Odd Evens and Other Things: A Small Dive Into Basic Number Theory
by Alexander Hwang
Let’s start with the basics. The things you would have learned in your first years of math: evenness and oddness. Walk up to a kid, give her a number, and she could probably tell you if that number was even or odd. You might say “seven,” “twenty-three,” or “eighteen.” Nothing difficult. Simple stuff. If you wanted to make it even harder, you’d step it up a notch. “One thousand three hundred thirty-seven.” Her mind implodes; she didn’t even know there were numbers bigger than one hundred. However, she still gets it. Then you decide to throw her a curve ball. “Zero,” you say. She’s stumped.
Zero is amazing, the thing that stands for there being nothing. One of the initial traces of zero’s progenitors can be found in the records of the Babylonians, where they used marks to indicate if a placeholder was empty. So instead of writing down that they had 110 sheep, they wrote something like 11- sheep. The Indians presented the next major step for our empty friend. At around 650 AD, famous mathematician Brahmagupta started to use zero, represented by dots, in arithmetic. Fast forward a few centuries and you’ll see zero pass through the hands of the Arabian mathematicians (the ones who made our numerals), Fibonacci, Descartes (with his Cartesian plane), Newton and Leibniz (with their calculus), and then to us.
Now, I digress. Let’s return to the problem at hand. Is zero even or odd? Generally, if we are asked if a number is even or odd, or brains first test if it’s even. If it’s not even, then it must be odd. So is zero even, or is there a third category especially for it?
The definition of an even number differs from place to place, but here are two variations: an even number is a number that is divisible by two; that is, a number that returns an integer when divided by two (6/2 returns the integer 3, so it must be even). Alternatively, an even number is a number that is a multiple of two, or, in other words, the product of an integer and the number two (6 = 2*x, where x is 3).
If you think about it now, zero should be even (it is). If you divide zero by two, you get zero back. Zero is an integer. Thus, zero is divisible by two. Similarly, if you multiply two by zero, you get zero.
If you need more proof that zero is even, consider this little rule: “The sum of two even numbers is even.” I think we can all testify to the truth of that statement. When you add 4 and 8, you get 12. When you add -16 and 24, you get 8. When you add -2 and 2, you get 0.
Speaking of the number two, we all know that it’s even, but did you know that it’s a prime number? Two is the only even prime. What’s a prime? Well, a prime is a number that can only be divided by exactly two integers: one and itself. Two fits this description; it’s divisible only by itself and one.
While we’re still talking about primes, let’s consider the number one, the magic number, the conqueror of primes, the one to divide them all. Well, one isn’t a prime. It cannot be written as a product of exactly itself and one. This is because both itself and one are the exact same. Usually, if a number isn’t prime, it’s composite (a composite number is the opposite of a prime, a number that can be written as a product of two numbers other than itself). However, one isn’t a composite number either, because it can only be written as a product of itself. The number one is actually considered a unit (a number whose reciprocal is also a whole number). Shocking, I know.
The fact that one isn’t a prime number is key to the Fundamental Theorem of Arithmetic, which states that “every whole number greater than one is a prime, or is a unique product of primes.” This means that 13 can be written as 1*13, and 18 can only be written as 2*3*3 (order doesn’t matter here). If the number one were a prime, it would disprove the theorem because numbers would have more than one way to be written as a product of primes. Using the same example, 13 could be written as 1*13 or as 1*1*13.
Going back, zero is also similar to the number one when talking about primes. Zero isn’t a prime number nor a composite number. It’s not prime because it can’t be divided by itself, and it’s not composite because it can’t be written as a product of two numbers other than itself. To get zero by multiplication, you always have to use zero (zero is called a zero-divisor, a number that gives you zero when you multiply it with any nonzero number).
Now we’ve arrived at a full circle: we started with zero and ended with zero. A prime is divisible by one, but one isn’t a prime. Two is an even but also a prime. Zero is an even, but isn’t a prime number or a composite number. Whether you knew this before or not, hopefully you found this interesting.