Counting bears

We can think of a number $n$ as a collection of $n$ bears.

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Even and odd numbers

An even number can be divided into two equal parts, while an odd number cannot.

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Definition (Even and odd numbers):


Let $n$ and $k$ be integers. We say:

  • $n$ is even if there exists $k$ such that $n = 2k.$
  • $n$ is odd if there exists $k$ such that $n = 2k + 1.$

Odd numbers have a middle

Three bears have a middle; four don't.1 However, if three bears form a triangle, it's not clear which is the middle.2 We can at least say that if a set $S$ has an odd number of elements that are arranged "in a line", then $S$ has a middle element. Such an ordering is called a linear order or a total order.3

Prime numbers

A number is composite if it can be rearranged into a rectangle; if not, it is prime.4 However, 1 is neither prime nor composite. The first prime numbers are 2, 3, 5, 7, 11, ....

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See also

Notes

#math

  1. Or an even number has two middles. ↩

  2. Though there is a concept of the geometric median. The geometric median of the three vertices of a triangle is called the Fermat point. ↩

  3. Examples of totally ordered sets: tuples ordered by index, the set of natural numbers $(\mathbb{N}, \le)$, and the set of real numbers $(\mathbb{R}, \le)$. ↩

  4. Or a prime number can only be rearranged into a rectangle with one side 1. ↩