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Quadratic Residues
The setting is that the modulus in question is an odd prime . Consider an integer such that and are relatively prime, i.e. having no common prime factor. The number is said to be a quadratic residue modulo if the equation has an integer solution in . Another way to say this is that is a quadratic residue modulo if there exists a square root of modulo (a square root would be a solution to the equation). If the integer is not a quadratic residue modulo , we say that is a quadratic nonresidue modulo . If the context is clear, the word quadratic can be omitted. We can then say is a residue modulo or is a nonresidue modulo .
Since every integer is congruent modulo to one of the numbers in the set , the integers can be considered from the set (the non-zero elements of ).
Let’s look at a quick example. Let . Squaring each number in produces the set . Reducing modulo 11 produces the set . Thus the integers 1, 3, 4, 5 and 9 are quadratic residues modulo 11. The square roots of each residue are the solutions to the equation . For , the solutions are x = 5 and 6. There are two square roots of 3 modulo 11, namely 5 and 6.
When the modulus is small, it is easy to find all quadratic residues simply by squaring all the numbers in . The focus in this post is on how to use the law of quadratic reciprocity to determine whether a given is a quadratic residue modulo a large odd prime .
To check whether the integer is a quadratic residue modulo an odd prime , the most important idea, before considering the law of reciprocity, is to check the modular exponentiation . If the result is congruent to 1 modulo , then is a quadratic residue modulo . If the result is congruent to -1 modulo , then is a quadratic nonresidue modulo . This is called Euler’s Criterion (proved here). For clarity, it is stated below.
Theorem 1 (Euler’s Criterion)
Let be an odd prime. Let be an integer that is relatively prime to . Then the integer is a quadratic residue modulo if and only if . On the other hand, the integer is a quadratic nonresidue modulo if and only if .
According to Euler’s Criterion, the task of checking for the status of quadratic residue is a matter of performing a modular exponentiation. This can be done using software or a calculator for modular arithmetic. If so desired, the exponentiation can also be programmed using the fast powering algorithm.
Though Euler’s Criterion (with a calculator for modular arithmetic) is a sure fire way for checking the status of quadratic residues, the law of quadratic reciprocity can simplify the task even further. As the examples below will show, checking the status of quadratic residues using the law of reciprocity may require no modular exponentiation at all and if exponentiation is required, it is of a much smaller size.
Euler’s Criterion also gives us several basic facts about quadratic residues.
Theorem 2
Let be an odd prime. The following properties are true:
Theorem 2 says that the product of two integers of the same types (both residues or both nonresidues) is always a residue modulo the odd prime . The product of two integers of different types is always a nonresidue modulo the odd prime . The theorem follows from Euler’s criterion and from the fact that .
In arithmetic modulo a prime , there exists a primitive root modulo and that any primitive root generates by powering all the integers that are relatively prime to the modulus . Let be a primitive root modulo an odd prime . It then follows that the quadratic residues modulo are the even powers of and the nonresidues are the odd powers of . A related fact is that when is an odd prime and when and are relatively prime, the equation either has two solutions or has no solutions.
Theorem 3
Let be a primitive root modulo an odd prime . For any that is relatively prime to , the following is true:
Theorem 4
Let be an odd prime. For any that is relatively prime to , the equation either has two solutions or has no solutions (proved here).
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Legendre Symbol
The law of quadratic reciprocity is stated using the Legendre symbol. For an odd prime and for an integer that is relatively prime to , the symbol is defined as follows:
One obvious and important observation is that Euler’s Criterion can be restated as follows:
Theorem 1a (Euler’s Criterion)
Let be an odd prime. Let be an integer that is relatively prime to . Then .
In the above definition, the lower argument of the Legendre symbol is always an odd prime and the upper argument is an integer that is relatively prime to the lower argument. To smooth out some statements involving the Legendre symbol and to make it easier to define the Jacobi symbol in the next post, we relax the Legendre symbol by making whenever and are not relatively prime, i.e. . The Legendre symbol can be broaden slightly by the following:
Here’s some useful basic facts about the Legendre symbol.
Theorem 5
Let be an odd prime. The following properties hold:
The first bullet point in Theorem 5 is easily verified based on the definition of the Legendre symbol. For the case , the other facts in Theorem 5 are easily verified. For the case , the second bullet point follows from Theorem 2, i.e. the fact that the product of two residues and the product of two nonresidues are both residues modulo an odd prime and that the product of a residue and a nonresidue is a nonresidue modulo an odd prime. The second part of the third bullet point is true since any integer that is a square is a quadratic residue.
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Law of Quadratic Reciprocity
The law of reciprocity can ease the calculation of the Legendre symbol when both the upper and lower arguments are distinct odd primes. Theorem 6 is the law of reciprocity and Theorem 7 and Theorem 8 are two supplements to the law that can round out the calculation. After stating the theorems, we demonstrate with some examples.
