15 votes

How to explain what is wrong in this "proof" that $\sqrt N$ must be irrational?

This is indeed tricky, and it seems to me the most effective way (in far more general, similar situations) is to show them the problem would be to have them apply their method to another, close ...
Benoît Kloeckner's user avatar
14 votes
Accepted

Good way to explain fundamental theorem of arithmetic?

The way I have explained the fundamental theorem of arithmetic in the past is by establishing what it means to be prime (has exactly two positive divisors) and then having students construct factor ...
Benjamin Dickman's user avatar
11 votes

Number theory for self-study students: books and computer languages

I would recommend Python combined with SageMath, as already recommended by Joseph O'Rourke, or rather SageMath and Python comes naturally. Python is a modern, and widely used, interpreted language (...
quid's user avatar
  • 7,682
11 votes

Greatest common divisor applications

Another classic is the following: A rectangular floor measures $300 \text{ cm} \times 195 \text{ cm}$. What is the largest square tiles that can be used to cover the floor exactly?
Dag Oskar Madsen's user avatar
11 votes

Good way to explain fundamental theorem of arithmetic?

To really understand why the integers $\mathbb{Z}$ have unique prime factorization, it helps to understand how unique prime factorization can fail in other settings. For example, $$(2 + \sqrt{10}) \...
Daniel Hast's user avatar
  • 4,893
11 votes

How to explain what is wrong in this "proof" that $\sqrt N$ must be irrational?

What's wrong is that most of the justification is missing. Why can $N = p^2/q^2$ only happen if $q^2 = 1$? This can be justified using the fundamental theorem of arithmetic (which states that integers ...
Daniel Hast's user avatar
  • 4,893
10 votes

Could students learn a lot more from school if they're only taught number theory until way later?

The way I understand the question is: If students are not taught fractions, but instead formal deductive proofs of properties of natural numbers, would they learn mathematics better? I find it ...
Tommi's user avatar
  • 7,039
10 votes

Is there a numerical base that is in any way “better” for simple mathematical calculations than others?

Clearly there is no historical data that addresses this question I want to know if there are any numerical bases that are notably well-suited for humans to learn and use at an elementary or ...
Ethan Bolker's user avatar
9 votes

When two equivalent algebraic statements have two "different" meanings

The two statements aren't literally "the same", because as the student observed, they say different things. However, they are logically equivalent: each implies the other. (Similarly, "4/2" isn't ...
Daniel Hast's user avatar
  • 4,893
9 votes
Accepted

Should Euclid's algorithm be taught as rigid or flexible?

There is some value beyond the algorithm to insist on the fact that quotient and remainder in an Euclidean division are uniquely determined as soon as one settles on some convention on the remainder....
quid's user avatar
  • 7,682
9 votes

Greatest common divisor applications

Going back to Euclid, I have found questions such as `Given a large supply of rods of length $15$ and $21$, what lengths can be measured?' can appeal to students. This also motivates the result $\gcd(...
Mark Wildon's user avatar
9 votes

Analogy for multiplying modulo N

Maybe the issue is that if the values you're multiplying have units, then the result of multiplying will have the product of those units. Therefore, your result can't really be equivalent to a value ...
Nick C's user avatar
  • 9,406
9 votes
Accepted

Geometrical interpretation of the identity $\operatorname{lcm}(a,b) \operatorname{gcf}(a,b) = ab$

I don't see anything wrong with what you were trying to say. In fact, I think that is a good way of saying it. Let's draw some pictures with some concrete numbers. Let's take $a=6$ and $b=9$. Then we ...
Brian Moths's user avatar
8 votes

Greatest common divisor applications

Some nice geometrical applications arise in the analysis of periodical curves such as Roulettes (Spirograph curves), Star Polygons, etc. Concrete experience with implementations in toys like ...
Bill Dubuque's user avatar
  • 1,038
8 votes

What is most motivating way to introduce Fermat's Little Theorem

I wrote an article for NCTM's grades 8-14 journal, The Mathematics Teacher, which is about building towards ${\rm F}\ell{\rm T}$ by thinking through ways in which the following statement can be ...
Benjamin Dickman's user avatar
8 votes
Accepted

What is the name of this discipline in mathematics education?

