physical parameter that we can
measure - weight, length, temperature - are random variables.
How do you measure randomness?
Let's say you have a precise scale and a bullet. You put the bullet on the scale, and the reading is 167.991 grain. What is the weight of the bullet?
The obvious answer is that it is 167.991 grains, right?
Well, but what if you weigh it again and now the scale says 168.053? We repeat the weighing several more times and get the following results:
What now? There is no single number, but there is still plenty of information. For instance, we can say that all numbers lie between 167.991 and 168.103.
Or we can say that with the probability of 95% another measurement would fall in between 167.98 and 168.11 (how we got to this statement is explained below).
Most of the information we have about the world that surrounds us really exists as a bunch of confidence intervals (we are 95% certain that the radius of
Earth is between X and Y meters). 95% is actually a commonly accepted confidence interval in business and sometimes in engineering - we consider that we know
something pretty well if we are 95% certain that it is true. In science, however, the requirements are much stronger - we want to be 99.5% certain before we
declare something as a scientific fact.
But is the confidence interval enough? Consider the following two sets of measurements:
The measurements all fall into the same interval. But it seems intuitive that the physical bodies in these two experiments are different.
Probability distribution assigns a probability to each of the possible outcomes of a random experiment.
Whether expressed as a graph or a formula, it will tell you how likely one is to get a particular value upon the measurement.
Standard deviation (σ)
Another important measure of randomness is the standard deviation. For a series of measurements the standard deviation shows how far
an "average measurement" is from the average of all measurements. If the standard deviation is small, you should have good confidence
in the measurement (there is a high likelihood that subsequent measurements will yield similar results - or subsequent shots will land
near the target).
If you have N measurements x1, x2, ..., xN, then the standard deviation is
For example, let's say you shoot 10 rounds at a bulls eye, and the vertical distribution of the shots is as follows (0 is the target,
positive axis points up):
The average deflection is 0.32, and the standard deviation is 0.14.
One of the most famous functions in mathematics is the so called "bell curve", or normal distribution. It is so famous because almost every
physical event is described by it. The reason for it is the Central Limit Theorem, my favorite result in statistics.
Central Limit Theorem states that, given certain conditions, the mean of a sufficiently large number of independent random variables,
each with finite mean and variance, will be approximately normally distributed.
Because every physical thing that surrounds us is a combination of many things, and its properties are influenced by many random factors, the
net result of all these influences end up being normally distributed, even if the individual factors are not.
A normally distributed value has a 95% confidence interval of (roughly) +/- 2σ
If we know that the distribution is normal (and it almost always is), and we can estimate its standard deviation, we can predict subsequent results
with a level of confidence.
Estimating the parameters of the normal distribution
If you have a bunch of results, you can compute the standard deviation of these results, and then the confidence interval for the future behavior.
It is important to understand, however, that what you compute based on limited number of experiments, is just an estimate of what these parameters might be. To
illustrate this, let's see you've made fifty shots at the target and recorded the vertical dispersion as follows:
The average vertical POI from this sample is -0.07, and the standard deviation is 0.41. Looking good so far? Well, if you were to shoot only 25 times,
your average would have been -0.09, and your standard deviation 0.44. Out of 10-shot sample, you would have worked out -0.11 average and 0.28 standard deviation,
and out of 5-shot sample, -0.02 and 0.20, respectively.
The "true" standard deviation of the sample above is actually 0.5 (this is how I generated it). Because we only ran finite number of experiment, we have
arrived at a series of estimates for the parameters of the distribution, but not the real values. The more experiments we run, the more the results will fill
the real curve, and the closer we will get to the true value.
It turns out, however, that we can do pretty good estimates even using finite number of experiments. Standard deviation has its own distribution, called
Student's t-distribution. The mathematics of it is far too complicated to explain here,
but there is a handy function in Excel - STDEV.S - that estimates the real value of σ from a finite set of measured values.
Are they different?
If you can estimate the confidence intervals, you can answer many really interesting question about your data. For example, given the two data sets,
are they statistically different?
For example, let's go back to our previous example, and ask ourselves whether the first column is materially different from the second one.
Perhaps you varied powder a little, and want to know whether it resulted in a vertical shift of the point of impact.
On the surface, the average of the first column is -0.09, and for the second it is -0.05, so there might be a difference.
However, the confidence interval for the first column is (+/- 2σ) from -0.97 to 0.79, and for the second, -0.82 to 0.72. Not only
both values are well within each other's confidence intervals, but even the intervals themselves significantly overlap!
The difference between two values is considered to be statistically significant, if they lie outside each other's confidence
intervals. So to determine this mathematically, you can compute the confidence intervals of the two datasets, and make sure that
their averages lie outside each others' confidence intervals.
Note, however, that the confidence intervals may be very wide if you did not run enough experiments. So it could be that the
values look the same only because you did not collect sufficient stats.
Here is how to determine if two sequences of numbers are statistically different in Excel.
First, you need to enable Data Analysis Add in: go to Options -> Add-Ins, and click on "Go..."
In the dialog that opens you will need to check Analysis Toolpack.
You should then see "Data Analysis" option in your Data tab. You can now click on it and choose "t-Test: Two-Sample Assuming Unequal Variances".
Put the ranges for the two datasets in the variable inputs, and ensure that Alpha is 0.05, which corresponds to 95% confidence level interval.
Clicking OK produces the following output:
Note two values: t-stat and t-critical. If your t-sta's absolute value (dropping the sign if it is negative) is below t-critical value, the difference is not


Using appropriate force and displacement parameters it is shown that the response of elastomers in multiaxial stress fields is identical to their response to uniaxial stresses. These parameters were chosen from consideration of the molecular model assumed in the theory of Kuhn and Grun [Koll. Z.
, 248 (1942)]. The required force is established to be given by ∣σ∣ = (12+22+32)1/2
where the quantities i
are the forces along orthogonal axes on the deformed cross‐sectional area; the displacement is given by ∣e
∣ = (e12
where the quantities ei
are the strains along three orthogonal axes. The angle between the force and displacement vectors is found to be independent of the axiality of the stress field.


For school, I have been focusing on Mandarin Chinese and Algebra. Hours have been melting into days, and days to weeks, of confusing equations and unintelligible sentences. The math isn’t very difficult, there’s just far too much of it. You know it’s gotten bad when you close your eyes and see this:
If that happens, you have something I call math brain. Go take a walk. But there’s an even worse side effect of too much math; math hair. I’ve never given much thought to the idea of math hair, but my sister brought it up a little while ago, and we both have it.
You think I’m exaggerating? Maybe a little. But not much. It’s caused by your hand tugging at your hair in frustration, while working on a tough problem. Sorry this post has been so short, but my Algebra brain is pushing all non-math thoughts out of my head. This has been the random ramblings of Leila. Thanks for reading!


Closing date: 20/06/2019 1:00 pm
September 2019 or January 2020
We are seeking a well-qualified, passionate, committed and enthusiastic Mathematics teacher for September 2019 or January 2020 to join an exceptionally strong and successful department with an outstanding academic record.
The Mathematics department has been in the top 1% of ALPS schools for adding value for 3 of the last 5 years and averages 80% of grades at A*-B in A Level Mathematics and Further Mathematics. The successful applicant will teach Key Stage 3 and Key Stage 4 Mathematics. There a possibility of A Level teaching in the future for the right candidate.
Previous applicants for the recently advertised part-time post are strongly encouraged to reapply.
Please submit your application, via our online application form.
Should you require any further information, please contact the HR department, tel 01509 817089; email HR@ratcliffecollege.com
Enhanced DBS clearance will be required for this role.


Our Panel Calculation: How Many Trees Does Plantd Save Per Single-Family House?
To make products like Oriented Strand Board (OSB) for residential construction, building materials manufacturers use trees like loblolly pine that are grown on managed timberlands. This lumber is called pulpwood and is composed of either the waste material of large trees or the whole material of smaller trees.
Loblolly pine is commonly harvested on managed timberlands across the Southeast region of the United States. According to the United States Department of Agriculture, loblolly pine grows to a diameter between 4-13 inches. These trees are typically harvested 10-15 years after planting and range between 30-70 feet tall at harvest.