Theorem 6 (the law of quadratic reciprocity)
Equivalently and more explicitly,
Theorem 7 (first supplement to the law of quadratic reciprocity)
Theorem 8 (second supplement to the law of quadratic reciprocity)
Theorem 6 shows how to flip the Legendre symbol if both the upper and lower arguments are distinct odd primes. As long as one of the primes is congruent to 1 modulo 4, we can safely flip the symbol. If both primes are not congruent to 1 modulo 4, we can still flip the symbol except that we have to attach a minus sign. Theorem 7 (the first supplement) indicates when -1 (or ) is a quadratic residue an odd prime . In words, -1 is a residue modulo if dividing by 4 gives the remainder of 1. Otherwise -1 is a nonresidue modulo . Theorem 8 (the second supplement) indicates when 2 is a residue modulo an odd prime . In words, 2 is a residue modulo if dividing by 8 leaves a remainder of 1 or 7. Otherwise 2 is a nonresidue modulo .
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Examples
Example 1
Evaluate , an example where both the upper and lower arguments in the Legendre symbol are primes.
The above derivation is a repeated use of Theorems 5, 6 and 8. The idea is to flip the Legendre symbols to make the lower arguments smaller. Then reduce the upper arguments and then factor the upper arguments. Then flip again until reaching a Legendre symbol of , which is easy to solve.
It then follows that 1783 is a quadratic residue modulo the odd prime 7523. The answer can be confirmed by using Euler’s Criterion. Note that .
Example 2
Evaluate where 1298351, an odd prime and 756479.
The number is not a prime and is factored as . We have the following derivation.
The evaluation of the Legendre symbols in this example does not start with a flipping since the upper argument 756479 is not a prime. Instead, we start with factoring the upper argument into prime factors and then proceed with a series of flipping, reducing and factoring.
Example 3
Evaluate where 569, an odd prime and 1610280.
This example can also be evaluated by first reducing 1610280 modulo 569.
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Comment
The law of quadratic reciprocity is a deep and powerful result. It guides the evaluation of Legendre symbols in an attempt to answer whether a number is a quadratic residue modulo an odd prime. The law of reciprocity as represented above requires that the lower argument in the Legendre symbol is an odd prime. When the upper argument is an odd number that is not a prime, there is no way to flip it (based on the law of reciprocity using the Legendre symbol). In Example 2, the evaluation of the Legendre symbol cannot begin until the top argument is factored. The factoring in Example 2 is possible since the number is small. When the number is large, factoring may not always be feasible. It turns out that the Legendre symbol has a generalization, called the Jacobi symbol, that is even more versatile.
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I always wish for more vacation time and this year, it was granted. I literally had a longer vacation than my friends.
I had to stop for one year because of financial reasons. So during that time, I HELPED in our little store.
AS MUCH AS POSSIBLE, when there are events in UST – which is where all of my best friends attend school – I go there to participate. I AM A FORMER THOMASIAN STUDENT MYSELF.
So as of now, November 29, I am currently reading a mathematics module I got from academic clinic. Thank you, you’re a lifesaver!
I’m doing this to pass the PUPCET. If I pass their entrance examination, my first choice would be BS Accountancy, second is BS Psychology, and lastly BS Mathematics.
After successfully surviving what will probably be the darkest part of my school life, my first semester as a freshman, I WILL TRANSFER TO UNIVERSITY OF THE PHILIPPINES. I JUST HAVE TO MAINTAIN A 1.5 average GRADE.
Well, I hope the odds are in my favor!
]]>MindShift author, Deborah Kris, did a spectacular job. She asked thought provoking questions that encouraged me to continue to research mathematics and creativity, and to reflect again upon my beliefs and practice.
I loved immersing myself in the research and reflection. As I did, I came to believe even more deeply that creativity in thought and action increases the power and beauty of mathematics. I hope you enjoy the article, and that it enhances your thought and teaching practice.
Using Creativity to Boost Young Children’s Mathematical Thinking
By the way, there is a link in the article to my paper – Managing the Classroom for Creativity. If you haven’t read it yet, give it a look.
James, M. (2015) Managing the Classroom for Creativity. Creative Education, 6, 1032-1043. doi: 10.4236/ce.2015.610102.
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Fear not, for this is where we enter the wonderful world of statistics! While I don’t claim to be an expert on the subject, we can work together in sorting out this complex world of data with a few interesting little tidbits.
There are many things in nature which follow something of a logarithmic distribution. This includes things such as earthquake magnitudes and the sizes of moon craters, but also characterizes human driven behaviour, such as the populations of cities or the popularity of last names. This might all seem somewhat strange, given the expectation that all events are equally likely. What we must realize is that this is a false assumption. A natural disposition of animals, people, and to some extent well-designed machines, is to follow the path of least effort. If one path to a goal is much easier than another it therefore becomes preferential over the others. This initiates something of a feedback loop, in which the value of the preferential choice is only amplified. If you find one particular coffee shop is more convenient to get to every morning you are going to organise your mornings around that particular convenience, rather than forcing yourself to look for other options. This is what’s known as a preferential attachment process, where the number of times you visit a coffee shop is proportional to the amount of times you have already been.
