Plane 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.

Man has interconnected arithmetic and geometry since ancient times. It continues to this day in advanced modern mathematics. There is such a powerful interconnection that the ancients could not but help to draw connections themselves.

Daniel Sullivan in An Introduction to Philosophy explains that the ancient Greek philosopher Pythagoras (572 – 497 BC) thought that “reality itself consists in numbers” (page 14). Interestingly, Pythagoras used dots to represent numbers. He “discovered that numbers could be fitted into regular patterns” (page 13).

A lot of fun is to be had when exploring plane numbers (a.k.a. polygonal numbers).

It’s a much-neglected topic…

What we nowadays call Nicomachus’ Theorem relates square and triangular numbers.

Before we can explore his theorem, let’s figure out what these numbers are all about. Nicomachus (60 – 120 AD) in Introduction to Arithmetic thought of numbers in terms of linear numbers, plane numbers, and solid numbers. Arithmetic “is the mother of geometry and more elementary than it” (Book II, Chapter VI). Geometric figures cannot be defined without arithmetic. Arithmetic as a science is “prior” to geometry.

A dot has zero dimensions. A line has one dimension. Two dimensions can yield a surface. And three dimensions can yield solids. In a similar manner, unity (the number one) occupies the place of a geometric point. Nicomachus wrote, “unity alone out of all number, when it multiplies itself, produces nothing greater than it self” (ibid). Obviously, it’s not just that one times one is one, this number is, in modern terms, a multiplicative identity because 1·x = x. No doubt, we can start with unity and sequentially add unity to then generate natural numbers: 1, 2, 3, etc.

Linear Numbers start with two and increase by the successive addition of one.

Euclid (around 300 BC) thought of Plane Numbers differently than Nicomachus. He just thought of them as produced by multiplying two numbers together, “where its sides are the numbers which have multiplied one another” (Book VII of The Elements).

For Nicomachus, Plane Numbers get their start with Triangular Numbers. He is following the ancient Greek philosopher Plato (429 – 347 BC)! In Timaeus it is said that triangular numbers are the most fundamental type of plane number.

When a number can be arranged in the form of an equilateral triangle, it is a Triangular Number. Numbers include 3, 6, 10, 15, 21, etc.

Each of these numbers relate to its side-length. The side-length of 2 corresponds to 3. The side-length of 3 corresponds to 6. And so on.

We form these numbers by adding consecutive numbers!
1 = 1
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15
…

Take the example of 10. It’s a triangular number. It’s the fourth triangular number, which corresponds to its side-length. In adding consecutive numbers, we stop at four. 1 + 2 + 3 + 4 = 10.

The next triangular number is formed by adding five to the last number. So, 1 + 2 + 3 + 4 + 5 = 15. And the next triangular number must be 15 + 6 = 21. Etc.

There are infinitely many triangular numbers.

The derivation is an arithmetic summation. Let’s derive a general formula!

This proof took some “creative” algebra. The formula will generate any nth triangular number. For example, the sixth triangular number will be 6(6+1)/2 = 6(7)/2 = 42/2 = 21.

Carl Gauss (1777 – 1831) is said to have done a similar technique, under a different context, at the age of seven! A schoolteacher wanted to keep the kids busy by adding all numbers from one to 100. He solved it within moments, getting the correct answer of 5,050.

Square Numbers are the next type of Plane Numbers. Square Numbers are of the form n^2 (where, as usual, n is a natural number). Hence, square numbers include 2^2 = 4, 3^2 = 9, 4^2 = 16, 5^2 = 25, 6^2 = 36, etc.

Square Numbers can be formed by adding consecutive odd numbers.
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
…

Nicomachus explains (Book II, Chapter IX) this visually. It is odd that odd numbers can form square numbers!

The orange “L” like shape represents a square’s gnomon. Those happen to be the odd numbers that we sum.

Let’s derive a general formula! There are different ways to go about this. Adding odd numbers is an arithmetic summation, like with our triangular numbers. That is, we’re consecutively adding a constant number to move onto the next. Consider the sequence itself we’re adding: 1, 3, 5, 7, 9, etc.

Those are just the odd numbers. Hence, 2n – 1 can describe it.
2(1) – 1 = 1
2(2) – 1 = 3
2(3) – 1 = 5
2(4) – 1 = 7, etc.

This yields the summation 1 + 3 + 5 + … + (2n – 1).
How does that equal n^2?

Proof by Mathematical Induction!
(1) Conjecture.

(2) Show it’s true for a “base case,” such as n = 1.

(3) Introduce the “inductive hypothesis” by assuming that the conjecture is true for an arbitrary case n, then use the “inductive step” to show that it is true for case n+1.