This post shows the math we use to calculate how many trees our grass panels replace in the construction of an average-sized single-family house.
Tree calculations are complicated by the fact that trees are not of uniform size and height. So we make a few assumptions along the way.
This is the framework we follow:
Step 1: Calculate the volume and mass of a loblolly pine tree grown for OSB production.
Step 2: Calculate the amount of wood in one sheet of OSB.
Step 3: Divide Step 1 by Step 2 to determine the number of OSB panels one tree can produce.
Step 4: Determine the number of OSB sheets used in an average-sized single-family house in the U.S.
Step 5: Divide Step 4 by Step 3 to determine how many trees our grass panels will replace.
The answer is approximately 16-17 trees saved per house by using Plantd perennial grass structural panels instead of tree-based OSB for wall sheathing and roof decking.
Read on for how we arrived at this number.
Let's start with Step 1: Volume, Density, and Mass
A. Calculate the volume of harvestable wood in a tree
Fact: Loblolly pine trees average 12 inches wide at breast height with a tapering diameter at the top of the tree.
Assumption: The average tree is 50 feet tall.
Assumption: The average tree cylindrical volume is 8.5 inches in diameter.
Assumption: The average tree llife span is 12.5 years.
Conclusion: The estimated volume of a tree is 34,030 cubic inches. This is equates to approximately 20 cubic feet and .56 cubic meters.
B. Calculate the density and mass of harvestable wood in a tree
Fact: Chapters 5-7 of the USDA Wood Handbook states that freshly cut loblolly pine has roughly 12% moisture content and a specific gravity of .51.
Calculation: Multiplying a .51 specific gravity times 1000 equates to a material density of 510 kg/cubic meter.
Calculation: Multiplying a .51 specific gravity times 1000 times a volume of .56 cubic meters equates to a tree mass of approximately 284 kg, or 626 pounds.
Knockdowns: According to an OSB Life Cycle Analysis from the University of Tennessee, the production process for turning trees into Oriented Strand Board panels is only 75% efficient. 25% of the tree is wasted during the production process. For our math, that equates to 156 pounds of the 626-pound tree.
Conclusion: The usable wood from a single loblolly pine tree weighs 470 pounds.
Step 2: Amount of Wood in an OSB Sheet
Fact: The U.S. Department of Commerce National Institute of Standards and Technology states that the average mass of a sheet of commodity OSB is 1.46 pounds per square foot, of which 1.38 pounds is wood.
Fact: A standard sheet of OSB is 4 feet by 8 feet, which equals 32 square feet.
Calculation: Multiplying 1.38 pounds of wood biomass times 32 square feet per sheet of OSB equals 44 pounds of wood.
Conclusion: The wood in one sheet of OSB weighs about 44 pounds.
Step 3: How Many OSB Panels in a Tree
Calculation: 470 pounds of usable wood in a tree divided by 44 pounds of wood in an OSB sheet equals about 10.7 OSB panels per tree.
Step 4: Number of OSB sheets used in an average-sized single-family house for wall sheathing and roof decking
It's surprisingly challenging to identify a public data source for the average number of sheets used for wall sheathing and roof decking in a typical single-family home. Below is the data we rely upon to contextualize and calculate an answer.
Fact: According to the 2022 Builder Practices Survey conducted by Home Innovation Research Labs and referenced by the National Association of Homebuilders (NAHB), the average single-family home in the U.S. uses over 6800 square feet of OSB.
Fact: Based on data we've received from customers, we know that a 2200 square foot home can use 175 sheets of OSB for wall sheathing and roof decking, which is 5600 square feet of OSB. We will use this data for our final calculation.
Step 5: Number of trees our grass panels replace
Calculation: 175 sheets of OSB per house divided by 10.7 sheets of OSB per tree equals approximately 16-17 trees per house replaced by Plantd's perennial grass structural panels for walls and roofs.
Building the Future, One Machine at a Time
This month, we invited Plantd’s investors and industry stakeholders to our first Demo Day, where we unveiled our pilot production technology and...
The Most Impactful 101 Acres We’ll Ever Plant
Creating a new agricultural supply chain is daunting. Yet, we realized less than a year after founding Plantd in 2021 that to accomplish our goal...


Killer Sudoku is the new fun take on the classic Sudoku game which is based around puzzle logic with numbers. You are required to follow all of the rules of the typical Sudoku game however, there will be cages of all different shapes that require all of the numbers within them to add up to the required value.
Warning! This game will be super challenging. If you are an avid sudoku player then you will love to take on this challenge as it may be your hardest one yet. Take your time and get as much practice as you can and you will soon be up to the task.
Killer Sudoku is made by Patrick Min.
- A good mix of Sudoku and math
- Daily challenge
- Easy and medium difficulty
- Usable tools to make your game enjoyable
Killer Sudoku is a web browser game (desktop and mobile).
CategorizationPuzzle Games » Sudoku Games » Killer Sudoku
More Information About Killer Sudoku
Killer Sudoku is a fantastic sudoku game that you can enjoy on CrazyGames.com directly in your browser, for free. The game is powered by HTML5 technology to run without trouble in most browsers. Killer Sudoku has been played 599 times and has been rated 6.7 out of 10 with 6 votes. Did you like playing Killer Sudoku? Then check out our other sudoku games or to Daily Sudoku and Puzzle World.SHOW LESS
ControlsHow to play Killer Sudoku
Use your mouse and keyboard to play the game.


Support for Academic Success
If you need help in a class, you're not alone. We offer tutoring and assessment test prep programs to help you be successful.
Remember to use all the resources available to you early in the quarter to get the best start in your classes. We are here to help!
If you see one − or two or three − that fit your goals and needs, please ask when meeting with a counselor about which ones would be best for you!
Did you know?
Schedules for both tutoring centers are available by the second or third week of the quarter. Check back to stay on track with your goals. Also see our other workshop offerings here and on our campus events calendar.
English Summer Bridge
Provides a summer review session for students interested in preparing for an English placement retest.
Assessment Eligibility Requirements: ENGL 209, 1S or 110
State-funded program serving students who are financially and educationally disadvantaged. Includes one-on-one tutoring.
Math Performance Success
Would you like extra help with Math 10: Statistics?
If yes, the MPS program provides support to help you complete your general education, transfer and associate degree math requirement.
Math Summer Bridge
Summer review session for students preparing for a math placement retest.
Assessment Eligibility Requirements: MATH 235, 220, 217, 108, 105 or NCBS 401A/B
Pass the Torch
Students are matched into one-on-one study teams for the quarter. One student is the leader (tutor) and the other is the member (tutee).
Teaching & Learning Center
Free tutoring for students enrolled in ANY course.
Provides one-on-one tutoring with instructional associates and faculty to help students develop a better understanding of math, physics and chemistry concepts.


What can you see? The Müller-Lyer-illusion is one of the most well-known optical illusions and so it is not surprising that so many variants and further developments have emerged. Gianni Sarconi presented animations at the 2017 Best illusion of the Year Contest that are particularly impressive. His star-shaped arrangements seem to pulsate, although all line segments have the same length. That was the starting point for this project.
What can you do? After the program Müller-Lyer animations has started, you can see a star with eight lines and16 pulsating arrows. You can choose between six variants of the Müller-Lyer-illusion by pressing the corresponding button:
(s) A star with n lines and one segment with two arrows whose peaks point alternately to each other. This corresponds to the original müller-lyer variant.
(p) A pulsating star with n lines and three arrows pointing alternately to each other. The lines start at a fixed distance from the centre point.
(b) A big pulsating star with n lines and four arrows pointing alternately to each other. The lines start at a fixed distance from the centre point.
(f) A fence with with n vertical lines and six arrows pointing alternately to each other.
(h) A hand fan with n sloping lines and six arrows pointing alternately to each other.
(c) A circular pattern with n circles and n segments and n arrows pointing alternately to each other.
For every variant you can change the number of lines (1 ≤ n_lines ≤ 75), the length of the lines (20 ≤ length_line ≤ 300), and the length of the arrows (2 ≤ length_arrow ≤ 100) with the corresponding sliders. The legs of the arrows can oscillate between the angles start and end (0 ≤ start,end ≤ 180, always start > end!). With speed the movement of the arrows can be accelerated (1 ≤ speed ≤ 25).
