Even vs. Odd Numbers

Arithmetic Tutorial from The Amateur Logician:
Even vs. Odd Numbers
Prime vs. Composite Numbers
Sieve of Eratosthenes
Deficient, Perfect, and Superabundant Numbers
Multiple, Superparticular, Superpartient and their Reciprocals
Plane Numbers: Triangular, Square, Pentagonal, etc.
Solid Numbers: Pyramid, Cube, Scalene, etc.

The Greco-Syrian mathematician Nicomachus (60 – 120 AD) in his Introduction to Arithmetic thought that the primary division of natural numbers is in terms of odd versus even.

The division goes back to ancient mathematics. Greek mathematician Euclid (around 300 BC) defines them in Book VII of The Elements.

A natural number (ℕ) is also called a “counting number.” It consists of the numbers 1, 2, 3, 4, 5, etc. Some modern textbooks allow for zero to be a natural number. We will here. Numbers like -2, 1/2, 0.22, pi (π), or √2 are not natural numbers.

Nicomachus defines an even number as “that which can be divided into equal parts without a unit intervening in the middle” (Book I, Chapter VII). Even numbers include 2, 4, 6, 8, 10, 12, etc. Notice that 2 = 1+1, 4 = 2+2, 6 = 3+3, etc. Four can be divided into equal parts (two plus two). Six can be divided into equal parts (three plus three). In contrast, three cannot be (with equal parts that are natural numbers) and is hence not even.

Using modern math, we’ll define an even number thus:
n is even when n = 2k, where n and k are natural numbers.

Put differently, n is an even number when 2|n
That is, n is divisible by two (such that there is no remainder).

n = 2k
2 = 2⋅1
4 = 2⋅2
6 = 2⋅3
Etc.

Nicomachus defines an odd number as “that which cannot be divided into equal parts because of the aforesaid intervention of a unit” (Book I, Chapter VII). Odd numbers include 1, 3, 5, 7, 9, 11, etc. Notice that 3 = 1+1+1, 5 = 2+1+2, 7 = 3+1+3, etc. Three cannot be divided by two without remainder. The “unit” Nicomachus speaks about is the number one. An odd number differs by a unit – that is, by one. For instance, consider when the odd number five is divided by two there is a remainder of one.

Using modern math, we’ll define an odd number thus:
n is odd when n = 2k+1, where n and k are natural numbers.

Put differently, n is an odd number when it is not the case that 2|n

n = 2k+1
1 = 2⋅0+1
3 = 2⋅1+1
5 = 2⋅2+1
Etc.

Math concerns patterns! It’s about relations. A most obvious relation is that “the odd … differs by a unit from the even in either direction” (Book I, Chapter VII). An odd number is equal to an even number plus one. The smallest even number is two (dyad). The smallest odd number is one (unit or monad).

Lang’s classic modern text Basic Mathematics invites us to explore even and odd theorems.

Direct Proofs! If P, then Q. P. Ergo, Q.
(Modus Ponens)
These are proofs in the modus ponens logical form.

If a is even and b is even, then a + b is even.
Direct Proof: Assume a is even of the form 2x and b is even in the form 2y. Then we have 2x + 2y. Next we can factor the two to get 2(x + y). But 2(x + y) is of the even form. Q.E.D.

If a is even and b is odd, then a + b is odd.
Direct Proof: Assume a is even of the form 2x and b is odd of the form 2y + 1. Then we have 2x + 2y + 1. Next we can partially factor the two to get 2(x + y) + 1. But 2(x + y) + 1 is of the odd form. Q.E.D.

If a is odd and b is even, then a + b is odd.
Direct Proof: We just proved if a is even and b is odd, then a + b is odd. Natural numbers are “commutative” when adding, which means that x + y = y + x. Order doesn’t matter with addition! Ergo, given even + odd = odd, it follows odd + even = odd. Q.E.D.

If a is odd and b is odd, then a + b is even.
Direct Proof: Assume a is odd of the form 2x + 1 and b is odd of the form 2y + 1. Then we have 2x + 2y + 2. Next we can factor the two to get 2(x + y + 1). But that is of even form. Q.E.D.

Remember, any number that’s even has the form 2k (where k is a natural number) and any number that’s odd has the form 2k+1 (where k is a natural number). Anything that can be, so to speak, “fitted” into one of those patterns will be even or odd respectively. For example, the expression 2(x+y+1) is even as it “fits” into the form 2k because 2 is multiplied by x+y+1 which will yield some natural number k. These direct proofs are general and abstract enough to prove these theorems for all cases. And there are infinitely many odd and even numbers!

Together let’s work on Lang’s Theorem 2…

If a is even, then a^2 = a⋅a is even.
Direct Proof: Assume a is even of the form 2x. Squaring that, we get 4x^2. We can factor this to get 2(2x^2). But that’s of the form of an even number. Q.E.D.

If a is odd, then a^2 = a⋅a is odd.
Direct Proof: Assume a is odd of the form 2x + 1. Squaring that, we get (2x + 1)^2 = 4x^2 + 4x + 1. We can partially factor a two out to get 2(2x^2 + 2x) + 1. But that’s of the form of an odd number. Q.E.D.

I invite you to try your hand on the exercise set to prove other properties (page 25 of Lang’s Basic Mathematics). Before moving on, however, utilizing pure logic, we can make use of contraposition to prove things here!

If a^2 is odd, then a is odd.
Proof by Contraposition: We have already proved that if a is even, then a^2 is even. By contraposition, it thereby follows that if a^2 is odd, then a is odd. Q.E.D.

Contraposition is a logical equivalency between the proposition “if P, then Q” and “if not Q, then not P.” Hence, the contrapositive of “if a is even, then a^2 is even” is “if a^2 is odd (not even), then a is odd (not even).”