What is the name of this subdiscipline in math education? In one of the comments, Dave L Renfro has a reference to Piaget, whose work was primarily done with younger schoolchildren. With respect to ...
Benjamin Dickman's user avatar
8 votes
Accepted

Source material to study number theory?

I've heard very good reviews of the 2017 book, "An Illustrated Theory of Numbers" by MH Weissman. The book's main site is here; a write-up, along with some reviews, by the American ...
Benjamin Dickman's user avatar
8 votes

Introductory book or other resource on $p$-adic numbers/number theory/analysis

A few recommendations: Fernando Q. Gouvêa's $p$-adic numbers: An Introduction. This text gives a good introduction to the $p$-adic number system and the properties of the space of $p$-adic numbers, ...
Xander Henderson's user avatar
  • 7,697
8 votes
Accepted

What is the terminology for integers with the same oddness or evenness?

Expanding on a comment by @TomKern: The word you're looking for is parity. You can see it in most standard English dictionaries, such as Merriam-Webster, here: 4 a: the property of an integer with ...
Daniel R. Collins's user avatar
7 votes

Should Euclid's algorithm be taught as rigid or flexible?

(Summary: I would suggest exploring it flexibly, but ensuring students also know the "rigid" version.) Rather than directly addressing Euclid's algorithm for the $\gcd$ of two whole numbers, I ...
Benjamin Dickman's user avatar
7 votes

Number theory for self-study students: books and computer languages

What computer languages might one recommend for, say, investigations in number theory? I find Mathematica ideal, e.g.: "Mod sequences that seem to become constant; and the number 316" "Does 53 ...
Joseph O'Rourke's user avatar
7 votes

How to introduce Wilson's Theorem?

Wilson's Theorem is very powerful for proving things related to quadratic residues, and I think also in general a good theoretical tool. Motivating it by just saying "hey let's multiply everything ...
kcrisman's user avatar
  • 5,980
6 votes

Greatest common divisor applications

Here's a word problem for the greatest common divisor: 12 boys and 15 girls are to march in a parade. The organizer wants them to march in rows, with each row having the same number of children, ...
JRN's user avatar
  • 10.8k
6 votes

Making modular arithmetic interesting for school kids

I would say probably not; the work (algebra mod n) and the payoff (those digits come out in a neat pattern) are not in a good ratio. If your goal is to get students to do some of the work (modular ...
Chris Cunningham's user avatar
6 votes

Geometrical interpretation of the identity $\operatorname{lcm}(a,b) \operatorname{gcf}(a,b) = ab$

The OP describes a geometrical illustration using the fact that the GCD is the side of the greatest square that tiles a $A \times B$ rectangle. I don't understand why the following is not a "...
user1815's user avatar
  • 5,475
6 votes
Accepted

How can I explain construction of the Bézout's identity to my kid?

I will answer with an example. I seek the Bezout coefficients for 99 and 707. First I execute the Euclidean algorithm: $$ \begin{align*} 707 &= 7 \cdot 99+14\\ 99 &= 7 \cdot 14+ 1 \end{...
Steven Gubkin's user avatar
5 votes

Explaining difference between natural numbers, integers, rationals, reals, complex numbers, Gaussian integers

I am not sure whether this really answers your question, but I could think of the following strategy of introducing these sets of numbers. It is not based on any kind of research and just based on ...
Christian's user avatar
  • 544
5 votes

Planning high school workshop on Goldbach Conjecture

I like to ask my probability students the question: If you pick an integer between 2 and 100 uniformly at random, what's the probability that it's the average of two (not necessarily different) ...
Pat Devlin's user avatar
5 votes

How to explain what is wrong in this "proof" that $\sqrt N$ must be irrational?

After writing for the students where they have gone wrong with their given proofs, I realized that it was just me thinking they are wrong, just because I was accustomed to seeing the fundamental ...
Amir Asghari's user avatar
  • 4,428
5 votes

What is most motivating way to introduce Fermat's Little Theorem

Look at patterns in decimal expansions: what is the period of the repeating decimal of $1/n$? From numerical data, the period is at most $n-1$, and you only get equality when $n = p$ is prime (but not ...
KCd's user avatar
  • 3,456

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