These sorts of patterns are the kinds of things which are unearthed by statistical analysis and occur everywhere in nature and our daily lives. One particularly well known instance is Zipf’s Law. Popularised by American linguist George Kingsley Zipf, it states that in any language the second most frequently used word will be used approximately half as much as the most frequently used word, the third most frequently used word will be used a third as much, and so on. Put concisely, number of times any word will be used in a language is the number of times the most frequently used word is used, multiplied by one over its frequency rank. In the case of English, the most frequent word is ‘the’ (which I have managed to use about 35 times in this article). According to wordcount.org it is used twice as much as the word ‘of’, which seems to have dejectedly accepted its very far behind position of second place.
While the numbers aren’t exact, they definitely indicate a very interesting natural trend. Those who already have a lot find it easier to get more, while those who do not struggle to get out of a slump. This biological snowball effect underlies the commonly accepted (in some communities) Pareto Principle, suggesting that 20% of the input is responsible for or receives 80% of the results. While this is only a rule of thumb, it is telling that 20% of our vocabularies accounts for more than 80% of the words we say.
So let’s break the norm every now and again. We all may have certain tendencies to one activity or another, but it is that remaining 80% of our vocabularies which provides the spice of our conversations. Who knows, maybe we can shake things up a bit.
]]>The reason the sky is red is not one which is of overall great complexity it is just nice to stop and think about what is really happening. Perhaps we should start with why the sky is normally blue? It seems foolish to explain why the sky is red before we know why it was blue in the first place. And then why sunlight is a warm yellow. Then finish up with red. It all comes down to light. The elusive, mysterious light. Light has wave-particle duality, but for the purposes of this explanation think of light as travelling in waves. When light hits something, one of three things can happen:
Now the earth of course has an atmosphere which is what allows us to breathe this delicious air. On a very small scale, air contains big things. There are lots and lots of tiny particles whizzing around doing their thing – as gaseous particles like to do. Light is an arrogant character and wont be considerate of others. What light tends to do is take the shortest path possible and go that way. Well of course what this means is collision. Air-light collision. Of course air isn’t really one gas at all, it is made up of lots of different gasses the most important being nitrogen and oxygen. When light hits these particles it is sort of taken in by these particles and then re-scattered in a different way.
This analogy is by no means perfect but it helps me to understand things in my mind (I would encourage you to think of it in whatever way you understand it best). I think of this a little bit like a watering can. The water is the beam of light, and the head is the atom. When it hits the head of the watering can the water is sprinkled out. If I were to put a head on the can with a different profile – different holes or different spacing then the can would sprinkle differently. Light and particles are very much like this – the light is absorbed by the particle and then scattered.
Richard of York gave battle in vain. The colours of the rainbow. Everyone knows this – red, orange, yellow, green, blue, indigo, violet. Now the colours in the second half of this progression, the greeney blues are associated with short wavelengths and high frequencies. The particles which exist in the air scatter light in all colours but not in equal proportions. Infact it is basically in reverse – most violet, then indigo, then blue etc etc. So the reason the sky looks to be the colour which it is is a result of the blend of these scatter patterns.
Okay I think that should suffice nicely for blue light. Now all this blue light has been scattered. What that means is we see Roy on Earth. That is Red, orange and yellow. Think about it carefully – light travels the shortest distance when it is directly above us – 12pm. It has longest to travel when it is sunset – because the sun has moved. All that is happening is more of the OYGBIV is being scattered – and more of the red. Sunsets (or sunrises – the earth is a sphere) scatter of more and more of the OYGBIV. What is left? Red.
]]>“The traditional school often functions as a collection of independent contractors united by a common parking lot.” ~Robert Eaker~
After a move to a new school in September and, what could only be described as an interesting fall, I’m starting to feel a little more settled with my (not so) new surroundings. Over the past few months I’ve had the chance to focus on getting to know the school community, the students, families and staff and see, more or less, how the school ‘works’.
It is often noted that there is greater consistency in practice across schools than within a given school. During 24 years in public education, working at 10 schools in various capacities; teacher, mentor and administrator I’ve had the opportunity to observe this phenomenon first hand.
The key challenge many schools and school systems face is one of both complexity and diversity. With so many variables and influences to factor; people, context and resources, trying to enact a change initiative is akin to the iconic cat herding commercial from a few years back. Often, we are feel we need to respond to complex problems with complex solutions; but the more I think, read and reflect upon this, the more it occurs to me that these problems actually demand simple solutions and that require a focus on relationships and doing less, more effectively.