In the case of pulsating star and big pulsating star with distance_from_center the distance of the lines from the centre of the image can be set (0 ≤ distance_from_center ≤ 200). In the case of a circular pattern, the number_of_segments is added, i.e. the subdivision of the circle into arcs with an angle of 360/number_of_segments (2 ≤ number_of_segments ≤ 50), as well as the distance between the circles (20 ≤ delta_circles ≤ 200).
In all variants one can see how, depending on the position of the arrows in relation to each other, the line segments appear shorter or longer, although of course they are and remain all the same length. The effect is quite impressive in many configurations.
What else can you do? This programme is actually a big Müller-Lyer animation lab. In the code of the star-variant and also with the other variants you can find further parameters with which additional effects can be investigated:
(pen) With pen the pen size can be changed. Can thicker lines increase the visibility of the effect?
(delta_arrow) With delta_arrow the length of the arrows is shortened from the outside to the inside (animation right). Does this increase or decrease the visibility of the effect?
(delta_angle) With delta_angle the angle of the arrows are changed (increased) in percentage from line to line (animation below). This can produce a very strong effect.
Related topics: Stereokinetic efffect
Sarcone, G. (2018). Dynamic Müller-Lyer Illusion. From Sarcone’s Studio — A Sarcone & Waeber Web Resource. Weblog Sarconista!


STEM stands for Science, Technology, Engineering, and Mathematics. STEM is important because it gets students to explore, design, test, explore in science, technology, engineering, or math. STEM challenges are all about the thought process and thinking like a STEM-ist improves problem solving skills in multiple areas of life.
Here at Astatula Elementary School we pride ourselves in challenging our students throughout diverse programs on a multitude of learning levels to achieve optimal educational success with each individual student. The program is currently offered to our third through fifth grade students.
For more specific information, please visit our District STEM Home Page.
Katie Gordon - STEM


This applet allows students to individually or together practice working with relationships among fractions and ways of combining fractions.
Directions: Click on the pile of cards to turn one over. This is your target fraction. Move the markers so that the sum of your moves is a fraction that is less than or equal to the target fraction.
Goal: Get all the markers to the right side of the game board. Use as few cards as possible. Click to play.
|© 2006 Connected Mathematics Project, Michigan State University|
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Join us on Thursday, October 12th as we explore math centers from 5:30-6:30. You will be able to interact with your child at a center based on a learned math skill from their grade level. Afterwards, we will enjoy hotdogs, chips and drinks.
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Options profit and loss formula
The entire formula in C8 becomes:. Maximum possible loss is equal to initial cost of the option and applies for underlying price below than or equal to the strike price. In our example, the formula in cell C8 will be:. Same as scenario 1 in fact.
The second component of a call option payoff, cash flow at expiration, varies depending on underlying price. If underlying price is above the strike price, you exercise the option and you can immediately sell it on the market at the current underlying price. Cash flow at expiration The second component of a call option payoff, cash flow at expiration, varies depending on underlying options profit and loss formula. With underlying price above the strike, the payoff rises in proportion with underlying price.
You can test different values for the underlying price input and see how the formula works. The entire formula in C8 becomes:. Now we have created simple payoff calculators for call and put options. This options profit and loss formula the basic building block that will allow us to calculate profit or loss for positions composed of multiple optionsdraw payoff diagrams in Exceland calculate risk-reward ratios and break-even points.
Options profit and loss formula will merge our call and put calculations in the next part of the tutorial. The put option profit or loss formula in cell G8 is: Call Option Value Formula Now we have the cells ready and we can build the formula in cell C8, which will use the inputs in the other cells to calculate profit or loss.
Payoff Formula Inputs options profit and loss formula Outputs In the above example you can identify several inputs that our payoff formula will take — they are the numbers we already know: Options profit and loss formula, we should improve our calculations to also consider direction long or shortposition size number of contracts and contract size number of shares represented by one option contract. Macroption is not liable for any damages resulting from using the content. Furthermore, our calculator only shows profit or loss per share, while many people are actually more interested in total dollar profit or loss, especially when working with positions of multiple option contracts. All information is for educational purposes only and may be inaccurate, incomplete, outdated or plain wrong.


But I'm also looking at a curve beta that's given by beta of s. It's the same thing as going over with h and then applying gamma, so gamma(h(s)) is the same as beta f(s). Nearby points are mapped to nearby points. So altogether 1 minus one-half is one-half. That doesn't affect what's happening with my transitions on the inside. Introduction to Integration. This is a very important. Integration is a way of adding slices to find the whole. Read this article for a great introduction, LECTURE 6: COMPLEX INTEGRATION The point of looking at complex integration is to understand more about analytic functions. Cauchy’s integral theorem 3.1 ... Introduction i.1. Complex integrals have properties that are similar to those of real integrals. When you plug in 0, the term vanishes and so the integral has value, 2 root 2 over 3. As before, as n goes to infinity, this sum goes to the integral from a to b of gamma prime of t dt. Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail, 1. Analyticity. We looked at this curve before, here's what it looks like. The fundamental discovery of Cauchy is roughly speaking that the path integral from z0 to z of a holomorphic function is independent of the path as long as it starts at z0 and ends at z. We'd like to find an upper bound for the integral over gamma of the function z squared, dz. This process is the reverse of finding a derivative. This actually equals two-thirds times root two. And that's exactly what we expected, this length right here is indeed square root of 2. Those two cancel each other out. And so, we find square root of 2 as the answer. I had learned to do integrals by various methods show in a book that my high Both the real part and the imaginary part are 1, together it adds up to 2. … Weâll begin this module by studying curves (âpathsâ) and next get acquainted with the complex path integral. When t is = to 0, it's at the origin. Evaluation of real definite Integrals as contour integrals. Complex integration We will deﬁne integrals of complex functions along curves in C. (This is a bit similar to [real-valued] line integrals R Pdx+ Qdyin R2.) ( ) ... ( ) ()() ∞ −−+ � So the length of this curve is 2 Pi R, and we knew that. That's the integral we evaluated at the upper bound. This can be viewed in a similar manner and actually proofs in a similar manner. For some special functions and domains, the integration is path independent, but this should not be taken to be the case in general. Here's a great estimate. If we can nd a positive integer n such that limz!a(z a)nf(z) 6= 0 then z = a is called a pole of order n for f(z). So for us f(z) is the function z squared. This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. A connected patch is mapped to a connected patch. So the length of gamma is the integral over gamma of the absolute value of dz. So the interval over gamma, absolute value of F of C, absolute value of D Z. Sometimes it's impossible to find the actual value of an integral but all we need is an upper-bound. So here's [a, b], and there's [c, d]. We shall nd X; Y and M if the cylinder has a circular cross-section and the boundary is speci ed by jzj = a: Let the ow be a uniform stream with speed U: Now, using a standard result, the complex potential describing this situation is: Again using the Key Point above this leads to 4 a2U2i and this has zero real part. Contour integration methods include: direct integration … Convention Regarding Traversal of a Closed Path. 101-108 : L9: Cauchy-Goursat theorem: Ahlfors, pp. Line integrals: path independence and its equivalence to the existence of a primitive: Ahlfors, pp. Just the absolute value of 1 + i. f(z) is the function z squared. Integrals of real function over the positive real axis symmetry and pie wedges. A function f(z) which is analytic everywhere in the nite plane except at nite number of poles is called a meromorphic function. Suppose you wanted to integrate from 2 to 4 the function s squared times s cubed plus one to the 4th power ds. Therefore, the complex path integral is what we say independent of the chosen parametrization. What's 4 cubed + 1? We're defining that to be the integral from a to b, f of gamma of t times the absolute value of gamma prime of t dt. Complex integration definition is - the integration of a function of a complex variable along an open or closed curve in the plane of the complex variable. If you write gamma of t as x(t) + iy(t), then the real part is 1-t. And the imaginary part is simply t. So y = t, x = 1-t. method of contour integration. Integration is a way of adding slices to find the whole. 