Sergen Lang’s classic modern text Basic Mathematics in the last chapter introduces how to engage in these proofs with summation series. (“Induction” is defined in a variety of ways in logic. See this entry for an overview.)

We’re going to re-prove the formula for triangular numbers with induction.

1 + 2 + 3 + … + n = n(n+1)/2
Proof via Induction: Our hypothesis is clearly true with the base case of n = 1, since the right-hand-side of the equation yields 1(1+1)/2 = 1. And 1 = 1. Our inductive hypothesis assumes the formula is true for an arbitrary case n. We’ll prove the hypothesis true by the inductive step to show it holds for case n + 1.

With the inductive step, n + 1 was substituted into n. This was done to both the left-hand-side and the right-hand-side of the equation. Both sides were shown equal, which proved the inductive hypothesis true. Notice that we’re able to make use of that hypothesis in simplifying the equation. That’s why we’re able to move from 1 + 2 + … + n to n(n+1)/2.

Now, let’s prove the formula for square numbers!
(Exercise #1 of Ch. 16, §1 of Lang’s Basic Mathematics)

The infinite nature of mathematics is reflected by there being an infinite number of patterns to discover. Soon we’ll take a look at the fact that adding consecutive triangular numbers are all squares. There’s a very cool visual proof for this! We’ll finally look at Nicomachus’ Theorem, which shows that adding consecutive cubes similarly brings us squares, and that these squares are of triangular numbers.

Before that, let’s consider Pentagonal Numbers and then generalize our findings.

The gnomons determine the arithmetic summation. (The yellow one’s make up the gnomon in the diagram.) Observe the numbers we add to get to a Pentagonal Number forms a sequence that increases by a constant three: 1, 4, 7, 10, 13, 16, etc.

Pentagonal Numbers!
1 = 1
1 + 4 = 5
1 + 4 + 7 = 12
1 + 4 + 7 + 10 = 22
1 + 4 + 7 + 10 + 13 = 35
…

The sequence of 1, 4, 7, etc. is 3n – 2.

3(1) – 2 = 1
3(2) – 2 = 4
3(3) – 2 = 7
3(4) – 2 = 10, etc.

Hence, the nth pentagonal number is the sum of the first n terms.

To derive the general formula, we’ll make use of our previous discovery with triangular numbers. Note the use of summation notation. The sum from 1 to n of k is thereby n(n+1)/2. That’s in purple. 3 was factored out. The sum from 1 to n of 1 results in adding 1 n times. That’s in red. 2 was factored out.

Generalization of All Plane Numbers (a.k.a. Polygonal Numbers). . .

Every plane number is generated by summing an arithmetic series of gnomons. The common difference – that is, we’re consecutively adding a constant number to move onto the next – is determined by the number of sides of the polygon.

We can now compute that summation for a general formula with n as the nth number of the plane number with r sides.

To be sure, this is not a “pretty” formula. It is, nevertheless, consistent with our findings. For example, consider the triangular number (where r = 3).

Beautiful Patterns and Visual Proofs…!

We can relate plane numbers to discover hidden secrets. For instance: when adding consecutive triangular numbers, the answers are all squares.

There must be some relation between triangular numbers and square numbers!
1 + 3 = 4 = 2^2
3 + 6 = 9 = 3^2
6 + 10 = 16 = 4^2
10 + 15 = 25 = 5^2
…

Add the first and second triangular numbers, and a square number emerges. Likewise, adding the second and third triangular numbers produces another square number—and so on ad infinitum.

But are we sure this goes on forever? We need a proof! Consider an intuitive “visual proof.” Let t represent a triangular number. There’s a staircase divider. On the top left of the divider, start counting the dots to see how they form triangular numbers. One, three (1+2), six (1+2+3), etc. It goes all the way to the nth triangular number, that is, t sub n. Next, on the bottom right of the divider, start counting the dots in the same manner upward. Notice that it stops at one less triangular number! That is, t sub n-1. However, together they form a square! The number of dots is n^2.

We’ll use a direct proof, which yields n^2 quickly and elegantly.

Nicomachus’ Theorem is in Book 2, Chapter 20 of Introduction to Arithmetic. Math is about discovery and creativity. Notice a new pattern? Add up consecutive cubes.

The answers are all squares. And, if we squint our eyes, we’ll see they are squares of triangular numbers.

9 is the 2nd triangular number squared.
36 is the 3rd triangular number squared.
100 is the 4th triangular number squared.

We need a proof!

It’s a cool inductive proof. All of them have been. It comes to show the importance of developing our algebraic skills to formulate these concepts. If you’re interested in some additional exercises (more advanced than Lang’s textbook), check out Problems 2.1 in Elementary Number Theory by David Burton as they concern plane numbers.

You’ll find the one and only reference to Nicomachus in this textbook with problem #1(c) on page 15.