|LateNightHacking Louis Projects 2011||Auth|
2011-11-16 : Some collected thoughts on creating a Rational numeric type.
In finance, we are always dealing with quantities of money, which usually have some number of digits after the decimal point, usually two or four. It also used to be common to see prices given in eighths. Unfortunately, rather than using a representation that can represent the numbers with complete accuracy, standard floating point numbers are usually used. This often leads to rounding issues.
Now, I'm not one of those people who is paranoid about floating point numbers and believes that every comparison also needs to include an accuracy epsilon. However, I do understand that standard floating point are numbers stored as fractions with a denomitor that is 2n. Therefore, even a simple number like 0.1 (1/10) must be stored as an approximation (838861/8388608, which has been rounded up by 0.2/8388608).
It would be nice if we could represent simple useful numbers lime 1/10, 1/8, and even 1/3 without rounding. These are all rational numbers.
Standard IEEE 754 encoding of floating point numbers basically uses bit fields within a standard 32-bit int / 64-bit long / etc. They encode the numerator (mantissa or more properly significand) and denominator (derived from the exponent) for a base 2 fraction in a single integer. It would be interesting to come up with an encoding that allowed arbitrary fractions. Obviously, there would be some trade-offs compared to floating point, but how good could you make it?
(Hand written notes: 2008-04-22 2008-04-24)
Here are some features that would be nice to have:
Some bit patterns are redundant, e.g., 2/4 == 1/2, 3/6 == 1/3.
I've looked how to encode the denominator using a variable number of bits. (See 2008-04-22/pg2.) One problem with this approach is that supporting a large number of "number of bits" itself consumes valuable bits. In the worst case, you are reduced to base 1 encoding. To have enough bits to actually encode a large denominator, you have to pick a small number of denominator bit-widths. That leads to the question of which widths are most useful. Bit-widths that can encode "even" powers of ten are definitely useful, since many calculations are performed on decimals.
Some interesting combinations for 32-bit ints are as follows. Here, the first group is the number of bits in the denominators. The second group shows in more detail the number of bits in the numerator, the denominator, and the control bits.
For 64-bit longs, consider these patterns:
The 10/20 for ints and the 10/20/30/40 for longs are interesting because, for the largest denominator, the numerator is smaller than the denominator. I'm not sure if this is useful, since while it allows you to represent smaller magnitude fractions (large denominator), the number of bits of precision goes down (small numerator).
It would be useful to have a denominator encoding that's the same for ints and longs, since you can then just cast from int to long and get the same value with greater precision (just like with regular integers).
The two most favorable candidates are
|Louis K. Thomas <loui sth@hotm ail.co m>||Auth||2012-07-30 (1938 days ago)|


All you need to know about vectors
In mathematics, vectors are objects that have both magnitude and direction. The size of the vector is determined by its magnitude. It’s represented as a line with an arrow, with the length of the line representing the vector’s magnitude and the arrow indicating the direction. It’s also called a Euclidean vector, a Geometric vector, a Spatial vector, or just “vector.”
When the magnitude and direction of two vectors are the same, they are said to be equal. It is crucial in the fields of mathematics, physics, and engineering. A vector can be added to another vector, head to tail, according to vector algebra. The sequence in which two vectors are added makes no difference because the outcome is the same. For further information, look at the laws of vector addition.
Addition, subtraction, multiplication, and transformation are the issues you must solve using vectors. You can start by learning to add and subtract two to three vectors, then on to multiplication. The most intriguing vector problem is converting a unit vector to a scalar. Vectors also support two-way multiplication, allowing you to multiply one vector with another or a scalar. There are other technical multiplications, such as the cross and dot product.
Table of Contents
Definition of Vectors
The term “vector” refers to an item that has both magnitude and direction. The movement of an item from one point to another is described by a vector. The directed line segment is a geometric representation of vector math.
The magnitude of a vector is the length of a segment of a directed line, and the angle at which the vector is slanted is the vector’s direction. A vector’s beginning point is called “Tail,” and its finishing point (which has an arrow) is called “Head.”
A mathematical structure is specified as a vector. It has a wide range of applications in physics and geometry. We know that the ordered pair may be used to represent the position of points on the coordinate plane (x, y). In the simplification of three-dimensional geometry, the use of the vector is quite beneficial.
We’ve all heard the terms vector and scalar used together. The “real numbers” are represented by a scalar. A vector of “n” dimensions is a collection of n items called “components” that is ordered.
All of these quantities have magnitude and direction. As a result, they must be calculated in their vector form. Furthermore, speed is a magnitude-only quantity with no direction. This is the fundamental distinction between velocity and speed.
Notation in Vectors
A vector has both magnitude and direction, as we well know.
Vector of 0
A zero vector is a vector having zero magnitudes. (0,0,0) are the coordinates of the zero vector, which is generally represented by 0 with an arrow () at the top or simply 0.
If ‘a’ is any vector, then the sum of any vector with zero vector equals the vector itself.
Note that the zero vector has no unit vector and cannot be normalized.
We learnt about the many operations we do on numbers in math class. Let’s study about vector operations including addition, subtraction, and multiplication.
There is a chapter that you must study if you have taken mathematics and physics in your higher education. It’s a vector, after all. To understand what a vector is, you must have a solid understanding of both primary algebra and physics. You could find studying the ideas of vectors difficult since they relate to the application of math and physics. Make sure to read over all of the ideas of vectors in your books, and then go to Cuemath online website to acquire detailed notes that will help you build an interest in the chapter and tackle difficult numerical problems.


Each player draws seven pieces. The highest double leads.
The object is to make all ends total five or a multiple of five and each player succeeding in doing this scores the number of points the "spots" total. For example, 5, 10, 15, etc.
Play can be made from both sides of the first double and both ends of the first double, giving four different points upon which to build.
When a player cannot lay down a domino which will match the same "spot" of those already showing on any one of the ends, he must draw from the surplus pieces until he secures one that will play. It is permissible to draw even though the player holds a piece which would play.
When a player fails to claim a score, any opponent can call "muggins" and add the count to his own score.
A player laying down his last piece calls "domino." All other players count the "spots" on the pieces remaining in their hands and the total score of all hands is given to that player.
If the game is blocked, then each player counts his "spots" and the player holding the lowest number is given the difference between his total and that of each opponent. Then the next lowest is given a score equal to the difference between his total "spots" and that of each of his higher opponents, etc.
100 points is the game.


Some Reloading Statistics
Here are a few statistical numbers we arrive at when reloading and using a chronograph. Most chronographs compute these figures for you, but they are nice to know how they are arrived at, and what they mean.
For this example and to keep it brief we will use a four shot string instead of ten shots which gives you a much better set of numbers.
Also I'm throwing in free :
how to calculate foot pounds of energy
Taylor Knock Out Value
as well as number of loads per pound of powder
Hv: the highest velocity recorded in the string
Lv : the lowest velocity recorded in the string
Es : extreme spread of the string ( the difference between the highest and lowest velocity in fps )
Av : average velocity of the string fired
Sd : standard deviation , a measure of how close each shot will be to the average
OK here we go with our four shot string
1. 2990 fps
2. 3010 fps
3. 2996 fps
4. 3004 fps
Hv : 3010 fps
Lv : 2990 fps
Es : 20 fps , the difference between the high and low velocities
Av : total of each velocity divided by the number of shots fired
2990+3010+2996+3004/4 or 3000 fps average velocity
Sd : is derived by multiplying the square root of the average velocity by the number of shots fired , and deduct it from the squares of all shot velocities -1 and then take the square root of this figure
73sq= 8.54fps , or each shot will be within 8.54fps of the average
On to the free stuff :
foot pounds of energy ( which to me is a totally worthless figure and means nothing . another story alltogether )
velocity x velocity x bullet weight/450240
for a 405 grain bullet at 2500 fps it will look like this :
2500 x 2500 x 405/450240 = 5622 ft lbs
Taylor Knock Out Value was derived by one of my dead heros John "Pondoro" Taylor , this value was originally used for solids and the arithmetic is for solids , but many use it to compare softs as well , with either softs or solids , to me , it is a better guide ..
As with foot pounds , TKO has guide numbers for various critters ..
bullet weight x velocity x diameter/7000
again we use the 405 grain .458 diameter bullet at 2500 fps
405 x 2500 x .458/7000
TKO of 66
Number of loads per pound of powder .. This is very basic but many people do not know how many grains are in one pound ..
There are 7000 grains in one pound of powder , so simply divide 7000 by the charge weight you are loading ..
Lets say you are loading 80 grains of powder in your .17 souper mag ele slayer
7000/80= 87.5 loaded rounds per pound of powder


How big is square feet? What is square feet in acres? This simple calculator will allow you to easily convert sq ft to acres.
A little bigger than the size of a baseball diamond (home plate, first, second, third base, then home plate). Actually, if you go to the edge of the.
How big is 10, square feet? Use the calculator to see approximate dimensions for 10, square feet. What are the sides of a 10, square foot rectangle?.
I would like to say that something is 1, square feet, as big as a "blah blah blah ," just for ~ rulers stacked on each other? posted by.
What's the conversion? Use the above calculator to calculate length. How big is square feet? How much land is it? What is square feet in hectares, .
How many acres are in 10, square feet? How much? What is 10, square feet converted to acres? Use the calculator above to convert between square.


