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

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● | ● | ● | |||||||||||

● | ● | ● | ● | ||||||||||

● | ● | ● | ● | ● | |||||||||

● | ● | ● | ● | ● | ● | ||||||||

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 = 5^{2}.

Our formula for the sum of the first N odd numbers is

1 + 3 + 5 + … + (2N – 1) = N^{2}

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.

○ | ● | ● | ● | ● | |

○ | ● | ● | ● | ● | ← M^{2} 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 M^{2} blacks and (2M + 1) whites. If the number of whites happens to be a perfect square N^{2}; then

(M + 1)^{2} = M^{2} + N^{2}

which is the **Pythagoras’ Theorem**. And we get the **Pythagorean triple** (M, N, M + 1).

Solving for M from N^{2}=(2M + 1), it can be shown that

[ | N^{2} – 1 |
] | ^{2} |
+ [ N ] ^{2} = |
[ | N^{2} + 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 ;-))