David Kirp, an American public policy researcher and author, wrote about this recently in his book Improbable Scholars. One of Kirp’s main assertions is that successful schools and districts avoid trendy, fancy or complex improvement strategies and instead focus on these three key areas:
Success, it turns out, is a matter investing more in the collaborative capacities of classroom teachers and less on the external factors and tools that we have come to rely upon in many of our schools.
For me the word that best describes a truly effective school is coherent. A coherent school is one where teachers direct their resources and focus towards the development of logical, well-organized, consistent and effective teaching practices across the school. As a school leader my task is to engage the professionals I’m working with to create this coherence. As an example, collaborative assessment of student learning is one area where digital tools can help immensely as we can use media tools to gather and analyse a wide range of authentic student work samples as our teachers make use of tools like Google Classroom and Dreambox to support both their classroom teaching and professional learning.
Though we may use new tools, technologies and strategies to accomplish this, it is not the intention to add ‘more’ to the work we do in schools but rather to reduce the use of ineffective or inefficient practices and establish the structures necessary for teachers to work together to create coherence and communicate in a meaningful manner with their students and families.
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Bob Weber Jr’s Slylock Fox for the 23rd of November maybe shouldn’t really be here. It’s just a puzzle game that depends on the reader remembering that two rectangles put against the other can be a rectangle again. It also requires deciding whether the frame of the artwork counts as one of the rectangles. The commenters at Comics Kingdom seem unsure whether to count squares as rectangles too. I don’t see any shapes that look more clearly like squares to me. But it’s late in the month and I haven’t had anything with visual appeal in these Reading the Comics installments in a while. Later we can wonder if “counting rectangles in a painting” is the most reasonable way a secret agent has to pass on a number. It reminds me of many, many puzzle mysteries Isaac Asimov wrote that were all about complicated ways secret agents could pass one bit of information on.
Ryan North’s Dinosaur Comics for the 23rd of November is a rerun from goodness knows when it first ran on Quantz.com. It features T Rex thinking about the Turing Test. The test, named for Alan Turing, says that while we may not know what exactly makes up an artificial intelligence, we will know it when we see it. That is the sort of confident ignorance that earned Socrates a living. (I joke. Actually, Socrates was a stonecutter. Who knew, besides the entire philosophy department?) But the idea seems hard to dispute. If we can converse with an entity in such a way that we can’t tell it isn’t human, then, what grounds do we have for saying it isn’t human?
T Rex has an idea that the philosophy department had long ago, of course. That’s to simply “be ready for any possible opening with a reasonable conclusion”. He calls this a matter of brute force. That is, sometimes, a reasonable way to solve problems. It’s got a long and honorable history of use in mathematics. The name suggests some disapproval; it sounds like the way you get a new washing machine through a too-small set of doors. But sometimes the easiest way to find an answer is to just try all the possible outcomes until you find the ones that work, or show that nothing can. If I want to know whether 319 is a prime number, I can try reasoning my way through it. Or I can divide it by all the prime numbers from 2 up to 17. (The square root of 319 is a bit under 18.) Or I could look it up in a table someone already made of the prime numbers less than 400. I know what’s easier, if I have a table already.
The problem with brute force — well, one problem — is that it can be longwinded. We have to break the problem down into each possible different case. Even if each case is easily disposed of, the number of different cases can grow far too fast to be manageable. The amount of working time required, and the amount of storage required, can easily become too much to deal with. Mathematicians, and computer scientists, have a couple approaches for this. One is getting bigger computers with more memory. We might consider this the brute force method to solving the limits of brute force methods.
Or we might try to reduce the number of possible cases, so that less work is needed. Perhaps we can find a line of reasoning that covers many cases. Working out specific cases, as brute force requires, can often give us a hint to what a general proof would look like. Or we can at least get a bunch of cases dealt with, even if we can’t get them all done.
Jim Unger’s Herman rerun for the 23rd of November turns confident ignorance into a running theme for this essay’s comic strips.
Eric Teitelbaum and Bill Teitelbaum’s Bottomliners for the 24th of November has a similar confient ignorance. This time it’s of the orders of magnitude that separate billions from trillions. I wanted to try passing off some line about how there can be contexts where it doesn’t much matter whether a billion or a trillion is at stake. But I can’t think of one that makes sense for the Man At The Business Company Office setting.
Reza Farazmand’s Poorly Drawn Lines for the 25th of November is built on the same confusion about the orders of magnitude that Bottomliners is. In this case it’s ants that aren’t sure about how big millions are, so their confusion seems more natural.