Mathematics provides diverse opportunities for students to further understand the discipline. Mathematics has various practical applications to many academic disciplines, including business, computer science, psychology, political science, music, chemistry, and physics. At Austin College, students learn to view many aspects of human history mathematically, thus preparing students for graduate study or a professional career.
Computer Science introduces students to theoretical and practical aspects of computing relevant to a liberal arts education. The rapid growth of technology in our society has required more analytical skills and experience in developing and using ever-changing technology. With preparation for continued success, students can achieve a variety of careers requiring problem solving, logical reasoning, and applications of current technology.


The expected value is a forecast of what an average would be if the experiment were repeated several times. It is an estimate of the mean of a random variable. What would the average result be, for instance, if we rolled the dice a thousand times? The anticipated value refers to this figure. An anticipated value provides a rapid understanding of a random variable’s behaviour without requiring knowledge of its discreteness or continuity. The probability distributions of two random variables with the same expected value might vary. You can now utilise the best-expected value formula calculator by calculator-online.net to find the expected value of the probability. Some of these calculators are mentioned as under!
The Expected Value Formula Calculator: An Introduction
The Predicted Value to be used at any time in the future is the Expected Value (EV). This number is sometimes referred to as the expectation, average, mean, or initial moment. You may simply discover an internet tool called an expected value formula calculator. Calculate your values and get precise answers with the expected value of the random variable calculator. Users may pick the scenarios they want to run and get their desired outcomes by calculating the anticipated value.
Expected Value Calculator Formula:
By multiplying the variable’s value by the likelihood that it will occur, one may use an expected value formula calculator to determine the expected value. Five participants can spin the bottle, for instance. Once the bottle has been spun, there is an equal chance that it will land on the first, second, third, fourth, or fifth player. This anticipated value formula is used by the expectation calculator.
Why To Find Expected Values?
A random variable’s expected value is always determined as the variable’s distribution center. The long-term average value of the variable is what matters most in this value. Single discrete variables, multiple discrete variables, single continuous variables, and multiple continuous variables are all used in the calculation of the expected value. Calculating the expected value of any sort of variable is done using the expected value of the x calculator.
By determining expected values, expected results of probabilities are computed using a collection of numbers, and the individual probabilities add up to 1, or 100%, in this case. Additionally, keep in mind that no probability for any given combination of integers is larger than 1. This computation is taken care of by the expected number calculator.
This is due to the fact that no event’s likelihood of occurring cannot be larger than 100%. So, the calculator displays an error notice if any of the event probabilities is larger than 1. Every time, the maximum outcome is 100%. As a result, there is no way that any individual event or the sum of all occurrences could have a probability greater than 1.
Benefits of Using an Expected Value Calculator:
There are various benefits to using expected value formula calculator probability distribution. Calculator for expected value probabilities
- Center of the probability measures Long-term investment measures for variable distribution
- Reduces Information to One Answer or Possibility, Which Allows Us To Predict Future Results
TOP 2 Best Expected Value Calculators:
Following are the most trustworthy calculator to calculate the expected values in a matter of seconds:
If you’re wondering “How to calculate expected value?” then the expected value formula calculator is the answer. The remainder will be taken care of when you provide the values and their probabilities.
Finding the probability expected value (mean) of a discrete random variable is made easier with the use of an online expected value calculator (X). Keep in mind that the anticipated value calculation aids in condensing the information into a single possibility or response.


This page offers free printable math worksheets for kindergarten levels counting & coloring numbers and objects from 20 to 100 numbers 21 to 30 spell. Reception year / kindergarten number worksheets practice have your children figure out the missing number in each number range and write it in the box.
Here you will find a selection of kindergarten printable worksheets designed to help children learn to write and color numbers up to 10. These printable kindergarten worksheets help you kids learn to recognize and write numbers, and to count the worksheets compliment k5 math, our animated . Preschool writing common worksheets letter writing for kindergarten kindergarten writing numbers printable worksheets learn to count and.
Kindergarten writing numbers worksheets give your child practice tracing numbers and writing them by himself with fun illustrations. Free kindergarten math worksheets to improve counting skills and number see , say, write, and count - look at the numerals 1-10, say each number, trace the.
Number kindergarten worksheets writing numbers worksheet for sense printable missing 1-10 pdf counting tracing | lccmadison. Have you seen my handwriting worksheets for numbers 1-10 since his older siblings do a lot with ten frames beginning in kindergarten, i like that he's getting an introduction to i love your worksheets i can write 1-10. Numbers worksheets to learn how to count and write numbers fpr preschool, kindergarden and early elementary.
Writing numbers worksheets are designed to reinforce number recognition and counting for children in kindergarten help your child practice their writing and. This math worksheet will help give your preschool, kindergarten, or 1st free number worksheets - great way to practice writing numbers 1-20, counting,. Bridges in mathematics kindergarten practice book blacklines rates the number corner, a collection of daily skill-building activities for students small sets of pages have been written to follow the instruction in key bridges sessions.
Familiar with upon entering kindergarten to ensure their success - answer general -can write their name (beginning with a capital letter and spelled correctly) -know that you practice writing your numbers on the lines provided 1 2 3. Free printable activity pages for children to learn math and numbers.
Number tracing and writing sheet - free | kinder math | pinterest #362151 kindergarten writing numbers worksheet kindergarten worksheets for. These free printable worksheets are perfect for all kids from pre-school to kindergarten (3, 4, and 5 year olds) browse two: writing and recognizing numbers. This kindergarten worksheet will produce a worksheets for practicing writing numbers between 0 and 9. Count and write numbers up to 10 confidently first we need to count the objects and then write the numbers in the box in this kindergarten math worksheet kids.


Full course description
This is a two segment High School course. Start here for Segment 1.
The fundamental purpose of this course is to formalize and extend the mathematics that students learned in the middle grades. The critical areas deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. The Standards for Mathematical Practice apply throughout and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations


