54 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

"Because we said so" is a bit of a conversation closer, I agree. But "Because some people agreed a long time ago to define it that way so we could have conversations where we all understood each ...
Matthew Daly's user avatar
  • 5,599
34 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

There was a multiplication table posted on the wall. Like this \begin{alignat}4 1 &\quad 2 &\quad 3 &\quad 4 &\quad\cdots\\ 2 &\quad 4 &\quad 6 &\quad 8 &\quad\cdots\\ ...
Gerald Edgar's user avatar
  • 7,499
11 votes

How can we explain intuitively the convergence and divergence of these two series?

Look at a simpler example first: $(1.000000000001)^n$ compared to $0.9999999999^n$. Do they accept that the first sequence tends to $\infty$ and the second to $0$ even though it would take quite a ...
KCd's user avatar
  • 3,456
10 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

FYI: here's some pro and con: http://primefan.tripod.com/Prime1ProCon.html One was originally considered prime. It is prime with the most convenient ("natural") definition. It got excluded from ...
guest's user avatar
  • 109
8 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

How should one talk about the question of 1 or 0 being prime ... with primary or middle school children? Depending on what you did before you will have an easy or a hard task: If the children were ...
Martin Rosenau's user avatar
6 votes

How can we explain intuitively the convergence and divergence of these two series?

For me, the intuition just comes from the integral test (which is itself intuitive since a series is just a Riemann sum of rectangles with unit width). The $n$th prime is asymptotically $n \ln n$ (...
Justin Skycak's user avatar
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

Explaining why (or whether) zero and one are prime, composite or neither to younger children

A good way to lead to the uniqueness of prime factorization and the convention that $1$ is not a prime is to build factor trees (that's common in elementary school these days in fourth grade, ...
Ethan Bolker's user avatar
5 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

This one is really very simple. First, tell them what a prime number is: A prime number has exactly two different factors. (If they don't know what factors are, and they ask about primes, the ...
skc's user avatar
  • 51
4 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

I'm kind of restating what other answers have said, but I wanted to practice expressing it in the clearest, most concise way I could think of. (Coincidentally, this came up with my partner tonight, so ...
Daniel R. Collins's user avatar
4 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

If you build each number n using n square blocks in rectangular configurations, there are multiple configurations for each composite number. (4 is 4 by 1 or 2 by 2.) The primes are the ones that can ...
Sue VanHattum's user avatar
  • 20.2k
3 votes

How can we explain intuitively the convergence and divergence of these two series?

Intuitively, to me, it means that if you take the positive number line, put a blue dot at every prime, and a red dot on all the the numbers of the form $n^{1.000000000001}$, then eventually, very far ...
Arthur's user avatar
  • 401
3 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

We don't need the full FTA upfront if we limit our discussion for the moment to obvious examples of what's necessary for a factorization to be unique. The FTA provides analogous sufficiency conditions ...
J.G.'s user avatar
  • 521
3 votes

Missing Step in Most Proofs of the Irrationality of $\sqrt{2}$

You are correct in that the coprimality of $a$ and $b$ is not used in its full strength. It is adequate to merely assume that they are not both even. But since people are so used to reducing a ...
user52817's user avatar
  • 10.6k
2 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

I would start by showing them, on paper, what they already know - that in the context of multiplication the number 1 is useless. It is the identity function. It simply reflects the original number. It ...
CramerTV's user avatar
  • 129
1 vote

How can we explain intuitively the convergence and divergence of these two series?

Consider the fact that $\sum_{n=1}^\infty n^x$ converges if $x<0$, diverges if $x>0$. Clearly the transition from just a little bit negative to just a little bit positive makes a big change to ...
Simon Crase's user avatar
1 vote

Explaining why (or whether) zero and one are prime, composite or neither to younger children

The problem here is defining primes in their own right rather than defining them in terms of factorization. Start with a number like 30. Writing 30 = 2 x 15 tells us something new; writing 30 = 1 × ...
Daniel McLaury's user avatar
1 vote

Explaining why (or whether) zero and one are prime, composite or neither to younger children

\begin{align} & \begin{array}{cccccccccccccccccccc} & & & & & & & & & 840 \\[12pt] = {} & & & & & 28 & & & & \times & &...
Michael Hardy's user avatar
1 vote

Explaining why (or whether) zero and one are prime, composite or neither to younger children

When I was at school, after being taught about integer division, I was told: "A natural number is prime if it has exactly 4 integer divisors" Then 2 is prime, as it can be divided by -2, -1, 1, ...
Mefitico's user avatar
  • 349
1 vote

Explaining why (or whether) zero and one are prime, composite or neither to younger children

(If your context does not include negative numbers, turn all the negatives below positive. This almost won't change the discussion.) Everything divides zero, so zero can't be prime. $0 \cdot 7 = 0$ ...
Eric Towers's user avatar
1 vote

Planning high school workshop on Goldbach Conjecture

Too top down and too proofy/hard in places. Do something more experiential and experimental. "Explain the conjecture, provide some simple examples to begin" Nope, run an exercise and have ...
guest's user avatar
  • 1,828
1 vote

A Non-Unique Factorization of Integers!

In addition to the $\sqrt{-5}$ example, there is also the $R=\mathbb Z[\sqrt{-3}]$ example, which fails to be unique factorization for the "secret" reason that it is not integrally closed: we should ...
paul garrett's user avatar
  • 14.3k

Only top scored, non community-wiki answers of a minimum length are eligible