The ants are also engaged in a fun sort of recreational mathematics: can you estimate something from little information? You’ve done that right, typically, if you get the size of the number about right. That it should be millions rather than thousands or hundreds of millions; that there should be something like ten rather than ten thousand. These kinds of problems are often called Fermi Problems, after Enrico Fermi. This is the same person the Fermi Paradox is named after, but that’s a different problem. The Fermi Paradox asks if there are extraterrestrial aliens, why we don’t see evidence of them. A Fermi Problem is simpler. Its the iconic example is, “how many professional piano tuners are there in New York?” It’s easy to look up how big is the population of New York. It’s possible to estimate how many pianos there should be for a population that size. Then you can guess how often a piano needs tuning, and therefore, how many full-time piano tuners would be supported by that much piano-tuning demand. And there’s probably not many more professional piano tuners than there’s demand for. (Wikipedia uses Chicago as the example city for this, and asserts the population of Chicago to be nine million people. I will suppose this to be the Chicago metropolitan region, but that still seems high. Wikipedia says that is the rough population of the Chicago metropolitan area, but it’s got a vested interest in saying so.)
Mark Anderson’s Andertoons finally appears on the 27th. Here we combine the rational division of labor with resisting mathematics problems.
]]>Host (Tom Evans)
The host of the day was Tom Evans. He opened up the session with two puzzling questions which the audience were asked to attempt during the breaks between speakers. The problems are below:
Happy birthday Fermat’s last theorem (Simon Singh)
The opener was Simon Singh talking about Andrew Wiles and Fermat’s last theorem. He opened up the talk with the start of the BBC Horizon episode (see here) he directed on the same topic. In the video Andrew Wiles expresses an analogy for mathematics whereby it can, for a time, feel like you are walking around a dark house stumbling into things all the time. Then, one day, the light switches on and suddenly you see. Wiles becomes very emotional when talking about Fermat’s last theorem, which as Simon explained in his talk is due to the sheer length of time and determination spent solving a problem which had become a childhood dream.
Simon gave us the problem and spent a large part of his talk discussing the back story and some of the people who tried to provide a proof for Fermat’s last theorem across some 300 years. He recommended purchasing his book for further reading (which I did!). If you are wondering the theorem states the following:
No three positive integers and can satisfy the equation for any integer value of greater than two.
There were also recommendations at the end, for further enriching mathematics, to take a look at Numberphile, Vi Hart and Martin Gardner. Do take a look at all if you haven’t yet!
How big is infinity? (Chris Good)
Chris focused a large part of his talk on infinity around the work of Georg Cantor. He opened the talk by asking a classic question; ‘Which is bigger 1 or 0.99999999999…?’ He used this to clarify that every number between 0 and 1 can be represented as a unique non-terminating decimal (due to one-to-one correspondence) which was introduced by Cantor first in 1874. As a result of much of Cantor’s work there was an implication that there are an infinite number of infinities (i.e. I can always choose a bigger infinity than the one you just chose!). Chris concluded his talk by discussing transcendental numbers.
Exam technique (optional session with Colin Beveridge)
I really liked the Venn diagram Colin used at the beginning of his short talk about exam technique to describe the ‘ideal’ experience maths A level students should have (i.e. extended subject knowledge and support with exam technique). One of the things I really liked, and have already used, is the ‘error log’ idea. This is where pupils are asked to write down their mistakes, take note of where they went wrong or got stuck and then include the steps they need/needed to take to rectify the problem. Colin had some really valuable things to say so I am glad I stayed to listen to this (even if my students didn’t!).
7 things you need to know about prime numbers (Vicky Neale)
Fact 1: 1 is not a prime number.
Fact 2: 2 is a prime number (and the only even prime!).
Fact 3: (Theorem) There are infinitely many prime numbers.
This was proved by both Euclid (c.300BC) and Euler (18th century). This theorem of the week blog post gives some detail on both.
Fact 4: (Fundamental theorem of arithmetic) Every integer greater than 1 can be expressed as a product of primes in an essentially unique way.
Fact 5: Every prime is one more or one less than a multiple of six if .
Therefore, could you prove the following statement (imagine delivering your explanation to a year 9 student)?
If is a prime number greater than 3, then is of the form , where is a natural number.
Fact 6: Let denote the number of primes less than or equal to . For example, or .
(Prime number theorem) Then where here is the number theory notation for .
Fact 7: (Twin prime conjecture) There are infinitely many primes such that is also prime.
This conjecture is yet to be proved, but there have been many recent developments!
Exit question:
If 3 and 5 are twin primes then let us call 3, 5 and 7 a prime triple. Are there any more prime triples?
Geometry and the art of optimisation (Richard Elwes)
Richard opened his talk by discussing how much of his work on optimisation is applicable in everyday life (e.g. traffic lights and train timetabling). He discussed an example of a toy factory where we needed to consider how to maximise the profit from the production line. The calculations made led to the consideration of the feasible set of values which we then needed to consider maximising. Richard then introduced us to the simplex algorithm found by George Dantzig in 1947. He concluded with the Hirsch conjecture to which a counter example was found in 2010.