another fun-packed week in Mousehole class! The class loved their first
orienteering lesson in PE and we worked hard on the importance of working
collaboratively to reach the end goal; something we will continue to work on
for this unit. The children loved discovering the different puzzle pieces
before being like ‘Teamwork Tiger’ to put them together. Can they tell you what
the map was of?
we have started our Myths unit of work. The children have loved learning our
WAGOLL: Theseus and the Minotaur. They have been creative with their own story
maps and actions.
we have been using short multiplication method to work out our multiplications
before using them to help us reason and solve problems.
discovered how Alexander the Great impacted the world in History and enjoyed
working collaboratively by using QR codes on the iPad to complete our own
reminder that if the children are in my spelling and reading group (not Read
Write Inc.) that our spelling tests are every Friday. This week is the first
set of spellings from the new booklet.
on Thursday, we will be celebrating World Book Day with lots of fun around
reading. The children can dress up as a book character. Please don’t go to any
expense for this and use things you that the children already have at home.
Check out some of the Greek pots that the children have worked hard on below:
A highlight from this first week back after half term has to be the Mousehole Machine dance. Watch Y3 perfrorm their final dance piece to you below. They have spent lessons perfecting their machine like moves and ideas to create this wonderful final piece. Well done Mousehole class I'm super proud of you!
We had lots of Archie fun this morning with different activities and enjoyed parading in our outfits for St Anne's this afternoon. The children's and resident's smiles beamed from ear-to-ear.
Today has also been Safer Internet Day and we have learnt about the importance of telling a trusted adult when needed.
Mousehole Class had great fun exploring the patterns that are created with multiplications.
Using a circle that
had been divided into 10 equal spaces, we connected the digits based on the ones
number in a given times table i.e. 3, 6, 9 the 2 of 12, the 5 of 15, the 8 of
18 etc, until we returned to 0 (our starting point). Once we had completed the
2, 3 and 4 times tables and examined the shapes we had made, we then predicted
which pattern the 6 times table would match. We thought the 6s would match the
3s as we had learnt in previous maths lessons that the 6 times table is double
the 3. It turns out the 6s match the 4s. This puzzled us for a while until we
had completed the 7s (which matched the 3s) and the 8s (which matched the 2s).
We had a theory that the number bonds to 10 had the same patterns, so we completed
the 9s and 1s to see if they matched. They did. Mathematical magic!
Have a look at some of our fabulous entries for the Fowey Festival that we created this week. We took our learning of Haiku poems a step further by creating some new Haikus with the competition topic of 'Beaches'. We will be sending off our entries soon!
Mousehole class have been super when multiplying and dividing by 3 and 4 this week. Keep rehearsing at home! 'Hit the Button' is a great free online game to help repeat times tables at home!
It has been
a super start to the Spring term. We have started our new topic-‘What’s so
amazing about the Ancient Greeks?’ by constructing a timeline as a class about
the different eras within it. The children worked hard in groups to do this and
then presented the facts about their era.
we have starting our learning about Japanese Haiku poems-can your child tell
you how many lines and syllables they have? We have started our multiplication and
division unit in maths and will have a particular focus around multiplying and
dividing by 3, 4 and 8 over the next week.
started to learn about the Hindu faith in RE and in art we are began our learning
by looking at who the sculpture Antoni Gaudi was.
started our machines topic dance and there were some great robotic moves this
Please see the document attached which shows our learning for this term.
We have been full of festive cheer these last couple of weeks.
We decorated the classroom with the wonderful winter paintings and cards as well as the very sparkly pharaoh heads that the children all worked so hard on. They bring smiles and joy to all who visit the classroom (and to lucky us who have them decorating our walls).
Hunt the Issy, singing carols at St Anne's, visiting our Brunel Christmas tree decorations at the festival, Christmas lunch and of course the Christingle have packed our timetables with joy.
We have also been working hard on our adapted Aladdin stories and enjoying finishing our learning about the Ancient Egyptians. On the final days we shall put our DT plans into action by creating a working Shaduf with mechanisms.
To all of year 3: What a fantastic first term we have had. Enjoy the break and we will look forward to seeing you in the new year.
Look who has earnt their pen Licenses!!! Super work ethic and resilience has ensured that these 5 children have been the first to earn theirs this year.
Another fabulous week in Mousehole class. We enjoyed planting this week, Christmas crafts and continuing to rehearse our songs for the Christingle. We have continued to learn and recap column addition and subtraction in maths. In English, we have been learning how to write a traditional tale-with an Aladdin theme. In DT, we have learnt all about mechanisms and the Ancient Egyptian Shaduf. We have of course been learning lots more too!
We are bold and innovative in our approach to find new solutions to the challenges we face.
We are inspired by the awe and wonder of the world.
We take responsibility for our actions in an environment of mutual respect.
We are passionate about learning.
We are the best we can be.
We overcome all barriers to reach our potential, developing a capacity to improve further.
Brunel is an Ofsted graded 'GOOD' school with 'OUTSTANDING' Personal Development and Early Years Provision - click here to see the full report
© 2024 Bridge Schools trust is a company limited by guarantee, registered in England and Wales. Registration number 7736425. It is an exempt charity.


California’s Department of Education is going to ban accelerated math courses for students who excel at math. You read that right – they are going to discourage students who are good at math from pursuing accelerated math classes.
The reason given for this insane move is “inequity,” or more specifically that some students may do better in math than others. This ignores the fact that this is how it is for any subject in school – some people are just naturally gifted.
Instead of focusing on math, California math courses will focus on “connecting math to social justice concepts like bias and racism” which makes no sense because math can’t be racist.
Radical Democrats can’t even get 2+2 right. Their policies make no sense and are actively harming our children’s’ futures.
Read more at Reason.com.


Let’s talk a little bit about math. Yes math. I think the history of math is really interesting. It started out with adding and subtracting then multiplication and division; today, quantum physics. And still mathematicians have ultimately uncovered that there is infinitely more to be known about math. The important part of math, and the reason mathematicians don’t find their work to be done in vain, is that it produces application along the way. Math has facilitated the most phenomenal inventions of the modern world and even many of the simplest things that we can’t seem to live without.
Consider it all joy, my brethren, when you encounter various trials, knowing that the testing of your faith produces endurance.
And let endurance have its perfect result, so that you may be perfect and complete, lacking in nothing. James 1:2-4
In the beginning of James we find that this pattern is not foreign to God’s design. We are commanded to consider trials a joy and this because trials produce endurance. The more that we are tested by God, the more endurance we have. And as a result of that endurance comes the perfect result. The result is the spiritual maturity that we gain from the trials—the result itself being the perfect result of the trial, not a perfect man.
Now the more we experience trials, the more endurance we get. The more endurance we get, the more perfect results we get, that is the more mature we become spiritually. And the more mature we are, the more trials we will have and so on and so forth, so that a perfect result is continually being produced in us.
All of this takes place, ultimately producing spiritual maturity, so that we may actually be [or become] perfect and complete. This will take place on the day of the Lord when we go to be with Him in eternity. We will have become, through His blood perfect and therefore be lacking in nothing. For He who created all things will be among us and we will have no needs: physical, spiritual or emotional.
This is ultimately the same as the mathematician. We, as Christians, work our way through trials in life, the same way mathematicians work their way through the trials or problems that math presents. In a micro-chasm, we attain maturity, but there is infinitely more maturity to be gained. In a micro-chasm, the mathematician succeeds as well, solving one problem at a time, but there is infinitely more problems to solve.
The take away from this passage is that we should find joy in trials. When we suffer in this way, we can be confidant that we will gain endurance and ultimately more maturity. And ultimately, we will attain perfection when we go to be with Christ for eternity. Therefore take joy in the trials that you encounter in life.
Photo(s)/Resource(s): Anthony Delgado


