Chapter 2 of “The Pythagorean Theorem — A 4000 Year History” tells us that Pythagoras founded a school when he settled in Greek around 530 BCE. His students — the Pythagoreans — studied mainly philosophy, mathematics and astronomy.
The Pythagoreans were fascinated by figurative numbers, a representation of numbers as dots arranged in a regular pattern. This really permitted a level of visualization that is very interesting. Let’s look at three examples here…
Summing Numbers 1 to N
Consider adding the first six positive numbers, i.e. 1 + 2 + 3 + 4 + 5 + 6. We learned this in school as the arithmetic series, or the sum of members of a finite arithmetic progression. What a mouthful.
The Pythagoreans had a simpler way using dots, as follows:
i) Arrange the numbers to be added as dots in a staircase pattern
| ● | |||||||||||||
| ● | ● | ||||||||||||
| ● | ● | ● | |||||||||||
| ● | ● | ● | ● | ||||||||||
| ● | ● | ● | ● | ● | |||||||||
| ● | ● | ● | ● | ● | ● | ||||||||
| 1 | + | 2 | + | 3 | + | 4 | + | 5 | + | 6 |
ii) Fill in the spaces with missing dots to get a rectangle
| ○ | ○ | ○ | ○ | ○ | ○ | ● | |||||||
| ○ | ○ | ○ | ○ | ○ | ● | ● | |||||||
| ○ | ○ | ○ | ○ | ● | ● | ● | 6 dots | ||||||
| ○ | ○ | ○ | ● | ● | ● | ● | high | ||||||
| ○ | ○ | ● | ● | ● | ● | ● | |||||||
| ○ | ● | ● | ● | ● | ● | ● | |||||||
| 7 dots wide | |||||||||||||
iii) Count the total number of dots and divide by two.
Thus, (7 x 6) / 2 = 21 = 1 + 2 + 3 + 4 + 5 + 6
Isn’t that neat? Notice that the width of the rectangle always has one more dot than its height. So, if we generalize for the sum of the first N positive integers, we’d get
| 1 + 2 + 3 + … + N = | N * (N + 1) |
| 2 |
which is the formula we learned by heart from Maths class!
Summing the First N Odd Numbers
Do you remember that when we sum a series of odd numbers starting with 1, we’ll always get a perfect square? For illustration, let’s just add the first 5 odd numbers. Thus, 1 + 3 + 5 + 7 + 9 = 25 = 52.
Our formula for the sum of the first N odd numbers is
1 + 3 + 5 + … + (2N – 1) = N2
The Pythagoreans saw that through different eyes…
| ● | ● | ● | ● | ● | ← 1 dot here only |
| ● | ● | ● | ● | ● | ← 3 dots in this “lane” |
| ● | ● | ● | ● | ● | ← 5 dots in this lane |
| ● | ● | ● | ● | ● | ← 7 dots in this lane |
| ● | ● | ● | ● | ● | ← 9 dots in this lane |
| 5 dots wide & high | = 25 dots in total | ||||
Such simplicity and elegance! Doesn’t it make you wonder if we can still find the same in today’s complex world?
Finding Pythagorean Triples
Naturally, the Pythagoreans loved their triples 🙂 But how did they find them using their unique powers of visualization? The dots do their magic once again…
Consider an array of black dots arranged in a square with each side having M number of dots. In the example below, we have M=4. If we extend this square — as shown by the white dots — by adding another row and column plus a dot at the corner, we’d get a larger square with sides of (M + 1) dots.
| ○ | ● | ● | ● | ● | |
| ○ | ● | ● | ● | ● | ← M2 black dots in this square |
| ○ | ● | ● | ● | ● | |
| ○ | ● | ● | ● | ● | |
| ○ | ○ | ○ | ○ | ○ | ← (2M + 1) white dots in this “lane” |
| (M + 1)2 dots in the enlarged square | |||||
This enlarged square has (M + 1)2 dots, formed by M2 blacks and (2M + 1) whites. If the number of whites happens to be a perfect square N2; then
(M + 1)2 = M2 + N2
which is the Pythagoras’ Theorem. And we get the Pythagorean triple (M, N, M + 1).
Solving for M from N2=(2M + 1), it can be shown that
| [ | N2 – 1 | ] | 2 | + [ N ] 2 = | [ | N2 + 1 | ] | 2 |
| 2 | 2 |
With this formula, simply assign N any odd number other than 1, and you’ll get a primitive Pythagorean triple from the terms in square brackets ([ ]). For example, when N=3, we get the triple (3, 4, 5); when N=5, we get another triple (5, 12, 13), etc. Wow, what an innocent looking yet powerful method to generate Pythagorean triples by hand! Who needs a computer?
Ancient Maths wisdom is really profound in all its simplicity. Learn to view things from other angles that can both be enlightening and amusing. Be WittyCulus. Don’t be square! (pun intended ;-))