Back to Nicomachus!

Unity “is the natural staring point of all number” (Book I, Chapter VIII). He continues to tell us that every number is the mean of the two numbers adjacent to it.

It’s the “Property of Consecutive Numbers.”

For now, let’s think more about even and odd numbers. An even number can be an “even-times-even” number, “even-times-odd” number, or “odd-times-even” number (Book I, Chapters VII – X).

The sequence of “even-times-even” numbers can be generated by taking the even number two, multiplying it by two, and then by continuing to multiply by two to generate new numbers ad infinitum. 2⋅2 = 4, 4⋅2 = 8, 8⋅2 = 16, etc.

A property of these numbers is that when we divide them by half you will always get another even number until you reach one. 16 is an even-times-even number because 16/2 = 8, 8/2 = 4, 4/2 = 2, 2/2 = 1. But even though 18 itself is even, it’s not an even-times-even number because 18/2 = 9 and 9/2 is not even!

The sequence of “even-times-odd” numbers can be generated by (1) listing odd numbers starting with three and (2) multiplying each by two. So, we’ll take the odd numbers 3, 5, 7, etc. and then multiply each by two: 3⋅2 = 6, 5⋅2 = 10, 7⋅2 = 14, etc.

These numbers have the property that we can only half them once (without fractions or running into non-natural numbers). That’s why 18 is an even-times-odd number! 18/2 = 9, and we cannot half 9 any further.

“Odd-times-even” numbers are a “mixed species” in that they are a mixture of even-times-even and even-times-odd numbers. In fact, such a number is a product of even-times-even and odds. (Notice the formula placed in the table: 2^k is a even-times-even and 2m+1 is odd.) Take an odd number greater than one and multiply it by an even number that’s even-times-even to produce an odd-times-even number.

For example, let’s stick with the odd number five. And then we’ll multiply five by each in the even-times-even sequence 4, 8, 16, etc. This gives us 5⋅4 = 20, 5⋅8 = 40, 5⋅16 = 80, etc.

When we happen to take such a number and half it, it is possible to half that generated number at least one more time before it becomes impossible without ever being able to stop the sequence to one. Take 80. 80/2 = 40, 40/2 = 20, 20/2 = 10, 10/2 = 5. We were able to half this number more than once, but we were unable to reach unity.

Odd-times-even numbers resemble even-times-even in halving multiples times and resemble even-times-odd in stopping before reaching unity.

Nicomachus provides us a table like this (Book I, Chapter X). . .

More patters can be discovered when examining these numbers. Notice that 3⋅4 = 12, 3⋅8 = 24, etc. Or notice that 5⋅4 = 20, 5⋅8 = 40, etc. The table is being filed in by generating the odd-times-even numbers.

Moving along “breadth” we have a so-called arithmetic sequence. That is, we can move from one number to the next simply by adding a constant. Notice that 12+8 = 20, 20+8 = 28, etc. Or that 40+16 = 56, 56+16 = 72, etc.

Observe that when moving along the “breadth” we’ll discover that the “extremes” will average to the middle number. And if there’s no middle number, just find the average of the two middle numbers.

Or observe that moving along “length” has it that the product of the “extremes” equals the product of the two middle numbers. And if there’s only one middle number, just square that middle number.

But what about odd numbers?! Nicomachus classifies odds into three species. Unlike an even number, he cannot classify odds by how they split into two.

To have mutually exclusive and exhaustive categories, there really are only two species of odd numbers, namely, those which are prime and those which are not.

Recall that Euclid defined a prime number as “that which is measured by a unit alone” (Book VII of The Elements). Put differently, it is a natural number greater than one where its only divisors are one and itself.

Natural numbers greater than one that are not prime are said to be composite.

No even number other than two is a prime! For example, 8 has unique divisors other than one and itself. Four and two divide eight.

By definition, all even numbers greater than two are divisible by two. Ergo, they are all composite. All other prime numbers other than the number two are odd, but not all odd numbers are prime. Nine is odd; yet, three is a divisor because it divides nine, and hence nine is a composite number.

Out of the odd numbers, we can find pairs that are “relatively prime.” To quote Euclid: “Numbers prime to one another are those which are measured by a unit alone as a common measure” (ibid). So, these numbers share no common factors other than one.

Consider 9 and 25. Both are odd numbers that happen to be composite. 9 has divisors 1, 3, and 9. And 25 has divisors 1, 5, and 25. The greatest common divisor (gcd) between them is one. This is what makes them “relatively prime.” Note that all prime numbers are relatively prime to each other! Nicomachus, however, has in mind an “intermediate” state where the odds are both composite but relatively prime to each other.

Preview of Square Numbers: Book I, Chapter IX of Nicomachus’ Introduction to Arithmetic covers “square numbers.” A square number is of the form n^2 (where n is a natural number).

What’s remarkable is that the addition of successive odd numbers can form these numbers! Consider that 1+3 = 4 or that 1+3+5 = 9. The diagram’s squares are made up of individual squares. The orange “L” like shape represents a square’s gnomon. Those happen to be odd numbers.

Plane numbers include square and triangular numbers. We’ll explore these elsewhere. For now, I want to point out that the number one (unity) is special. “For every origin,” Nicomachus teaches us, “is elementary … into which everything is resolved and out of which everything is made, but the origin itself cannot be resolved into anything or constituted out of anything” (Book I, Chapter XI).

Nicomachus’ point is more profound than someone might at first expect. Numbers can be used as building blocks for other numbers! Elsewhere we will explore divisibility in modern number theory. Using the greatest common divisor of two numbers, we can construct integer combinations. Bezout’s Theorem reveals that with two relatively prime numbers, we can derive every other natural number. And… with prime numbers, we’ll be able to reach the Fundamental Theorem of Arithmetic.