This page was co-written by the author of this site Thomas DeMichele based on work by Carter Bastian.
Can Everything that is True Be Proven True?
Everything is either true or not true, but not everything that is true can be proven true, and not everything false can be proven false. Despite this however, we can, in some cases, calculate the probability that something is true or not true, even when we can’t know for sure.
The Technologies of Logic and Reason
The above statements can be shown with logic and reason by looking at concepts like:
- The Principle of bivalence which says, “every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false.” In other words, every statement structured properly has a definite binary truth-value (two-value True or False logic).
- The laws of thought, specifically the law of identity, contraction, and excluded middles. These are “the laws” that logic and reason are based on, noted by Plato and Aristotle, which in simple terms also states that “everything” is either true or not true. This leads us to the rule-sets of logic.
- Constructivism (in the philosophy of mathematics) which shows that, even though everything is either true or false, there are some things we don’t know. So we have to consider 3-value logic: True, False, and Unknown.
- Since there are unknowns, from the human perspective there is probability. Thus, we have to also state certainty (using multi-value logic to make statements like likely true, or unlikely true, or 75% likely). This brings us to some important theorems for dealing with truth and probability.
- Gödel’s Incompleteness Theorems, which shows, using mathematical proofs, that there are things that are true that we can’t prove true (Gödel’s theorems are a set of equations that works out the logic behind a paradox like the sentence “this statement is a lie”).
- Bayes’ Theorem, which shows that we can compare conditional probabilities to find the “likelihood” something is true or false (even when we can’t know it for sure). We can use inductive logic to find likelihood.
- The Scientific Method, which shows that we can create a rock solid and usable theory based on a collection of related facts, testing, and a strong hypothesis. We don’t need to know F=ma directly, we just need to know it works every time when subjected to rigorous test. We can also use hypothesis and theory to great effect.
- The Rule-sets of Logic and Reason in general, which include all the above rules and show how we can deal with propositions (as semantic language or as mathematical equations). In general, the rule-sets of logic and reason (like the rules of a syllogism and deduction, and the rules for building a case with induction) can help us to decode truth-values, including certain truths and probabilities.
The above technologies of reasoning can help us understand many truths, and some extra heady logic that expands on concepts like those above can help us show many more. Still, the end result will be “not everything that is true can be proven true, and not everything false can be proven false.” We explain the logic behind this below.Gödel’s Incompleteness Theorem – Numberphile. A video on Gödel explaining the gap between “truth and proof.” The reason to start with this video is that the other methods, the fundamental laws of truth, Bayes’ laws, the scientific method, and the rule-sets of logic and reason in general all point toward the idea that we can create a solid system for “knowing all their is to know.”… but Gödel’s theorems show us otherwise.
The Three Fundamental Laws of Thought
Before moving on, lets cover the basic logic of epistemology (what we can know).
(1) The Law of Identity—
Whatever is, is;
Every A is A.
(2) The Law of Contradiction—
Nothing can both be and not be;
Nothing can be A and not A.
(3) The Law of Excluded Middle—
Everything must either be or not be;
Everything is either A or not A.
From there one only needs to consider theories that deal with induction and probability, like Gödel’s theorems, Bayes’ theorem, and the Scientific method (which uses a mix of deduction and induction), to infer that there is some information we can know, some we know we can’t know, and some we can know with different degrees of certainty.
That can get complex, but for our purposes, we only need to accept the basics of the logic to show, “Everything is either true or not true (that there is objective truth and certain truth values), but not everything that is true can be proven true (that we can’t know for sure), not everything false can be proven false (again, we can’t know for sure), but we can prove some things are probable (despite what we can’t prove for certain, there is much we can).”
Known Knowns, Known Unknowns, Unknown Unknowns, and Formulating Good Theories
One major point here is that despite all we can know, there are always going to be things we don’t know for sure… and this can and should cast doubt on the world and turn us into skeptics (lovers of wisdom who search for truth, not sophists who are sure we’ve already found it).
Luckily, conversely, despite what we can’t know for sure, there are many things we can know with degrees of certainty (so again, as lovers of wisdom, we should take pride in our ability to know the objective truths we can know).
Thus, despite the uncertainty, we can still (for practical purposes) label many statements true or false based on current accepted knowledge, philosophical lines of reasoning, experiment and mathematical proofs, empirical evidence (experience-based evidence, rather than reason-based logic), and importantly by applying the scientific method.
To rephrase, the main goal then is to hunt for truth using the proper rule-sets, state certainty, make logical arguments, and show proofs… all while remaining skeptical (knowing better theories can come along and that our knowledge is always somewhat limited).
TIP: Here it is vital to understand, all a theory is a probable truth, with many facts pointing at it, that hasn’t been proven wrong. If it works, then it is accepted as true until proven false. The thing that saves us from becoming dogmatic is skepticism!
The Key to Understanding
The key to “approaching knowing and understanding” is asking well-phrased questions, knowing what technology we need to find the answers, and accepting the difference between theory, fact, hypothesis, justified belief, opinion, and conjecture (by understanding the nature of epistemological truth).
The scientific method allows for what is shown true today to be labeled false tomorrow, and math- and logic-based theories back up the idea that this should be expected. Beyond this, truth as a concept is somewhat semantic, with much of what we call truth is actually a form of BS; but such is the nature of language and rhetoric.
TIP: Our ability to confirm and disprove a premise or inference differs depending on what types of truths we are talking about (for example whether we are talking about the formal or material in terms of the empirical, logical, or metaphysical) and how we frame our inquiry.
NOTE: United States Secretary of Defense Donald Rumsfeld gave a speech in 2002 about WMDs, in it he said, “as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don’t know we don’t know.” He was skewered for the riddle of a sentence… but, language aside, his core point is 100% correct (his point about epistemology, not the WMDs). This should remind us Language is complex, truth can seem paradoxical, and politicians aren’t always concerned with “the nature of truth” (they are also concerned with “rhetoric”). But this is another lesson, we can cite Bayes all day, but outside of some small circles “cuz’ (i.e. Bayes’ theorem) alone is not going to win the average argument…. although the same argument structured as a syllogism (which in this case uses conditional reasoning) might:
- Major premise: Most of the data points to A being true.
- Minor premise: If A is true then B is not true.
- Conclusion (or “inference”): Therefore it is likely B is not true based on the data.
TIP: Instead of trying to break out complex and heady logic to win arguments (for example instead of trying to go through each data point plugged into a Bayesian formula and explain the results), it is useful to transpose the results onto a simpler form of reasoning and use generalizations (while having the more heady proofs in your back pocket to cite). For example, I’m not going to go through Gödel’s theorem, the scientific method, or Bayesian theory line-by-line here, I’m going to simply tell you Gödel’s theorem proves to us that some truths can’t be known for sure, while the scientific method and Bayes’ theorem show us we don’t have to know something for certain for what we know to be useful and “true enough.”
Knowing and that Which We Can’t Know for Sure
There are some things that can’t, by their nature, be proven with today’s technology (technology meaning broadly all collective knowledge, tools, and computing power), this includes (but isn’t limited to) that which is metaphysical or ontological philosophy and that which is a current theory. Physics is one such realm where widely accepted theories remain unprovable in every sense (the goal is to find theories that work, improve them, and prove them false through rigorous testing to find theories with predictive power, not to prove absolute true).
Political theory is an example of another realm where we are sure to find ourselves unable to find absolute truth. There may be a “best system” (best model) for any given point in spacetime, all things considered. However, any solution will be at least partially subjective and include “unknowns” as well as room for argument and debate. Even when truth exists, some truths are dependent on other factors and are bound to be so complex that truth might be elusive, impractical, and even require “minimum mutilation” of one’s ideals or truths to find any common acceptable ground.
Although we can find half-truths, fuzzy logic, vagueness, and degrees of truth all around, and sometimes in practice we will have to sacrifice ideals and truths, we can still use reason and logic to prove the law of excluded middles.
Life is a complex matrix of gray areas, but the nature of truth is black and white regardless of our abilities to perceive and prove absolute specifics. Below we cover epistemological and statistical proofs by Bayes, Gödel, and others to better understand truth, but first let’s let Richard Feynman teach us about the Scientific method, because all our heady statistics theories aside, we will inevitably be falling back to on the scientific method to define useful truths and ideas for the world we live in.Feynman on Scientific Method.
NOTE: A lot of the logic used to define truth is mathematics based, or proved with mathematics (although often using symbols to represent values like A, B, I, O, E, T, F, p, q, etc. more than using numbers), other proofs are purely semantical arguments of reason. With that said, we will just be giving a simple conceptual math-free overview to ensure that anyone can get the gist. Check the links and videos for explanations of the proofs.
TIP: Speaking of mathematics based proofs, Gödel, and logic… 1910’s Principia Mathematica (PM) (a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell) was an early attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven (a logic-and-proof-based Theory of Everything). However, in 1931, Gödel’s incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them. In other words, the only real ToE in logic mathematically speaking is Socrates’ [paraphrasing], “all I know is that I still don’t know for sure”.
How to Prove That There are Statements That Can’t Be Proven True – Gödel’s Proof
In 1931, Kurt Gödel published two of the spookiest and most important results in mathematical logic: his self-named Incompleteness Theorems. In intuitive terms, what these two theorems prove is that “In any logical system, there will be statements that are true but which cannot be proven.” This is the conceptual takeaway, like Bayes theorem, this is proved with somewhat complex mathematics that goes outside the scope of this site. See the math behind Gödel’s Incompleteness Theorem is explained “simply” here.