Cryptography (Keith Martin)
Keith was a very engaging and entertaining speaker, and whilst his talk didn’t contain much ‘maths’ he was able to discuss the importance of cryptography. Some elements of security are ‘lost’ in the cyber world and we need to ensure our information holds its confidentiality, integrity and authentication. Cryptography is a tool used and built by mathematicians to help with encryption and security in cyberspace. Keith discussed ciphers, our data integrity (using ISBN codes as an example) and authentication:
On the internet no one knows if you are a dog.
Keith recommended taking a look at cryptool.org if you are interested in looking further into cryptography. His alternative suggestions were Piper and Murphy’s Introduction to Cryptography book or Simon Singh’s The Code Book.
]]>When we see things like e^iπ = -1 or the Banach Tarski Paradox, we go ‘blah blah blah blah blah x equals blah blah.. I can’t do this anymore..’ as it takes time for us to wrap our heads around such ideas. Often, it is not when we understand new ideas that we learn something but when we alter old ideas. As we progress, we learn so many things about mathematics that we were thinking about in the wrong way. For instance, a better way to think about addition and multiplication is to picture a line with numbers on them. Imaginary numbers then become concepts that we understand with ease since we then define them to be results of rotational operations. But not everything do we have such explanations for. Getting an intuitive sense of advanced math is a task that no one has done. We need to remember that negative numbers came into existence after thousands of years. (At that time, people didn’t need negative numbers. You wouldn’t need to describe buying two apples and selling three!)
Infinite sums get me. After pondering over sums for some time, I just accept that it is the way nature works. I know there might be some other efficient way to think about such things but I give up for now.
Often, people tell me that Math is nonsensical and that it is useless in real life. I’m here to tell you that this is completely wrong. Mathematics is a language we use to describe the Universe. I’ll just tell you something marvelous. Any circle you take, the circumference when divided by the radius always results in π. It doesn’t matter where the circle is and who is holding it. The ratio is always π. It is not just fascinating but also inspiring. It doesn’t matter who draws the circle and who observes the circle, the result is the same for every person. We are one. United by logic and reason, yet separated by ideology. Math isn’t biased. It is a language for all. One can’t argue with that fact. Counting is a skill we have mastered. And every person can count. That is where Math was born. Math as I would like to call it, is the ‘best invention’ we humans have ever made. Mathematics might not be a part of the universe. Math might just be our best way of viewing things as they are. The physical world works just like math dictates. Math does break down at certain instances like when we try to describe the singularity of a black hole or when we try to describe the big bang. Maybe it is because math is in its initial stages or maybe because there is some piece of the puzzle missing. Or maybe because something better than math is available. We clearly can’t figure that out. Math is our only hope for now and it is thrilling to know that it works for almost every single thing we know.
At school, we learn math in a different manner. We take in what the books give us instead of using various other concepts to make a little more sense out of some ideas in the subject. People usually have this idea that only people who are ‘gifted’ will be able to do math and that is not the case. I’d repeat this from my previous article: Most people treat Math as a set of rules. It is true that Math follows a set of a rules but that doesn’t restrict one from extending its boundaries. Math is a language. And learning any language takes effort. If you are going to learn a new language, it would take you years of practice before you can fluently speak. And living in an environment where everyone else knows the language speeds up your learning. Imagine that level of exposure to Math. Imagine where we could be! Math is rewarding!
Making sense of mathematics is difficult. It isn’t unattainable. A fresh mind ready to correct oneself is all it takes to understand math. Learning takes effort but basic knowledge is a necessity for progress in any field.
So let’s get ready to be wrong and learn. Anyone can learn and understand anything.
Keep looking up!
Cheers!
(I’d like you to know that I don’t have a proper background in mathematics and I have just begun my journey.)
]]>Make certain you comprehend the guidelines and expectations associated with trainer. Often these guidelines is going to be found in the syllabus and quite often these are typically published individually. They might be in an announcement or course message. Discover them and read them. In a lot of colleges, trainers are permitted to make their own guidelines regarding later part of the work, grading, participation, along with other important concerns. Only because you had a course from just one trainer finally month and you comprehended his or her guidelines doesn’t suggest the present trainer will have the same guidelines. In a few cases may very well perhaps not see any teacher guidelines. If it takes place and you’ve got read most of the notices and course emails, after that perform perhaps not be scared to ask the trainer any specific concerns you’ve probably regarding what is expected in the course.
Particularly, you want to understand the attendance requirements, the publishing or involvement requirements, and how your work will likely be scored. In many cases, students have to upload a note on a particular wide range of times to be counted present for the week. In many cases, students must react to prompts when you look at the class with articles which are counted for a participation level. The sheer wide range of articles which have to be made can vary by teacher or program. Written projects could have scoring rubrics that offer extra information concerning needs when it comes to assignment to be acceptable.