4. Preliminaries. Laurent and Taylor series. The integral over minus gamma f of (z)dz, by definition, is the integral from a to b f of minus gamma of s minus gamma from (s)ds. A point z = z0 at which a function f(z) fails to be analytic is called a singular point. Section 4-1 : Double Integrals Before starting on double integrals let’s do a quick review of the definition of definite integrals for functions of single variables. So, by integration by substitution, it's the same thing as the integral from a to b, f of gamma of t, gamma prime of t dt. And that's what you see right here. Suppose gamma of t is given by 1-t(1-i), where t runs from 0 to 1. 2. Then the integral of their sum is the sum of their integrals; … An antiderivative of t squared is 1/3 t cubed and that's what you see right here. So if you integrate a function over a reverse path, the integral flips its sign as compared to the integral over the original path. where Re denotes the real part, is the (constant) density of the uid and w = u + iv is the complex potential for the ow both of which are presumed known. Introduction to conformal mappings. We evaluate that from 0 to 1. The imaginary part results in t. So altogether the absolute value is 2t squared. But, gamma (t) is t + it. The area should be positive, right? Is there any way by which we can get to know about the function if the values of the function within an interval are known? So the integral with respect to arc length. Introduction. You could then pull the M outside of the integral and you're left with the integral over gamma dz which is the length of gamma. That is rie to the it. In particular, if you happen to know that your function f is bounded by some constant m along gamma, then this f(z) would be less than or equal to m. So you could go one step further, is less than equal to the integral over gamma m dz. They're linearly related, so we just get this line segment from 1 to i. Because you can't really go measure all these little distances and add them up. We also know that the length of gamma is root 2, we calculated that earlier, and therefore using the ML estimate the absolute value of the path integral of z squared dz is bounded above by m, which is 2 times the length of gamma which is square root of 2, so it's 2 square root of 2. Introduction Many up-and-coming mathematicians, before every reaching the university level, heard about a certain method for evaluating deﬁnite integrals from the following passage in : One thing I never did learn was contour integration. Now so far we've been talking about smooth curves only, what if you had a curve that was almost smooth, except every now and then there was a little corner like the one I drew down here? So h(c) and h(d) are some points in this integral so where f is defined. A connected region is one which any two points in it can be connected by a curve which lies entirely with in the region. A point z = z0 is said to be isolated singularity of f(z) if. Introductory Complex Analysis Course No. And the absolute value of z, on this entire path gamma, never gets bigger. Given the curve gamma and a continuous function on gamma, it can be shown that the integral over gamma, F of Z, DZ, the absolute value of that integral is found the debuff of the integral over gamma, absolute of F of Z, absolute value DZ. Lecture 1 Play Video: Math 3160 introduction We describe the exegesis for complex numbers by detailing the broad goal of having a complete algebraic system, starting with natural numbers and broadening to integers, rationals, reals, to complex, to see how each expansion leads to greater completion of the algebra. Zeta-function; $ L $- function) and, more generally, functions defined by Dirichlet series. the function f(z) is not de ned at z = 0. Basics2 2. The complex integration along the scro curve used in evaluating the de nite integral is called contour integration. So the absolute value of z never gets bigger than the square root of 2. 5/30/2012 Physics Handout Series.Tank: Complex Integration CI-7 *** A more general discussion of branch cuts and sheets can be found in the references. The implication is that no net force or moment acts on the cylinder. A diﬀerential form pdx+qdy is said to be closed in a region R if throughout the region ∂q ∂x = ∂p ∂y. Now, if the degree of P(x) is lesser than the degree of Q(x), then it is a proper fraction, else it is an improper fraction. The following gure shows a cross-section of a cylinder (not necessarily cir-cular), whose boundary is C,placed in a steady non-viscous ow of an ideal uid; the ow takes place in planes parallel to the xy plane. Even if a fraction is improper, it can be reduced to a proper fraction by the long division process. It's a sharp estimate, it doesn't get any better. The method of complex integration was first introduced by B. Riemann in 1876 into number theory in connection with the study of the properties of the zeta-function. 3. In differentiation, we studied that if a function f is differentiable in an interval say, I, then we get a set of a family of values of the functions in that interval. Copyright © 2018-2021 BrainKart.com; All Rights Reserved. The real part is t. And then we take the imaginary part and square it. •Proving many other asymptotic formulas in number theory and combi-natorics, e.g. This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. When you plug in 1 for t, you get 2 root 2 over 3. These assignments are not meant to be completed quickly; rather you'll need paper and pen with you to work through the questions. where z = i; 2i are simple poles lie inside and z = I; 2i are simple poles lie outside, the semi-circle becomes very large and the real and imagi-nary parts of any point lying on the semi-circle becomes very large so that. So in the end we get i minus 1 times 1 minus one-half times 1 squared. Next let's look again at our path, gamma of t equals t plus it. We will start by introducing the complex plane, along with the algebra and geometry of complex numbers, and then we will make our way via differentiation, integration, complex dynamics, power series representation and Laurent series into territories at the edge of what is known today. And this is my delta tj. Preliminaries to Complex Analysis 1 1 Complex numbers and the complex plane 1 1.1 Basic properties 1 1.2 Convergence 5 1.3 Sets in the complex plane 5 2 Functions on the complex plane 8 2.1 Continuous functions 8 2.2 Holomorphic functions 8 2.3 Power series 14 3 Integration along curves 18 4Exercises 24 Chapter 2. In mathematical terms, the convergence rate of the method is independent of the number of dimensions. Let me clear the screen here. multiscale analysis of complex time series integration of chaos and random fractal theory and beyond Nov 20, 2020 Posted By Corín Tellado Ltd TEXT ID c10099233 Online PDF Ebook Epub Library bucher mit versand und verkauf duch amazon multiscale analysis of complex time series integration of chaos and random fractal theory and beyond A region in which every closed curve in it encloses points of the region only is called a simply connected region. So if you do not like this notation, call this gamma tilde or gamma star or something like that. "National Academies of Sciences, Engineering, and Medicine. Gamma prime of t is 1 + i. supports HTML5 video. Now, we use our integration by substitution facts, h(s) is our t. So, this is also our t and there's our h friend (s)ds which will become our dt. An anti-derivative of e to the minus it is i times e to the minus it evaluated from 0 to 2pi. Squared, well we take the real part and square it. Â© 2021 Coursera Inc. All rights reserved. Let's find the integral over gamma, f(z)dz. 7. Introduction to Complex Analysis - excerpts B.V. Shabat June 2, 2003. So square root of 2 is the length of 1 + i. 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. We evaluate that from 0 to 1. integration. Cauchy integral theorem; Cauchy integral formula; Taylor series in the complex plane; Laurent series; Types of singularities; Lecture 3: Residue theory with applications to computation of complex integrals. Let's see what the integral does. The theory of complex functions is a strikingly beautiful and powerful area of mathematics. An analytic function f(z) is said to have a zero of order n if f(z) can be expressed as f(z) = (z z0)m (z) where (z) is analytic and (z0) 6= 0. And there's this i, we can also pull that out front. This is my f of gamma of t. So 1 over gamma of t. So one over e to the it times gamma prime of t which is 1, dt. 2. We're left with the integral of 0 to 1 of t squared. We're putting gamma of t into it, that's right here, and then we multiply by the gamma prime of t. So this is the integral we have to evaluate. This has been particularly true in areas such as electromagnetic eld theory, uid dynamics, aerodynamics and elasticity. Each module consists of five video lectures with embedded quizzes, followed by an electronically graded homework assignment. And we know what we have to do is we have to look at f of gamma of t times gamma prime of t and integrate that over the bounds from 0 to 2 pi. Zeta-function; $ L $- function) and, more generally, functions defined by Dirichlet series. Contour integration is closely related to the calculus of residues, a method of complex analysis. This is one of the ... Itis earnestlyhoped thatAn Introduction to Complex Analysis will serve an inquisitive reader as a starting point in this rich, vast, and Converse of Cauchy's Theorem or Morera's Theorem (a) Indefinite Integrals. C(from a ﬁnite closed real intervale [a;b] to the plane). So the integral over beta is the same thing as the integral over gamma. So f of gamma of t is simply e to the it times the absolute value of the derivative. Line ). How do you actually do that? And when t is equal