# Tag Info

### 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 ...
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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 ...
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### 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?
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### 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 ...
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### 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 (...
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### 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 ...
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### 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) ...
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### 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 ...
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### 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 ...
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