For non-statisticians, the videos below will work just fine for an introduction to the concept of Gödel’s incompleteness theorem as it applies to truth.Kurt Godel: The World’s Most Incredible Mind (Part 1 of 3). This video explains Gödel the man, how he is connected to greats like Einstein and Turing, and how his studying of systems, sets, and statistics led to some leaps forward in some interesting areas. This video mentions the different types of infinities (a related and interesting subject). Godel’s Incompleteness Theorem (Doubting Math).
MYTH: Gödel’s incompleteness theorems disprove the idea that “everything that’s true can be proved”. Ironically enough, there’s a thorough and rigorous logical proof that there are lots of statements that are true, or are likely true, but can’t be proven true. Proving things false is actually much easier than looking for infallible absolute truths.
How to Prove That There are Statements That Can’t Be Proven True, Even Though They Are? – Gödel, Bayes, and the Scientific Method
Gödel’s incompleteness theorems (of which there are more than one) imply that there are theories that are true, even though we can’t prove them, but we can use Bayes’ theorem and the scientific method to find theories that are true enough, based on what we currently know, likely know, or can prove through testing.
A great example of the scientific method coming to our aid is special relativity and general relativity. We can’t prove these things true, but they work over and over again to make predictions, and thus are useful theories that are “true enough”. We didn’t prove them true; we have only proven them not to be false.
Employing Bayes’ theorem, a statistical theory of conditional probabilities, we can be pretty darn sure that a given theory is true (although we can’t prove it absolutely with statistics any more than we can with the scientific method).Bayes’ Theorem for Everyone 01 – Introduction.
TIP: See our page on Bayesian thinking (which we explain without much math), or see An Intuitive (and Short) Explanation of Bayes’ Theorem from betterexplained.com.
How to Prove That There are Statements That Are Likely True, Even Though They Can’t Be Proven
Again, to prove something is likely true even though it can’t be proven we can employ the same tactics as above. We can’t measure light speed in a vacuum, but we have deduced light’s speed with math and have applied the results with repeated success. This hints we have likely found truth, although we can’t prove it.
In the famous Bayesian example, we can never be sure if a person has cancer without direct testing, but if we have enough data-points that point for or against cancer, we can compare these confidently and assume a diagnosis (with little chance of error).A visual guide to Bayesian thinking.
How to Prove a Statement is False
If a statement contains any aspect that can be proven true but is proven false, then the whole statement is false. These fallacies are typically half-truths, but not always. I would consider alchemy and aether half-truth theories while I’d consider most rhetoric from politicians a form of semantic half-truths. These are just examples, but just like the scientific method insinuates, it is much easier to prove something false than true. This is why we use falsifiable statements as hypotheses in science. The method is one of deduction and reason.The Scientific Method.
Complex Truths and Multiple-Valued Logic
Some truths are complex and have more than one value to compare (Many-valued logic). We can consider three values: true, false, and unknown (which still keeps with the law of excluded middles), or we can expand this concept to consider complex situations (like creating a constitution for a new nation who values liberty, justice, and truth).
This sort of logic helps to show us that there are truths we can’t know, and also begins to present us with complexity and contradictions that need to be overcome to find truth.
Some things are only true at a given point in spacetime and are contingent upon many factors. For instance, I can tell you where Jupiter is in the sky at any instant with the right equipment, but it is dependent on my reference frame at a point in spacetime. The truth is there, but it changes and is dependent on view point. We can aggregate viewpoints over space, or over time, but it is complex and on some level has aspects of subjectiveness in practice.
Although there is always the truth, finding “truth” that applies to all people, especially in complex social situations when considering ideals, will require the sacrificing of greater truths, and the accepting of lesser truths, to seek “the truest” outcome. This leaves us with complex factors to consider. Luckily, there are equations for this too. This logic requires a sort of mix of Bayesian thinking and moral reasoning. Thus, it can require philosophical reason as much as a logical mathematical approach, and may require that we break down complex truths down into more simple systems. See Modern Uses of Multiple-Valued Logic.Many Valued Logics. Everything is either true or not true, but applying this to complex truths and belief systems requires a higher order of critical thinking. Three or More Value Logic. Sometimes we can’t know an answer; this is why, from a scientific angle we focus on asking the right questions.
TIP: Many-valued logic is a propositional calculus in which there are more than two truth values; it just so happens to be a type of logic that applies well to belief systems. See Gödel’s logics Gk and G∞ for examples of many-valued logic.
Proving a Statement True that is False, and Other Logic-Based Truths Like Contradictions
One of the more interesting things we can do with mathematics and logic is prove something that is false to be true or something that is true to be false, we can also find truth based on a false statement and vice versa, and we can seem to conclude that certain propositions are both true and false. Regardless of how complex things we get, the rule of excluded middles applies, and there is always a specific true or false answer (even if we can’t prove it).
Good examples include paradoxical statements such as: “This sentence is a lie.” This is known as the liar paradox and this is the paradox that Gödel’s First Incompleteness Theorem solves by translating the semantic puzzle into mathematics. The paradox is: if “this sentence is false” is true, then the sentence is false, but if the sentence states that it is false, and it is false, then it must be true, and so on.This Statement is False.
The video below explains how to solve contradiction paradoxes like the Liar Paradox (a type of paradox that seems both true and false). We know the law of excluded middles tells us that everything is either true or false, so it is our logic (and not the universe) that needs to do more work. The key to solving this is properly wording the question or employing a specific type of propositional logic related to contradictions (demonstrated in the video).Propositional Logic: Contradictions.
We won’t get into every logic-based proof for truth here, but you can see Three Ways to Prove “If A, then B” for some additional mind-bending logic related to direct proof, contrapositive proof, and proof by contradiction. Or see this Stack Exchange forum for more musings on proving true things false and false things true.
You don’t have to master logic to get the gist. Everything is either true or false, but that doesn’t mean we can prove it, and that doesn’t mean we should stop looking. The world is complex.
How to Prove a Statement is True
Most statements can be proved true or false with logic, but not all of them. This is reinforced by the following video.Introduction to Logic.
On a deep philosophical level there isn’t much, if anything, we can prove 100% true from the inference we make down to the root (as it would require us to prove that what we observe is 100% true and subsequent reasoning is valid).
In science we aren’t looking to prove truth (we are looking to falsify; to prove something isn’t true). We are, in other words, looking for theories that work 100% of the time (not gospel truth).
This is as close as we get to truth.
When we employ math, logic, and reason, we can find those close enough truths we call mathematical proofs and scientific theories.
Descartes said “I think therefore I am,” but his proof was ontological. Socrates [sort of] said “I know one thing: that I know nothing,” these are some very wise and insightful statements (but they aren’t 100% by every measure).
We can make true statements about what we consider truth, but at the root we have to question rationalization, observation, and thus the inductive, deductive, and other reasoning we use to process the concepts we conceive.
Still, we don’t need to prove every aspect of philosophical certainty to have good and useful knowledge about the worlds of ideas and facts (and this is the point I’m trying to make here; the goal is, for practical proposes, to find truths that work, the ones we can’t prove false on a practical level no matter how hard we try; behind that lies the other truth, that everything is either true or not despite any degree of uncertainty we have).Logic –The Structure of Reason (Great Ideas of Philosophy).
But For Practical Purposes, All Well Constructed Statements Worth Exploring Scientifically are Falsifiable
Whether it is FactMyth.com fact-checking statements or the scientific method testing a hypothesis, it typically isn’t proper form to ask a question that can’t be proven false.
We wouldn’t research a statement like “there are aliens,” because we know we don’t have the proof to back this up. Instead, we would research an answerable statement like “there are likely aliens.” We could employ the Fermi paradox and Drake equation and give a satisfactory answer.
The same goes for the scientific method; if we start with a hypothesis we can’t prove false, then we set ourselves up for a junk theory.
It isn’t that unanswerable questions are not worth exploring, Gödel ensures us that there is truth that we can’t know (and all the philosophers generally agree), it is that we work best when we use all the technology we have to answer the questions and fact-check statements and questions that can result in useful data. In this, we unlock more keys, and uncover deeper truths, and better questions can be asked.
If you have made it this far, you are in for a treat, below is one of the most mind-blowing metaphysical-mathematical-cosmological videos on YouTube. An introduction to Gödel, Escher, Bach (a Pulitzer Prize-winning book by Douglas Hofstadter that inspired many the philosophy/math nerd).MIT Godel Escher Bach Lecture 1. This lecture series from MIT expands on the concepts on this page. If you found any of the above interesting, this lecture is a must watch. Check out the link for the full series.
- Principle of bivalence
- “Law of thought” Wikipedia.org
- “Gödel’s Incompleteness Theorems” Wikipedia.org
- “An Intuitive (and Short) Explanation of Bayes’ Theorem” Betterexplained.com
- “Modern Uses of Multiple-Valued Logic” books.Google.com
- “Degree of truth” Wikipedia.org
- “Gödel’s Incompleteness Theorems” plato.Stanford.edu
- “Many-valued Logic” Wikipedia.org
- “liar paradox” Wikipedia.org
- “I know that I know nothing” Wikipedia.org
- “Fermi paradox” Wikipedia.org
- “Drake equation” Wikipedia
One would assume truth is more provable than it is, but ask someone who studies statistics and philosophy and you will begin to understand the complexity of “what we can know” which is looked at through a branch of philosophy called epistemology, while the rest is proven with logic (a branch of philosophy and mathematics that employs statistics).