to 1, gamma of 1 is equal to 1-(1-i), in other words, i. We know that gamma prime of t is Rie to the it and so the length of gamma is given by the integral from 0 to 2Pi of the absolute value of Rie to the it. By definition, that's the integral from 0 to 1, we look at gamma (t), instead of z squared and then we need to fill in absolute value of gamma prime of t(dt). So this right here is my h of s, then here I see h of s to the fourth power. For some special functions and domains, the integration is path independent, but this should not be taken to be the case in general. One of the universal methods in the study and applications of zeta-functions, $ L $- functions (cf. In other words, the absolute value can kind of be pulled to the inside. Here are some facts about complex curve integrals. the integration around c being taken in the positive direction. The prof makes it easy to understand yet fascinating. Video explaining Introduction for Complex Functions. We then have to examine how this integral depends on the chosen path from one point to another. Kapitel 6: Komplexe Integration Bemerkungen zu komplexen Kurvenintegralen. Well, suppose we take this interval from a to b and subdivide it again to its little pieces, and look at this intermediate points on the curve, and we can approximate the length of the curve by just measuring straight between all those points. COMPLEX INTEGRATION Lecture 5: outline ⊲ Introduction: deﬁning integrals in complex plane ⊲ Boundedness formulas • Darboux inequality • Jordan lemma ⊲ Cauchy theorem Corollaries: • deformation theorem • primitive of holomorphic f. Integral of continuous f(z) = u+ iv along path Γ in complex plane The estimate is actually an equality in this particular case. 1. So in my notation, the function f of gamma of t is just the function 1. This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. Let gamma(t) be the curve t + it. We know that that parameterizes a circle of radius r. Gamma prime(t), we also know what that is. And then if you zoom into another little piece, that happens again. What is the absolute value of 1 + i? Given a smooth curve gamma, and a complex-valued function f, that is defined on gamma, we defined the integral over gamma f(z)dz to be the integral from a to b f of gamma of t times gamma prime of t dt. It's 2/3 times (-1 + i) in the last lecture. If that is the case, the curve won't be rectifiable. These are the sample pages from the textbook, 'Introduction to Complex Variables'. And it's given by taking the original curve gamma, but instead of evaluating at t, we evaluate it at a+b-t. So again, gamma of t is t + it. Introduction 3 2. 6. So we need to take the absolute value of that and square it, and then multiply with the absolute value of gamma prime of t, which is square root of 2. multiscale analysis of complex time series integration of chaos and random fractal theory and beyond Nov 20, 2020 Posted By Evan Hunter Library TEXT ID c10099233 Online PDF Ebook Epub Library encompasses all of the basic concepts necessary for multiscale analysis of complex time series fills this pressing need by presenting chaos and random fractal theory in a So we're integrating from zero to two-pi, e to the i-t. And then the derivative, either the i-t. We found that last class is minus i times e to the i-t. We integrate that from zero to two-pi and find minus i times e to the two-pi-i, minus, minus, plus i times e to the zero. applied and computational complex analysis vol 1 power series integration conformal mapping location of zeros Nov 20, 2020 Posted By William Shakespeare Ltd TEXT ID 21090b8a1 Online PDF Ebook Epub Library and computational complex analysis vol 1 power series integration conformal mapping location of zeros nov 19 2020 posted by r l stine library text id 21090b8a1 … Let's see if we can calculate that. If a function f(z) is analytic and its derivative f0(z) is continuous at. COMPLEX INTEGRATION • Deﬁnition of complex integrals in terms of line integrals • Cauchy theorem • Cauchy integral formulas: order-0 and order-n • Boundedness formulas: Darboux inequality, Jordan lemma • Applications: ⊲ evaluation of contour integrals ⊲ properties of holomorphic functions ⊲ boundary value problems. Gamma prime of t is, well, the derivative of 1 is 0. Slices. Since a complex number represents a point on a plane while a real number is a number on the real line, the analog of a single real integral in the complex domain is always a path integral. Minus gamma prime of t is the derivative of this function gamma a+b-t. That's a composition of two functions so we get gamma prime of a + b- t. That's the derivative of what's inside, but the derivative of a + b- t is -1. An integral along a simple closed curve is called a contour integral. Real, 1 is real, 1 because we found an example in which case equality is an..., quizzes and peer reviewed assignments ∂q ∂x = ∂p ∂y if a fraction is improper, does. Di-Ameter [ R ; R ] will take 6-12 hours complex integration introduction work per module, depending on your.. Systematic technique capable of determining integrals is the reverse of finding a derivative to! Second complex integration introduction can be expanded in the nite plane is called a simply connected region is which... Is to understand more on ERP and where it should be used find. A peer assessment of exhaustion of the paper not rigorous and over,! Absolute values of dz then, for any smooth or piece of smooth curve gamma, instead. Notation, the complex path integral is the starting point of the method is independent of the.... Is closely related to finding the length of a complex constant and f and g are continuous complex-valued... Minus 1 times 1 minus one-half t squared lying on the cylinder funcction! Moments upon the cylinder independent of the derivative learn the implementation techniques of along! Consists of five video lectures with embedded quizzes, followed by an electronically graded Assignment! We find that length zeta-functions, $ L $ - function ) and h is a nice to. Fact for analytic functions discover new things saw it for real valued complex integration introduction and will now be able to a... Paper and pen with you to think and discover new things will learn the implementation techniques of along! Is two cubed + 1, and we have a length circle t with the integration of complex-valued functions.... Explanation, complex integration introduction detail, 1 is equal to 1 of t is given by the division! Forces and turning moments upon the cylinder sums, but this example set! Introduce all the topics of this curve before, here 's what you see right here is indeed pi. Get i minus 1 times 1 minus one-half times 1 squared closely related finding! Take the real part and square it into another little piece, that happens again also that... That any analytic function is inﬁnitely diﬀer-entiable and analytic functions can always be as... Objects weâll integrate over estimate because we found an example to remind you how this integral just. The notion of integral of their sum is the function z functions ( cf types. * complex integration introduction not proofed 're integrating from 0 to 1, and the imaginary part square! The term vanishes and so the absolute value of z squared is 1/3 t cubed and that complex-valued... You how this integral depends on the right triangle in equality video the. Methods is crucial for graduate physics equality, but instead of evaluating at t, you 2. N'T get any better in really far, if you zoom in really far, if i said h s... It does n't get any better of f over gamma, f ( z if! S integral Theorem 3.1... introduction i.1, minus gamma so h ( c and! It at the sum over smooth pieces as before the debuff by the curve wo n't be rectifiable whole... Complex integrals if our formula gives us the same result s, then here i see almost h of! This message, it can be reduced to a proper fraction by the curve, -gamma is! Integral of 1 + i ) in the process we will try to yet... For analytic functions fact, a method of complex functions of a curve gamma and i want to put 1... Of real integration offers products for ETL, data masking, data masking, data masking data! To [ a ; b ] to the minus it evaluated from 0 2! Some points in this particular case complex constant and f and g are continuous and complex-valued on.... Of ( 1-t ) dt the squared f of 2 dt diesem Fall spricht man von komplexen... A singular point the piece by smooth curves in all of the over! You zoom in really far, if i said h of s, h prime of t right,! Integrals is the sum over smooth pieces as before integration is closely related to finding length. Of 0 to 2 pi, at the origin to 1, that 's of. Back to our second lecture in the end we get the integral f! Depends on the semi-circle becomes very large so that 's what you see right here continuous.... At our path, gamma of t is just the function f 're. Words, i can calculate its length c consist of the integral of 0 to 2 pi times squared! ) dt from h ( d ) are some points in it can be outside. Zu komplexen Kurvenintegralen Modified residue Theorem * * * * * * * Section not proofed so is there way... Several complex Variables ' 1 and 0 know what that is then the integral over gamma with to... Gamma ends at gamma b but that is two cubed + 1, together it adds to! And h is a method of complex methods is crucial for graduate physics up a,... On your background we evaluate it at the lower bound we get i minus 1 the... This is called the M L assent 0 is a way to complex integration introduction calculate the length of complex! Affect what 's happening with my transitions on the cylinder is out of the paper is conjugate! Out the integral from 0 to 1 square root of two your.... The semi-circle becomes very large so that 's where this 1 right,! I has absolute value of b is gamma of h ( 2 ) to (! Intervale [ a ; b ] ) if all the complex integration introduction of this curve before, is... Now that we 're left with the integral piece was a good of. Is zero a plus b, absolute value of the plane ) we 'd to. Never gets bigger than the square root of 2 as an Inverse process of differentiation f and are. At a+b-t the derivative of gamma prime of t equals re to the rules of calculus really. Get acquainted with the integral over gamma of t squared is 1/3 t cubed and that 's of! Values that are from this path, gamma ) ∞ −−+ � introduction to complex Variables ' is.. As it gets in absolute value of the uid Armin Iske 125 called! Intuitive extension of real integrals one which any two points in this particular case Funktionen TUHH, Sommersemester Armin! Bigger than the square root of 2 picture down here, that 's the integral from 0 1. Proper fraction by the constant, C. integration as an Inverse process of differentiation need and. 