Take a look at class and find the materials, exactly where in actuality the tasks are detailed, and just exactly how to distribute the assignments. Some internet based classrooms are really easy to navigate as well as others take a whilst to discover the structure. You will definitely most likely have actually links to viewing materials such as a text. You may possibly have links to many other sources. You might discover questions to which you may be supposed to respond. There may most likely be once a week projects that may also have specific due times. Make sure you know just what those dates are. It’s wise to mark all of them on your own calendar.
If it is maybe not apparent, discover how you distribute written projects. Do maybe not wait until the assignment is because of to ask simple tips to distribute. The trainer might not be readily available at the final moment for questions, so think in advance. When you believe you’ve got posted a project, check to see if there is a method you can easily validate everything you have submitted. All too usually, students publish a rough draft that has been conserved utilizing the title of this project rather than the last task. In a few instances, pupils have actually posted totally empty assignments. If there is an option to check your assignment to see exactly what you presented, do so.
Ultimately, you need to understand your progress into the training course. You will many likely have some type of quality guide in which you can keep up with your levels. You have grades for composed work and levels for involvement. It is advisable to view your levels which are published and then render sure they concur using the grades you obtained through the trainer on returned work. Mistakes can be built in recording the levels and it is a great deal far much better to get those errors before the end of course. If grades are not published weekly, communicate with all the teacher concerning how you will determine if you may be fulfilling the requirements. Whenever the class has ended, trainers frequently share the last class so that you could have a look at before in fact publishing that quality just in situation you believe the grade is wrong. When you do believe there’s an issue, tell the teacher instantly. It is a lot easier to repair a class before it offers been completed than following the final distribution.
Guaranteeing your very own success in your internet based course is essential. While the instructor has an obligation to instruct the content material in a method you can discover, you’ve got the obligation to ensure that you understand the expectations and they are after all of them. Each class is a brand new one in addition to expectations can be various. It is your job as a pupil to understand those expectations and follow all of them, asking concerns when you are not certain.
]]>Sol. Let a be any integer. Let q be the quotient and r be the remainder, when a is divided by 4.
By Euclid’s division algorithm, we have : a = 4q + r, where r = 0, 1, 2, 3
For r = 0, a = 4q and a2 = 16q2 = 4(4q2), which is of the form 4m, where m = 4q2.
For r = 1, a = 4q + 1 and a2 = 16q2 + 8q + 1 = 4(4q2 + 2q) + 1, which is of the form 4m + 1, where m = 4q2 + 2q.
For r = 2, a = 4q + 2 and a2 = 16q2 + 16q + 4 = 4(4q2 + 4q + 1), which is of the form 4m, where m = 4q2 + 4q + 1.
For r = 3, a = 4q + 3 and a2 = 16q2 + 24q + 9 = 4(4q2 + 6q + 2) + 1, which is of the form 4m + 1, where m = 4q2 + 6q + 2.
Thus, a2 is either of the form 4m or 4m + 1 for some integer m.
]]>Judy C – Librarian
TDSB Professional Library
]]>As a teacher, I have had variants of this conversation many times. The specific details, however, are fictional, for this changes, somewhat, each time it happens.
…At least I try. Also, sometimes, the educational outcome is better than in this fictionalized example.
[Image source: http://www.decorationnako.tk/birthday-cake/]
]]>The Standard model and its Lagrangian form a vast topic . I will attempt to give relevant and accurate information about it.
The story of the Standard Model started in the 1960s with the elaboration of the theory of quarks and leptons , and continued for about five decades until the discovery of the Higgs boson in 2012.
For a timeline of the history of the Standard Model see the Modern Particle Theory timeline .
The formulation of the Lagrangian of the Standard Model with its different terms and parts mirrored the theoretical and experimental advances associated with particle physics and with the Standard Model.
The Lagrangian function or Lagrangian formalism is an important tool used to depict many physical systems and used in Quantum Field Theory . It has the action principle at its basis .
In simple cases the Lagrangian essentially expresses the difference between the kinetic energy and the potential energy of a system .
The Standard Model of particle physics describes and explains the interactions between the essential components and the fundamental particles of matter , under the effect of the four fundamental forces: the electromagnetic force , the gravitational force , the strong nuclear force , and the weak (nuclear) force.
However , the Standard model is mainly a theory about three fundamental interactions , it does not fully include or explain gravitation .
The Standard model (or SM) is a gauge theory representing fundamental interactions as changes in a Lagrangian function of quantum fields. It depicts spinless , spin-(1/2) and spin-1 fields interacting with one another in a way governed by the Lagrangian which is unchanged by Lorentz transformations.
The Lagrangian density or simply Lagrangian of the Standard Model contains kinetic terms , coupling and interaction terms (electroweak and quantum chromodynamics sectors) related to the gauge symmetries of the force carriers (i.e. of the elementary and fundamental particles which carry the four fundamental interactions) , mass terms , and the Higgs mechanism term .