Φ symbolizes the ancient Greek idea of the golden ratio
the divine proportion
in mathematics, it speaks of symmetry
in art, it measures beauty and balance
Φ is an inherent part of nature
To find the balance between the ancient and the modern,
pragmatism and imagination, health and indulgence.
To find the essence of Greece.
Reaching towards the golden ratio is for us a process of love and a celebration of life.
Come along for the journey!
Look for us online, and in select stores around Greece.


Statistics Research Guide
Statistics is "The science of collecting, analyzing, presenting, and interpreting data. Governmental needs for census data as well as information about a variety of economic activities provided much of the early impetus for the field of statistics. Currently the need to turn the large amounts of data available in many applied fields into useful information has stimulated both theoretical and practical developments in statistics.From Encyclopedia Britannica Online. http://search.eb.com/eb/article?eu=115242 [Accessed 15 October 2007].
Sub-topics may include: variance, standard deviation, normal distribution, estimation, hypothesis testing. A related term is PROBABILITY.
Search the Public Access Catalog (PAC) by subject or keyword. Try some of the following subject headings: statistics, statistics -- dictionaries, statistical methods, commercial statistics, correlation (statistics), educational statistics, industrial statistics, mathematical statistics, medical statistics, political statistics, probabilities, vital statistics. Try a title search for statistics.
Statistical Data and Information Resources from the Library's Reference and Circulating Collection:
- The Illustrated Book of World Rankings
- Statistical Handbooks on various subjects
- Statistical Abstracts of local, national and international scope
- Statistical Abstract of the United States, one of the most heavily used resources for national statistics is available in print and online versions from the Office of Government Publications.
Introductory and Self-learning
- Basic Statistics
- Basic Business Statistics
- A First Reader In Statistics
- The Statistical Exorcist: Dispelling Statistics Anxiety
Explanatory Resources to the Field of Statistics Found in the Circulating Collection:
Interpretation and Use
- How To Lie With Statistics
- How To Use (And Misuse) Statistics
- Statistics For People Who (Think They) Hate Statistics
- Use And Abuse Of Statistics
- Against All Odds Inside Statistics [Videorecording]
- Sampling and Estimation [Videorecording]
- Statistics & Sports Performance [Videorecording]
- Statistics At A Glance [Videorecording]
- Statistical Applications--Basic [Videorecording]
- Statistics, An Audio-Tutorial Approach [Sound Recording]
These databases are research tools that cover thousands of printed publications.
You can find statistics on almost any subject, therefore, you should be able to find articles with statistics in almost any of our databases depending on your subject. So for example, if you want business statistics, look in our business databases. If you want crime statistics, look in our criminal justice database. We also have many databases that cover many different subjects so you might try some subject specific databases and the multiple subject databases. If you can find your topic in the CQ Researcher database, you should find a bunch of statistics. Do a really broad search in this database. For example, search for "autism" or "alcohol". The database is small enough that you won't have too many results to look through, but it is well worth trying.
To construct your search, you should try typing the subject into the database and then add "and statistics". For example, you could go to our criminal justice database and search for "murder and statistics". You may also find some articles that have statistics by searching for your subject without adding the word "statistics". For example, you might search for "Chicago and murder". In this case, you will need to look at each individual article to determine if there are statistics in them.
Web Resources for Statistical Info and Data
U.S. Government Sites
- Statistical Abstract of the United States - is a yearly government publication and one of your best "first choice" resources.
- American FactFinder 2 - Raw data from the decennial (10 years) US Census, and more. (Watch this video to learn how to use it.)
- FedStats and USA.Gov - Two great places for finding U.S. statistics and other U.S. Government info.
- Data Bank - Data from Census 2000, product overviews, and assistance in acquiring and using census information.
- The CIA World Factbook - The basic skinny on every nation on Earth is in the CIA World Factbook.
- Other Statistical Sites on the World Wide Web - (Scroll down page to International Statistical Agencies) List of links to federal and international statistical agencies, including the official statistics departments of around 60 countries. Maintained by the U.S. Bureau of Labor Statistics.
Other Internet Resources
- RobertNiles.com - A source for general stats information organized by category. As it claims, "Scroll down to the links below and your days of coming up empty while reporting are over. You'll soon be checking facts and downloading reputable data on everything from public safety to campaign contributions...you name it."
- University of Michigan's The Document Center - The Documents Center is a central reference and referral point for government information, whether local, state, federal, foreign or international. Its Web pages are a reference and instructional tool for government, political science, statistical data, and news.
- OFFSTATS - Compare a Southern Hemisphere source with a Northern one for accuracy! The Official Statistics on the Web is now located at the University of Aukland, New Zealand. OFFSTATS lists web sites offering free and easily accessible social, economic and general data from official or similar "quotable" sources, especially those that provide both current data and time series. In the country lists, these are mainly Web pages provided by statistical offices, central banks and government departments and agencies, whereas the topics list is comprised of links to the statistics pages of international organizations and associations and a few commercial sites.
- POPULATION STATISTICS - Here you will find a historical, demographic and statistical overview of the population of all the countries in the world, their administrative divisions and their important cities.
- Statistical Resources on the Web Foreign Government Data Sources - From the University of Michigan's the Documents Center, a compilation of Web links to statistical information on countries outside United States.
Web Tutorials and Self-learning for Statistics
- Animated Statistics Demonstrations Although these pages were written for behavioral statistics at Ohio University, they should be useful for anyone taking introductory statistics or in need of basic statistical information. View the animated demonstrations and practice your computational skills by trying the problems. Now you can complete the problems online with a new interactive format that allows you to check your work as you go.
- Statistics Every Writer Should Know Here are clear presentations of concepts such as mean, median, standard deviation, margin of error, etc. Of related interest, the Chance Project seeks to educate readers about the uses of probability and statistics in current news reporting. Chance offers lectures and videos (requiring RealPlayer 5.0), teaching aids, online texts and interesting articles demonstrating concepts. Still want more? Try the online course in Introductory Statistics.
- Tutorials: Dix: Statistics Statistics tutorials by Alan Dix, professor of Computing Department, Lancaster University, Lancaster, UK. Includes an animated "coin toss race" for probability. A discussion of seven common statistical errors.
- DAU Stat Refresher Module This module is an interactive tutorial which gives a comprehensive view of probability and statistics. This interactive module covers basic probability, random variables, moments, distributions, data analysis including regression, moving averages, exponential smoothing, and clustering. A great feature of this tutorial is a click-on a-z index of terms.
- Hyperstat Online Introductory hypertext statistics textbook. Also has links to related online resources. Produced by a statistics professor at Rice University in Texas.


At 09:04 12/07/2018 -0400, Carl Winerich wrote:
Existing cell (for the month of February) has the following formula; =$G30-$H30+$D32+$E32+COLUMN(A5)That stuff points to various parts of the spreadsheet and returns the correct numeric day number of the month. When I try to concat it with, say, =DAYS then Err:510 appears in the cell.
I'm not surprised: you concatenate references or expressions in a formula, not multiple formulae. What's that extra equals sign doing there? That's like writing =A1+=B2 instead of =A1+B2.
Yes: excise that superfluous equals sign.Brian Barker


This is going to be short, and the main portion is going to be about weird math stuff. Buckle in.
So! I’ve been busy lately, and one of the things I like to do when I’m stressed out is do math in my head. Not like, multiplication and algebra (though that too, sometimes), but mostly figuring out weird math patterns. For example, earlier I was trying to figure out what the smallest common multiple of 12 and 16 that was also 2 to the power of something. (Meaning 2*2*2*2 until I got a number that was divisible by both 12 and 16). Turns out, no such number exists, so that was an exercise in futility. Point being, I’ve been fiddling with numbers.
That being said, I spent a lot of time developing counting and money systems for the nations of my world. I talked a lot about dozenal in my last worldbuilding post, which much of my world uses. But I also fiddled around with a Base-8 system that had been used by another empire previously, several thousand years before my book begins.
Now, last week, the two came together. I liked the idea of an entirely non traditional counting system: something that used cool math instead of just going up. So, I thought it would be cool for one of the bigger nations in my world to use money that counted up exponentially. Instead of having a column for each unit, or tens of units, or hundreds of units, they would a different symbol for each exponent. Like this:
1 unit will be 0
2 units will be 1
4 units will be 2
8 units will be 3
Math-heads in the audience might realize this is just 2 to the power of something.
2^0 is 1
2^1 is 2
2^2 is 4
I think it’s really cool because even if each symbol is a completely different number, you can still count up.
0 is 1
1 is 2
10 is 3
2 is 4
20 is 5
21 is 6
210 is 7
3 is 8
This fascinates me especially because order doesn’t matter. 210 is the same as 021, because each number represents an amount regardless of position. Really nerdy folk may realize this system is similar to binary, or counting with just two symbols, and there may be a name for it but I haven’t found one (not that I’ve looked too hard.)
Enough about the math, though. One of the nations in my world is called Búandiš, and for a while they had a really stable trading community. So stable, in fact, that when Myria (the largest nation currently) came to full power, they pegged their own dollar to the Búandiš money, which is called Gill. Myrian money (called Kronkel), counts in twelves like the rest of my world, but Gill uses this weird counting system I showed you up above.
Anyway, that’s a bit of math and a bit of my worldbuilding process. I get bored or something and decide to make my world more complicated. Whoo!
Besides just sharing some math nerdiness with you, I also want to get more comfortable actually talking about the world I’ve made. Because, when you create something as dorky and wonderful as a whole new world, it can be a bit embarrassing to show that to people. You’re baring a part of your mind to the world, for them to love or hate, compliment or critique, and that’s scary. All this to say that creativity should be shared. If you put the time and effort into making something, then it is worth seeing and enjoying. No matter what it is, someone out there will be delighted to see/hear/feel/be a part of your creations.