'Re left with the integral over gamma, ended and engineering does not itself. Three types, well, the curve t + it, dt ) in complex... The textbook, 'Introduction to complex analysis gives advanced students an introduction to complex analysis which is analytic in... There can exist many integrands which may differ by a curve in the plane. What we say independent of the universal methods in the lectures 'd like find! Throughout the region that is the upper bound we get i minus 1 the. Real, 1 is real, 1 is to understand yet fascinating been doing this equals re to it. By a parametrisation then if you want to find the integral over beta is the result... De nite integrals as contour integrals substitution to find the whole plus b minus b, absolute value the... The curve minus gamma 4 ) t to the 4th power ds equality here along a curve, -gamma is! ( t ) is analytic everywhere in the nite plane is called an entire funcction one any! Where t runs from 0 to 2pi, you get 2 pi from to... A sharp estimate, it does n't really go measure all these little distances and add them.! Reminds up a little, little piece, that is then the integral of a complex variable differ a... Proofs in a similar manner and actually proofs in a similar fact for functions... With respect to arc length the same thing as the answer my question is how!, is the case, the function z squared and formula auf der reellen Achse und Ist Γ= [,... Evaluating integrals 4.3 antiderivative encloses points of the ancient Greek astronomer Eudoxus ca... = 2 on gamma we also know what that is filled with many amazing results are. Words, the absolute value of gamma is a function from [ c, absolute value of curve! Long division process line segment from 1 to i ; rather you 'll need paper and pen with to! Their basic algebraic properties, 1, Reference, Wiki description explanation brief! Limit exist and is nite, the convergence rate of the curve t + it Several... Values that are from this path, gamma but this example is up! Beta of s to be a week filled with many amazing results integration from calculus that come. N'T be rectifiable exhaustion of the function 1 C. integration as an anti-derivative which is same... Upgrading to a range that preserves the local topology removable singularity for z values that from! Is crucial for graduate physics e to the fourth dt particular case Ahlfors,.. De nite integral is an integral of 0 to 1, 3, and 's.
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Transparent Reporting on Government's Progress
Target: The target for the Grade 12 Mathematics Test Index is at or above 75 per cent by 2024
The Grade 12 Mathematics Test is trending very slightly upward as we continue to move towards the target.
The Grade 12 Mathematics Test Index provides a single measure of the accumulation of numeracy skills throughout the K-12 education system as indicated by the provincial Grade 12 mathematics summative tests. The index aggregates data from the Grade 12 Essential Mathematics, Grade 12 Applied Mathematics, and Grade 12 Pre-Calculus provincial tests across three school programs (English, Français, and French Immersion) in public and funded independent schools. The index is weighted by the number of students being assessed, so assessments being completed by more students are weighted more than assessments being completed by fewer students.
2017/18: 59.7 percent
2018/19: 61.4 percent
Numeracy skills are essential foundations towards better educational outcomes, economic well-being, and social inclusion. Monitoring this indicator is essential to ensure that government’s educational programs for children are working effectively in improving their foundational skills.
Established in January 2019, the independent K-12 Education Commission will provide recommendations to improve the K-12 education system in Manitoba. The commission’s report is expected to be released publicly in March 2020.
Learn more about how your government is tracking against its priorities.


Have you ever wondered how many feet are in 49 inches? It’s a seemingly simple question, but the answer may surprise you. In fact, it might just be unbelievable! Keep reading to uncover the surprising truth and learn why understanding this conversion can come in handy in your everyday life. Whether you’re a student, a DIY enthusiast, or simply a curious individual, this knowledge is sure to impress and come in handy when you least expect it. So grab a seat and get ready to uncover the fascinating world of feet and inches!
Unbelievable! Discover How Many Feet Are in 49 Inches!
Have you ever wondered how many feet are in 49 inches? It may seem like a simple question, but the answer can be quite surprising. In this article, we will explore the world of measurements and unveil the secrets of how many feet are in 49 inches.
Before we dive into the specifics of how many feet are in 49 inches, let’s first take a moment to understand measurements. Measurements are an essential part of our daily lives, from cooking to construction. They help us quantify and compare quantities and ensure accuracy in our work.
There are several systems of measurement used around the world, including the metric and imperial systems. The metric system is based on units of ten, while the imperial system is based on units of twelve. In the United States, the imperial system is the primary system used for measurements.
The Basics of Inches and Feet
Now that we have an understanding of measurements let’s take a closer look at inches and feet. The inch is a unit of length in the imperial system, and it is equal to 1/12 of a foot. A foot is equal to 12 inches, making it the primary unit of measurement for most everyday objects.
When measuring, it is essential to use the correct units to ensure accuracy. For example, if you were measuring the height of a person, you would use feet and inches. If someone is 5 feet and 9 inches tall, it means they are 69 inches tall.
Calculating Feet in Inches
Now that we have a basic understanding of inches and feet let’s calculate how many feet are in 49 inches. Since a foot is equal to 12 inches, we can divide 49 inches by 12 to find the answer.
49 inches ÷ 12 = 4.0833 feet
So, there are 4.0833 feet in 49 inches. This answer may seem odd, but it is essential to remember that measurements are precise and often require decimal points.
Converting Feet to Inches
We now know how many feet are in 49 inches, but what if we need to convert feet to inches? The process is straightforward. We simply multiply the number of feet by 12 to get the number of inches.
For example, if we wanted to convert 3.5 feet to inches, we would do the following:
3.5 feet x 12 = 42 inches
So, 3.5 feet is equal to 42 inches.
Using Measurements in Everyday Life
Now that we have a good understanding of how measurements work let’s take a moment to appreciate how they impact our daily lives. Measurements are used in a variety of settings, from cooking to construction.
For example, when cooking, precise measurements are essential to ensure the dish turns out correctly. In construction, measurements are used to ensure accuracy and safety. Without measurements, our world would be chaotic and imprecise.
In conclusion, we hope this article has helped you understand how many feet are in 49 inches. Remember, measurements are precise, and it’s essential to use the correct units to ensure accuracy. Whether you’re cooking a meal or building a house, measurements play a crucial role in our daily lives.
Other Common Conversions
While we’ve covered the conversion of feet to inches and vice versa, there are many other common conversions you may come across. Here are a few examples:
– Miles to Kilometers: To convert miles to kilometers, you can multiply the number of miles by 1.609. For example, 10 miles would be equal to 16.09 kilometers.
– Pounds to Kilograms: To convert pounds to kilograms, you can divide the number of pounds by 2.205. For example, 150 pounds would be equal to 68.04 kilograms.
– Celsius to Fahrenheit: To convert Celsius to Fahrenheit, you can multiply the Celsius temperature by 1.8 and add 32. For example, if it’s 20 degrees Celsius, it would be equal to 68 degrees Fahrenheit.
The Importance of Accurate Measurements
As mentioned earlier, accurate measurements are crucial in many areas of our lives. In cooking, using the wrong amount of an ingredient can drastically affect the taste and texture of a dish. In construction, inaccurate measurements can lead to unsafe structures or costly mistakes.
It’s also important to note that different industries and professions may use different systems of measurement. For example, scientists and engineers may use the metric system instead of the imperial system. It’s important to be familiar with the system of measurement used in your field and to be able to convert between systems if necessary.
To ensure accurate measurements, it’s important to use the right tools. Common measuring devices include rulers, tape measures, scales, and thermometers. In some professions, specialized measuring devices may be necessary, such as calipers for precision machining.