Explicitly , the parts forming the entire Lagrangian generally consist of :
Free fields : massive vector bosons , photons , and leptons.
Fermion fields describing matter.
The Lepton-boson interaction.
Third-order and fourth order interactions of vector bosons.
The Higgs section.
Leptons are the elementary particles not taking part in strong interactions.
All leptons are fermions. They include the electron , muon , and tauon , and the electron neutrino , muon neutrino , and tauon neutrino.
All leptons are color singlets , and all quarks are color triplets.
In the Standard model , the Higgs mechanism provides an explanation for the generation of the masses of the gauge bosons via electroweak symmetry breaking.
Different reference works , books , e-books or textbooks use different or slightly different notations and symbols to describe or designate the entities and terms within the Lagrangian of the SM .
Below is a detailed image of the Lagrangian of the Standard Model (Source: http://einstein-schrodinger.com/Standard_Model.pdf ).
However I have rearranged it and modified it with the help of Photoshop to make it look more presentable and more readable.
The Lagrangian function in the Standard Model , as in other gauge theories , is a function of the field variables and of their derivatives.
is the gauge field strength of the strong SU(3) gauge field.
Gluons are the eight spin-one particles associated with SU(3).
A particle which couples to the gluons and transforms under SU(3) is called ‘colored’ or ‘carrying color’.
Gluons and quarks are confined in hadrons.
is the gauge field strength of the weak isospin SU(2) gauge field .
The field strength tensor is given by :
where is the electroweak coupling constant , a dimensionless parameter.
The charged and bosons and the neutral Z boson represent the quanta of the weak interaction fields between fermions , they were discovered in 1983 .
is the gauge field strength of the weak hypercharge U(1) gauge field.
The field strength tensor is given by :
In the Standard model , electrons and the other fermions are depicted by spinor fields .
The group U(1) is the set of one-dimensional unitary complex matrices .
U(1) represents the symmetry of a circle unchanged by rotations in a plane.
SU(2) is called ‘the special unitary group of rank two’. It is a non commutative group related to SO(3) , the sphere symmetry in 3 dimensions.
SU(2) is the set of two-dimensional complex unitary matrices with unit determinant.
SU(3) , the special unitary group of rank three , is used in quantum chromodynamics (QCD) .
SU(3) is the set of three-dimensional complex unitary matrices with determinant equal to 1 .
The natural representation of SU(3) is that of 3×3 matrices acting on complex 3D vectors.
The generators of the group SU(3) are eight 3×3 , linearly independent , Hermitian , traceless matrices called the Gell-Mann matrices . These generators can be created from Pauli spin matrices (which are used with the group SU(2) ) .
The SM Lagrangian displays invariance under SU(3) gauge transformations for strong interactions , and under SU(2)xU(1) gauge transformations for electroweak interactions.
The electromagnetic group is not directly the U(1) weak hypercharge group component of the standard model gauge group. The electric charge is not one of the basic charges carried by particles under the unitary product group SU(3)xSU(2)xU(1) , it is a derived quantity.
All the masses vanish in the absence of the Lagrangian term related to the Higgs , due to the invariance of SU(3)xSU(2)xU(1) .
In some texts the gauged symmetry group of the SM is written with subscripts such as:
In the notation above , the subscript ‘c’ denotes color.
The subscript ‘L’ denotes left-handed fermions.
The subscript ‘Y ‘ distinguishes the group related to the quantum number of weak hypercharge , expressed by the letter Y , from the group associated with ordinary electric charge, expressed by Q .
denotes the electromagnetic group.
The Higgs field in the Standard model is a complex scalar doublet. It is generally represented by :
In the image of the SM Lagrangian above , the Higgs field has the form
The field h(x) is real .
In the SM Lagrangian image above , is equal to v .
As an additional note , the equation of the Lagrangian is usually made of a definite number of terms and Lagrangians.
In order to make such an equation look less like a big behemoth and make it more compact , it would be simpler to view it or write it first as the sum of Lagrangians :
or equivalently :
Then each Lagrangian in the equation could be expanded and explained.
Some helpful resources about the Standard Model and its Lagrangian :
Standard Model
The Standard Model of Particle Physics
Standard Model (mathematical formulation)
http://arxiv.org/pdf/hep-ph/0304186v1.pdf
Gauge Theory of Weak Interactions: Walter Greiner, Berndt MÃ¼ller: 9783540878421: Amazon.com: Books
The Standard Model: A Primer: Cliff Burgess, Guy Moore: 9781107404267: Amazon.com: Books
Here is also a link to one of the important papers in the history of the Standard Model written in 1967 by Weinberg and entitled ‘A Model of Leptons’ :
http://physics.princeton.edu/~mcdonald/examples/EP/weinberg_prl_19_1264_67.pdf
Best wishes
Lis Goodwin, your voivce coach
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