In statistical hypothesis testing, two hypotheses are compared. These are called the null hypothesis and the alternative hypothesis . The null hypothesis is the hypothesis that states that there is no relation between the phenomena whose relation is under investigation, or at least not of the form given by the alternative hypothesis. The alternative hypothesis, as the name suggests, is the alternative to the null hypothesis: it states that there is some kind of relation. The alternative hypothesis may take several forms, depending on the nature of the hypothesized relation; in particular, it can be two-sided (for example: there is some effect, in a yet unknown direction) or one-sided (the direction of the hypothesized relation, positive or negative, is fixed in advance).
Experimental methods are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable). Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship. That is to say that if changes in one variable actually cause another to change.


Students fared better this year with regards to mathematics in their high school leaving exams, the Education Ministry and exam organizer Cermat confirmed. According to the ministry, the number of those who succeeded in the math section rose by 1.5 percent compared to last year. Overall, 21.5 percent of students failed the section while 8.4 percent failed in the Czech language. The high school leaving exams, known as ‘maturita’ were attended by 62,000 students this year. Students prepare for weeks and months for the final exams.


Learn to Count
Learn to Count acts as a fun introduction to mathematics and counting from an early age. Children match the numbered squares to the dots on the base of the wooden puzzle. Consists of 55 play pieces.
Out of stock
Children match the numbered squares to the dots on the base of the wooden puzzle; then count the brightly coloured discs as they are placed onto their individual pegs. The bright colours encourage young learners to work on number and colour recognition as they develop fine motor skills and hand/eye co-ordination. A great tool for parent/child interaction. The ideal introduction to counting for preschoolers. Helps to develop dexterity and concentration. Made from high quality, responsibly sourced materials. Consists of 55 play pieces.
Hubby Made Products:
Proudly Made in Melbourne for almost a decade, we love hearing from our customers, as we pride ourselves on our high level of customer service. No matter what the question, we are always here to help and make your experience with Hubby Made a positive one.
|Dimensions||430 × 120 × 120 mm|
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Thursday, 5th December 2013 @ 9:55 am
k-Wave (1.0), a MATLAB toolbox for the time-domain simulation of acoustic wave fields, is now available for use.
This software suite/toolbox inherits all of MATLAB's features (array jobs, parallel and using GPU when applicable, etc.). Necessary job submission script can be generated via
An optimized C++ version of k-Wave that maximizes computational performance for large simulations will soon be made available as well.


A scientific specification to identify the quality of an individual sample, an entire population, or an individual test instance. As a practical term, an experiment. As nouns, adjectives relating to the sample: a rigorous scientific test; a controlled experiment. Derived from the Greek term demos, which means “of a group”.
A set of t-tests used to compare two sample sets in order to identify whether one variable is significantly associated with another and whether the other variable is significantly related to the first one. There are two concepts involved in t-testing: independence of variables and sample size. An example of a set of t-statements would be: set 1 | set 2 | t-statements | probability | variable} A hypothesis test states that there is a relationship between a specific variable and its dependent variable. In a common practice, there are two types of hypothesis testing: the null hypothesis and the alternative hypothesis. For a null hypothesis, there is no reason to assume that a variable does not exist, whereas for an alternative hypothesis, there could be a valid relationship between the variable and its corresponding sample.
A chi-squared test calculates the probability density function of the data set used in a particular study. The chi-square statistic is formulated as: where N is the number of sample points, I is the intercept and chi is the distribution function for the data set. The area under the curve of I is called the chi squared value. Correlated confidence intervals are calculated by connecting the data points for each variable, and their squares with the corresponding confidence interval. The chi-squared value is a continuous function that evaluates the value of the correlated variable I against the corresponding time variable t, where I represents the time variable and t represents the independent variable.
For cases where multiple t-statements are needed to reach the expected result, the multiple t-test approach can be applied. For example, if N is the number of items to be tested and C is the number of items to be excluded in the test case, then N x C will be the number of new item values that will be generated when C is run through the t-test model. The expected results will be the expected number of new items when C is entered into the model; however, when C is run, it may generate an item value of zero, hence the use of the accept maximum expectation approach. Accept maximum expectation approaches assumes that there will be a significant number of false results, and therefore there is a need to accept at least ten characters in the input data set in order to obtain statistically significant results.
When performing t-tests, the null hypothesis should always be rejected at p>0.1 unless there is strong evidence to suggest that the null hypothesis is true. Other than this, all other variables should be analyzed in the same way as described above. There is no need to perform more than one test under the same conditions as described above for the same variable, unless there are significant differences between the null hypothesis and alternative forms of t-tests. Additional information about multiple tests and their sample sizes can be obtained from vendors. Test examples can be found on the web at the statistics reference sites.


Sadly, tech geniuses in Silicon Valley have yet to design a computer that can weigh you … so until they have, here’s a simple calculation to help you find the plan that is right for you.
A word of encouragement … we do not starve you on either of our plans. You will find there is a lot of food and plenty of snacks to help keep cravings at bay … and you will still lose weight. Fab huh?!
Simply work out your BMI … (we say simply, you may need a calculator!)
- Measure how tall you are (in metres please)
- Weigh yourself (in kg)
- Calculate your height squared (eg – if you are 1m72, the sum is 1.72 x 1.72)
- Calculate your weight divided by your height squared (so for example 62kg / 2.96 (1.72 squared) = 21)
If your BMI is less than 29, we recommend signing up to our Lighter Menu Plan. If you have a BMI of 29 or above, it is probably best, initially, to try our Market Menu Plan.
If your BMI is less than 29, we would recommend signing up to our Lighter Menu Plan.


FANCY STONE GRADING
Then you measure the stone’s dimensions-width, length, and depth-in millimeters, interpolating to the nearest 0.01 mm.
Width in round stones means diameter: it is the average of the minimum and maximum girdle diameters. In square stones width is the average of the two side-to-side measurements.
On rectangular shapes, length is the distance from end to end measured along the longest side (not diagonally). Width is the distance between (and at right angles to) the longest sides.
With most other fancy cuts, length is the longest dimension, and width is the widest dimension perpendicular to the length.
On hearts, however, width is the measurement between the widest parts of the lobes, while length is the perpendicular distance from the point to an imaginary line across the top of the lobes.
And on triangles, width is the distance from corner to corner on the shortest or longest side (if all are equal, any side), while length is the perpendicular distance from the center of the width side to the opposite point.
These diagrams show where you measure the length and width of different fancy shapes.


Goal 8: Solve Problems
Use algebraic and analytic methods to identify and describe patterns and relationships in data, solve problems, and predict results.
The Activities provide practical applications. After the students have developed a math competency, they can keep it in active use by applying it in science and social studies contexts.
The Data section includes table for use with the activities and as a resource for your own project development.
|Ecology Activities||Technology/Physics Activities||Systems Activities||Discoveries Activities|
|Chicago Ecology Data
|Chicago Technology/Physics Data||Chicago Systems Data||Use any relevant data for discoveries projects.|
|Geography Activities||Culture Activities||Civics Activities||Economics Activities
|Chicago Geography Data||Chicago Culture Data||Chicago Civics Data||Chicago Economics Data|


The MIAA has adopted a new power ranking formula to determine what teams will play in the state tournaments. The top 32 teams in each division along with all other squads that finish with a .500 or better record qualify for the postseason.
Schools will be seeded by their overall power rating. A major piece of the power rating is the MOV – Margin of Victory. The MOV is capped at a certain amount, meaning that any score above this maximum amount will just be the maximum number.
The first piece that is important is a school’s individual overall margin of victory. That is the simple piece of the equation as you just go through all games and simply get the +/- number for a school’s margin of victory.
For example, a team plays 6 games:
W 2-0, L 3-2, W 5-1, T 2-2, W 2-1, & L 4-2
Overall record is 3-2-1 with an overall power rating of +0.5:
Margin of victory equals 3 divided by total number of games, 6, equals a power rating of +0.5.
The second piece is the average power rating of all of a school’s opponents. In order to get that number one must get the average margin of victory of all opponents (remembering to add an opponent’s rating twice if they play them twice).
Add the two numbers:
Team’s average margin of victory + average of opponents’ ratings = overall rating
Schools are then seeded from highest overall rating to lowest in each division.