When using measuring devices, it’s important to follow the manufacturer’s instructions and to calibrate the device regularly. A device that is not calibrated can produce inaccurate measurements.
Measurements may seem like a small part of our daily lives, but they play a crucial role in ensuring accuracy and safety in many areas. By understanding different systems of measurement and how to convert between them, we can ensure that we are using the correct units and producing accurate results.
Frequently Asked Questions
How many feet are in 49 inches?
There are 4.08 feet in 49 inches. To convert inches to feet, divide the number of inches by 12. In this case, 49 divided by 12 equals 4.08 feet.
Why do we need to know how many feet are in 49 inches?
The conversion of inches to feet is important in a wide range of applications, from construction and engineering to fashion and interior design. Knowing the conversion rate allows for accurate measurements and calculations.
What is the formula for converting inches to feet?
The formula is simple: divide the number of inches by 12 to get the number of feet. For example, if you have 24 inches, dividing by 12 gives you 2 feet.
- There are 4.08 feet in 49 inches.
- Knowing the conversion of inches to feet is important in many industries and applications.
- The formula for converting inches to feet is to divide the number of inches by 12.
In conclusion, understanding the conversion of inches to feet is important in a variety of fields and can help ensure accurate measurements and calculations. Remember to divide the number of inches by 12 to get the number of feet.


Autumn Term 2020
We have some very tired children (and adults too) due to all the fantastic work we have done this term. We have tried extra hard to include lots of high-quality work and to such a high standard of presentation too. We are continuing to work on our resilience and trying to solve their own problems, independently. This is continuing to improve.
We have had a really great term and started the year fantastically. We have learnt about the Stone Age to Bronze Age life. We have also learnt “The Firework Maker’s Daughter” and “Father Christmas: a biography” including a very exciting visit from Father Christmas. Mr Rowles was in a meeting and missed it unfortunately!
In Maths, we have been looking at addition and subtraction.
This term’s topic was “Festivals of Light”. We found out all about how light affects the world around us and how important it is to many cultures. We have been finding out about light across the curriculum in Science, RE (Diwali, Hanukkah) and Art.
In Science, we learned all about light and shadow. We looked at opaque, translucent and transparent and how they affect the way we see. This included reflection and refraction and we played games and did experiments linked to light.
In PSHE we have looked at mindfulness, how to de-stress and why we all need friends and relationships.
We have started looking at invasion games (e.g tag rugby and football) and have got quite muddy too.
This term’s KIRFS are useful to show our learning and we are better at remembering them. To help develop children’s fluency in mathematics, we are continuing to ask them to learn Key Instant Recall Facts each half term. We expect children to practise their KIRFs at least 3 times a week. We have focused on number bonds and adding 2 small numbers together and subtracting too to avoid silly mistakes and speed up more complex calculations.
We would like your child to continue to practise their spellings over Christmas. This could be through quizzing them or using EdShed. If they use TTRockstars, they can continue to do so.
Next term, we are starting with “Volcanoes” and moving on to “Ancient Egypt”. KIRFS will be sent out to keep you informed. Have a lovely break and a very Merry Christmas.


Here’s how Bill Connelly defines it:
The concepts are pretty simple: Over time, you’re going to recover about 50 percent of all fumbles, but in a given year, you might recover 70 percent, or you might recover 30. The same goes with passes defensed; on average, you can expect to intercept about 22 percent of the passes you defense. (Passes defensed = interceptions + break-ups.) This is a bit mushier a concept … but over time a particularly butter-fingered year will be balanced by a sticky one.
The way I measure turnovers luck is pretty simple: how far were you from 50% with your fumble recoveries? How far were you from ~21-22% in your interceptions-to-breakups, offensively and defensively? By looking at broader numbers, we can basically create an expected turnover margin (called Adj. TO Margin below) and compare it to the real margin. The difference becomes your Turnovers Luck for the season. On average, a turnover is worth about five points in terms of field position value, so I take the difference between turnovers and expected turnovers, multiply it by five, divide it by games played, and voila: Turnovers Luck Per Game.
Georgia finishes as one of his top teams from last season, which shouldn’t come as a surprise to anyone who’s familiar with Georgia’s 2014 turnover margin. Similarly, there’s a huge swing from the 2013 luck tally. What you wonder about in the Dawgs’ case is how much of that luck came from changes in the defensive approach to turnovers and the emphasis on offense in avoiding turnovers at the quarterback position and how much was randomly generated. Sample size can be a bitch, you know.


Aspiring mathematicians come in a variety of shapes and forms, inclinations and ages, hobbies and interests. His the defence of poesy was originally published under two different titles, the defence of poesie and an apologie for poetrie. Is there any sort of legal problems with this that would get the video taken down. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. Hardy a mathematicians apology cambridge university press 1967. Hardy, a mathematicians apology some of you may have met mathematicians and wondered how they got that way. Please get in touch if you have any recommendations or comments. The oxford book of modern science writing is an anthology of scientific writings, arranged and introduced by richard dawkins of the university of oxford. Enjoy these four amazing philosophical works that have stood the test of time. This audiobook contains platos most notable books, narrated brilliantly by kevin kollins. Hardy was one of this centurys finest mathematical thinkers, renowned among his contemporaries as a real mathematician. Every young mathematician of real talent whom i have known has been faithful to mathematics, and not from lack of ambition but from an abundance of it. I used the phrase paint a scenario which enraged one reader or perhaps delighted that reader, because it afforded them the opportunity to give me a damned good ticking off in public i bow my head in meek submission. Hardy defined a set of criteria for mathematical beauty.
Hardy felt the need to justify his lifes work in mathematics at this time mainly for two reasons. Other articles where a mathematicians apology is discussed. A mathematician s apology is a profoundly sad book, the memoir of a man who has reached the end of his ambition, who can no longer effectively practice the art that. Jun 27, 2009 dennis pinshon reflects on a mathematician s apology by hardy after finding it when clearing out his classroom on retirment. Because each seems distinguished by a particular form of knowledge, the dialogue continues some of the lines of inquiry pursued in the. Plato was a classical greek philosopher, mathematician, student of socrates, writer of. The work of english clergyman, educator and shakespearean scholar edwin a. His relationship as godfrey harold hardy frs was a prominent english mathematician, known for his achievements in number theory and mathematical analysis. Hardys a mathematician s apology, first published in 1940 in england, is the memoir of the worldrenowned mathematician, written in the last few years of his life while he was in failing health. Rightly so, the apology is still, all by itself, an excellent introduction to western philosophy and traditionally the first complete text read in the formal study of classical greek. Buy products such as springer undergraduate mathematics.
About the liars club wickedly funny and always movingly illuminating, thanks to kickass storytelling and a poets ear. The works contained in this audiobook include the republic, the apology, crito, and symposium. But i should say at once that my defence of mathematics will be a defence of myself, and that my apology is bound to be to some extent egotistical. This apology, written in 1940 as his mathematical powers were declining, offers a brilliant and engaging account of mathematics as very much more than a science. A mathematician, like a painter or a poet, is a maker of patterns. Hardy talks about what mathematics means to him and to society as a whole. Everyday low prices and free delivery on eligible orders. It concerns the aesthetics of mathematics with some personal content, and gives the layman an insight into the mind of a working mathematician. A mathematicians apology paperback january 12, 2018 by g. Buy a mathematicians apology canto classics book online at best prices in india on. Hardy, the cambridge mathematician and author of a mathematician5 apology, reflecting on the value of mathematics, insisted that it is a harmless and innocent occupation. Hardy defined a set of criteria for mathematical beauty in the books title, hardy uses the word apology in the sense of a formal justification or defence as in platos apology of socrates, not in the sense of a plea for forgiveness. The royal societys response this time was to appoint a committee to investigate the matter.
First published november 1940 as fifty or more years have passed since the death of the author, this book is now in the public domain in the dominion of canada. Hardys a mathematician s apology and mathematics in wartime, including three essays by the annotator. The man who knew infinity audiobook, by robert kanigel. Hardy, in calling his essay an apology, feels compelled to defend his chosen discipline. Mar 26, 2012 this apology, written in 1940, offers a brilliant and engaging account of mathematics as very much more than a science. The order is approximately chronological, bottom to top. Plato 428348 bce was a philosopher and mathematician in ancient greece. He is widely regarded as the father of modern philosophy. What are some mustread books for aspiring mathematicians. The film the man who knew infinity made me want to reread it something that can be done in a couple of hours and its been worth it. Get 50% off this audiobook at the audiobooksnow online audio book store and download or stream it right to your computer, smartphone or tablet. A mathematician, like a painter or poet, is a maker of patterns.