Pattern recognition and classification theory
Statistical learning theory: a tutorial sanjeev r kulkarni and gilbert harman february 20, 2011 abstract in this article, we provide a tutorial overview of some aspects of statistical learning theory, which also goes by other names such as statistical pattern recognition, nonparametric classi cation and estimation, and supervised learning we focus on the problem of two-class pattern. Chapter 1 pattern classification 11 what is pattern recognition it is generally easy for a person to differentiate the sound of a human voice, from that of a violin a handwritten numeral 3, from an 8 and the aroma of a rose, from that of an onion. Introduction to pattern recognition system 3 i representation - it describes the patterns to be recognized ii classification - it recognizes the “category” to which the patterns provided belong.
Subject area: pattern recognition topics covered fundamentals of pattern classification supervised learning methods o bayesian decision theory. Bayesian decision theory pattern recognition, fall 2012 dr shuang liang, sse, tongji minimum-risk classification • the general decision rule a(x) tells us. Course description this course will introduce the fundamentals of statistical pattern recognition first, we will focus on generative methods such as those based on bayes decision theory and related techniques of parameter estimation and density estimation. 11 a binary classification problem pattern recognition (or classification or discrimination) is about guessing or predicting the unknown class of an observation an observation is a collection of numerical measurements, represented by a d-dimensional vector x the unknown nature of the observation.
Bayes decision theory sargur srihari cse 555 introduction to pattern recognition cse 555: srihari 1 reverend thomas bayes 1702-1761 bayes set out his theory of probability in essay towards solving a problem in the doctrine of chances published in the philosophical transactions of the royal society of london in 1764 the paper was. Request pdf on researchgate | pattern recognition and classification an introduction | the use of pattern recognition and classification is fundamental to many of the automated electronic systems in use today however, despite the existence of a number of notable books in the field, the subject remains very challenging, especially for the. A probabilistic theory of pattern recognition (stochastic modelling and applied probability) [luc devroye, laszlo györfi, gabor lugosi] on amazoncom free shipping on qualifying offers a self-contained and coherent account of probabilistic techniques, covering: distance measures, kernel rules. Statistical learning/pattern recognition an approach to machine intelligence which is based on statistical modeling of data with a statistical model in hand, one applies probability theory and decision theory to get an algorithm this is opposed to using training data merely to select among different algorithms or using heuristics/common. Eece 667 – pattern recognition catalog description: the course provides an overview of the theory, principles and algorithms used in machine learning to construct.
This book considers classical and current theory and practice, of supervised, this chapter introduces pattern recognition as the scientific discipline with the goal of classification of objects into a number of categories or classes the chapter discusses the basic philosophy and methodological directions in which the various pattern. Introduction to pattern recognition wei-lun chao graduate institute of communication engineering national taiwan university, taiwan october, 2009. Classification: an introduction we human beings do pattern recognition everyday we “recognize” and classify many things, even if it is corrupted by noise.
Considers classical and theory and practice, of supervised, unsupervised and semi-supervised pattern recognition, to build a complete background for professionals and students of engineering this read more. The book offers a thorough introduction to pattern recognition aimed at master and advanced bachelor students of engineering and the natural sciences besides classification - the heart of pattern recognition - special - selection from pattern recognition [book. Support vector machine (and statistical learning theory) tutorial jason weston nec labs america 4 independence way, princeton, usa [email protected] Pattern classification and scene analysis richard o duda 37 out of 5 stars 44 hardcover 35 it was an excellent reference pattern recognition which was a hot topic what made this book so great is that it was a compendium of all of the methods used at the time little has changed except the speed of the computers to implement.
Other typical applications of pattern recognition techniques are automatic speech recognition, classification of text into several categories (eg, spam/non-spam email messages), the automatic recognition of handwritten postal codes on postal envelopes, automatic recognition of images of human faces, or handwriting image extraction. Ece662 pattern recognition and decision making processes fall 2016 purdue school of engineering and technology, iupui, indianapolis sarah koskie. Pattern classification and machine learning the theory of supervised learning in artificial neural networks will be compared with statistical pattern recognition and approaches from the theory of generalization the course is jointly taught by w gerstner and m seeger in addition to a series of interactive computer exercises in java that.
Pattern recognition using fuzzy sets, which is discussed in this section, is a technique for determining such transfer functions a good overview of the material discussed is “a a good overview of the material discussed is “a. This book is an excellent reference for pattern recognition, machine learning, and data mining it focuses on the problems of classification and clustering, the two most important general problems in these areas this book has tremendous breadth and depth in its coverage of these topics it is. This book considers classical and current theory and practice, of supervised, unsupervised and semi-supervised pattern recognition, to build a complete background for professionals and students of engineering.
Pattern recognition theory is demonstrated to be an effective means by which complex travel/activity patterns can be transformed into a structurally simpler space for purposes of planning and analysis. Classification techniques in pattern recognition lihong zheng and xiangjian he faculty of it, university of technology, sydney po box 123, broadway nsw 2007, sydney, australia. It’s good that you want to learn pattern recognition you will get an exposure to artificial intelligence through this subject, and trust me, it will be really helpful in the long run even years back, i had opted pattern recognition as an electiv.


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EducationBrown University - Current Undergrad, Neuroscience
ExperienceI am a Brown University senior graduating this May with a bachelors of science in neuroscience. For the past year and a half I have spent numerous hours as a volunteer for Let's Get Ready, a SAT and college prep program. I began my involvement in this organization purely on a whim, in order to give back to the community around me. I quickly discovered my passion for teaching, however, and began serving as the head math coach within a few short months. Teaching has allowed me to take part in the development of future intellects. To this purpose, I teach only mathematics and science, as those are my personal favorite subjects. I also focus on improving one's ability to think through a problem, as opposed to learning simply by base memorization. I do this because when a student's focus is solely on memorization, they often quickly forget the material due to time or stress. However, the skills one gains from learning how to think through problems are much more stable and flexible. This flexibility allows for those skills to be used in multiple subjects, and towards more advanced learning.
Now, a little about me. In the future, I plan to go to medical school and become a general surgeon, due to the growing need in our current healthcare system. I enjoy drawing, salsa dancing, and bachata dancing. I'm also the self-proclaimed FIFA king of my university, and will defend my title at any non-working moment. All being said, I hope to see you in my classroom as we work together towards & beyond your goals.
HobbiesSoccer, Rugby, Salsa Dancing, and Drawing
Algebra, Algebra 2, Algebra 3/4, AP Calculus AB, AP Statistics, Biology, Calculus, Geometry, Pre-Calculus, SAT Math, SAT Mathematics, SAT Subject Test in Biology E/M, SAT Subject Test in Mathematics Level 1, SAT Subject Test in Mathematics Level 2, SAT Subject Tests, Statistics, Test Prep, Trigonometry
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Bulgarian participants in the International Olympiad in Informatics in Tehran won one gold and three bronze medals. We are proud that two of the champions are Telerik Academy School students.
Encho Mishinev,12 grader in “Atanas Radev" High School of Mathematics in Yambol, won the gold medal. Bronze medals were awarded to Telerik Academy School alumni Victor Terziev - 11 grader in Sofia High School of Mathematics, Radoslav Dimitrov - 10 grader in "Acad. K. Popov" High School of Mathematics in Plovdiv and Peter Nyagolov – 10 grader in “Baba Tonka” High School of Mathematics in Ruse.
Thus, Bulgaria ranks among the top five countries in the world with over 100 medals in Informatics - 25 gold, 41 silver and 34 bronze. As a result of the brilliant performance of our national team, Bulgaria now ranks 6th in medals won at the International Olympiad in Informatics.
Did you know that:
We wish a lot of success to the Bulgarian National Team in Informatics and the next generation of contestants!
Monika is an Associate Software Engineer at Software AG. It is the job she always wanted. A job she landed without any previous work experience in the IT industry and while still studying at Sofia University St. Kliment Ohridski. How was this possible? With Telerik Academy Alpha Java! Read Monika’s inspiring story.Read the story
Liliya Burlakova graduated from Telerik Academy Alpha Java in May 2020, during the first COVID-19 wave. Less than 2 months later, she started working as a Junior Software Developer. This is her exciting journey.Read the story


What Makes Us Special
Erickson is a neighborhood school where students walk to campus on a daily basis and the community is a supportive force. The school has created a culture of trust with parents and staff, which enables the school to concentrate on academic achievement.
Erickson has a strong math and reading components and at each grade level. Teachers conduct daily math reviews and mental math activities and focus on the procedural and conceptual sides of math. Reading is encouraged at all levels, and tutors are available for students who might need extra help. Erickson also offers a strong band and orchestra program, access to the arts and other fun activities.