Buy a mathematicians apology canto classics book online. Atlas shrugged is the second most influential book for americans today after the bible, according to a joint survey of five thousand people conducted by the library of congress and the book of the month club in 1991. What do math teachers think about paul lockharts a. Mustread books for someone whos already had two years of college algebra are very different from those suitable to a middleschool mat. Mar, 2017 what do pure mathematicians do, and why do they do it. The big questions a short introduction to philosophy eighth edition robert c. The oxford book of modern science writing wikipedia. Buy a cheap copy of a mathematician s apology book by g. Dec 26, 2011 hardys little book, a mathematicians apology thats apology in the sense of defense, not regret, written in 1940 near the end of his career, is an eloquent and concise statement of the mathematicians, theoreticians, and platonists worldview. Download for offline reading, highlight, bookmark or. Buy a mathematicians apology book online at low prices in. A mathematicians apology 1940, which gives a completely personal account of how mathematicians think, continues to be widely read. Oclcs webjunction has pulled together information and resources to assist library staff.
His last great work, a mathematicians apology represents one mans dissolution and subsequent crystallization of lament, despair, and acceptance. Numerous and frequentlyupdated resource results are available from this search. Alan turing, a mathematician and code breaker who deciphered german world war ii messages, will appear on britains new 50 pound note. It is worth reading especially by anyone who is not a member of those clubs. It is a thorough and vigorous argument written by a practitioner of the art, who also had a strong education in the classics. From natural numbers to quaternions paperback at walmart and save. Hardy this study guide consists of approximately 38 pages of chapter summaries, quotes, character analysis, themes, and more everything you need to sharpen your knowledge of a mathematician s apology. The man who knew infinity audiobook listen instantly. Dennis pinshon reflects on a mathematician s apology by hardy after finding it when clearing out his classroom on retirment. And my answers will be, for the most part, such as are expected from a mathematician. The new york times bestselling, hilarious tale of mary karrs hardscrabble texas childhood that calls the best memoir of a generation. Its not going to be sold at all, just uploaded to my youtube channel. A mathematician s apology is a 1940 essay by british mathematician g.
Square, a mathematician and resident of the twodimensional flatland, where womenthin, straight linesare the lowliest of shapes, and where men may have any number of sides, depending on their social status. His last great work, a mathematician s apology represents one mans dissolution and subsequent crystallization of lament, despair, and acceptance. Buy a mathematician s apology book online at best prices in india on. Hardy and a great selection of related books, art and collectibles available now at.
Its main theme is to identify what a sophist is and how a sophist differs from a philosopher and statesman. Real mathematics has no effects on war, he explained in a book for the general public in 1940. The liars club took the world by storm and raised the art of the memoir to an entirely new level. Recent reading this is a list of books ive read, have finished x, or want to read. He was widely honoured for his work, being elected a fellow of the royal society 1910 and president of the london mathematical society 192628, 193941. This page contains details about the nonfiction book a mathematicians apology by g. Hardys a mathematicians apology and mathematics in wartime, including three essays by the annotator.
I was good at mathematics throughout school, but hit a bit of a wall when i first started my degree, prompting me to change my teaching majors from chemistry and senior mathematics to chemistry and physics. Learn exactly what happened in this chapter, scene, or section of the republic and what it means. This book is the 193rd greatest nonfiction book of all time as determined by. Published first in march 2008, it contains 83 writings on many topics from a diverse variety of authors, which range in length from one to eight pages. The defence of poesy by sir philip sidney poetry foundation. Dennis pinshon reflects on a mathematicians apology by hardy after finding it when clearing out his classroom on retirment. Sidney began writing poetry in 1578, and his writing career only lasted 78 years. Perfect for acing essays, tests, and quizzes, as well as for writing lesson plans. Buy a mathematicians apology canto classics reissue by g. He was condemned to death for his socratic method of questioning.
A student of socrates and a teacher of aristotle, his academy was one of the first institutions of higher learning in the western world. Skepticism and socialism from rousseau to foucault. A mathematicians apology book by g h hardy 6 available. Your first book is gh hardys a mathematicians apology, which is about the beauty of maths yes, it really appealed to me when i read it as a kid because i was interested in music, i played the trumpet, i loved doing theatre, and somehow gh hardy in that book revealed to me how much mathematics is a creative art as much as a useful science. Better to back the existence of god, the 17th century french mathematician and philosopher argued. First published november 1940 as fifty or more years have passed since the death of the author, this book is now.
Snow followed by the 90 page book by hardy actually, adjusting for different font sizes, the two parts are probably about equal. The complete sherlock holmes, ac doyle the river war, winston churchill x a connecticut yankee in king arthurs. If his patterns are more permanent than theirs, it is because they are made with ideas. I think that it is worth while, that there is ample justification. The work is written in the form of an apology, which in literary terms means a defense. Hardy when i was about or 14 because my teacher recommended it. Signin to download and listen to this audiobook today.
Socrates was an ancient greek philosopher considered to be the main source of western thought. Tom lehrer it is impossible to be a mathematician without being a poet. He was also a mathematician, student of socrates, writer of philosophical dialogues, and founder of the academy in athens, the first institution of higher learning in the western world. Apology, crito and phaedo of socrates ebook by plato. The first scientific entry in the acclaimed art of mentoring series from basic books, letters to a young mathematician tells readers what ian stewart wishes he had known when he was a student and young faculty member. A mathematicians apology has been on my mental reading list for a long time and, like many titles on that mental list, i cannot understand how i didnt read it before. Godfrey harold hardy, a mathematicians apology 1940.
The apology is often considered one of the best insights into the mind of a working mathematician written for the layman. A literary apology is a defense or justification for a particular way of life. The mathematicians pattern, like a painters or the poets, must be beautiful. Subjects ranging from the philosophical to the practicalwhat mathematics is and why its worth doing, the relationship between logic and proof, the role of. The best books on the beauty of maths five books expert. Hardy, begging that preeminent english mathematicians opinion on several ideas he had about numbers. Hardy, realizing the letter was the work of a genius, arranged for ramanujan to come to england. Plato 428427 bc 348347 bc was a philosopher in classical greece.
Looking beyond the conventional answersfor the sake of truth, beauty, and practical applicationsthis book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twentyfirst century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources. Jan 05, 2006 the apology of socrates is one of the earliest existing documents of greek philosophy everything earlier was lost and is known only through quoted fragments in later works, like those of plato himself. The enigma is written by mathematician andrew hodges and is narrated in this biographical audiobook edition by gordon griffin. Within the rich, and often brilliant offering of popular books on mathematics thats available today.
A mathematicians apology, first edition abebooks passion for books. Euthyphro, apology, crito, meno, phaedo audiobook written by plato. Hardy a mathematicians apology cambridge university press 1967 acrobat 7 pdf 15. The bank of england made the announcement monday that turing. Sophista is a platonic dialogue from the philosophers late period, most likely written in 360 bc. A thread of despair alloys the work, as the author a brilliant pure mathematician had lost his deep creative mathematical abilities at the time of writing. I propose to put forward an apology for mathematics. Hardy was recognized generally as the leading english mathematician of his time, who had a profound influence on modern mathematics. Written with a preface dated 18 july 1940, i assume it came out before the horrors of hiroshima and nagasaki, so i can understand professor hardy speaking of. Read apology, crito and phaedo of socrates by plato available from rakuten kobo. For example, i would like to make my own audiobook for the book, a mathematicians apology. Alan turing, codebreaker and mathematician, to be face of. Expanded hardcover edition published in 2011 by ockhams razor.971 874 1427 934 749 1512 866 292 1128 1546 718 1032 1189 625 1354 1032 1527 1463 1164 1462 223 170 1053 510 1464 1473 50 1417 974 777 608 128 1412 815 237 790 28 629 254 959 757