Triangle & Square Numbers

Triangle & Square numbers are part of properties of numbers.

Properties of numbers is a part of Number Theory.

Though number theory has its practical applications, it is pursued more for enjoying the patterns & relations between numbers, which are often hidden.

The main purpose is to bring out the beauty of the patterns & relations between numbers.

Prof G H Hardy, mentor of Srinivasa Ramanujan, said – “"A MATHEMATICIAN, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. ... The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics." "The best mathematics is serious as well as beautiful--'important' if you like, but the word is very ambiguous, and 'serious' expresses what I mean much better."

For a very long time, mathematics was seen to consist only of arithmetic & geometry.

Greeks developed geometry to a very high level which resulted in the axiomatic system. Axiomatic system is one in which you start with a few propositions or axioms, whose truth you accept at face value, and develop an entire framework using deductive logic.

They did not focus as much on arithmetic. They used letters from the Greek alphabet as numerals. They did not develop numerals which were distinctly different from alphabets. This situation also existed in Tamil.

Greeks explored numbers and their properties through geometry.

They explored ways of arranging numbers in various designs.

These explorations developed into the important topic of Number Theory.

Mathematician Karl Friedrich Gauss said that “Mathematics is the queen of the sciences and number theory is the queen of mathematics” -

Greeks invented numbers & number pairs with special properties like odd/even numbers, prime numbers, deficient numbers, abundant numbers, perfect numbers and amicable numbers.

Two of the major ways numbers could be thought of were as triangles & squares.

Square Numbers

1, 4, 9, 16 & 25 tokens can be arranged as a geometrical square. o O o o o o o o O o o o o o o o o O o o o o o o o o o O o o 1 2X2 3X3 4X4

All the sides of the squares would be equal. They can be called the “square roots” of the number which is arranged in a square.

Relation between consecutive square numbers

We can geometrically explore the relation between consecutive square numbers.

In the diagram above we have a 4 by 4 square. How do we make it into a 5 by 5 square?

By adding a 4 to one of the sides and a 5 to the other side!

So 4^2 + 4 + 5 = 25 = 5^2

Extending this idea to a N by N square we have

So N^2 + N + (N + 1) = 〖(N+1)〗^2

In Algebra this is written in a standard form as (N+1)^2 = 〖(N)〗^2 + 2N + 1

In the original form it can make several computations easy.

For example, (101)^2 = 〖(100)〗^2 + 100 + 101 = 10,201

Sum of consecutive odd numbers, starting from 1, is a square number

1 + 3 = 2^2

1 + 3 + 5 = 3^2

Triangle Numbers

Any number which can be arranged in the form of a triangle, was called a Triangle Number.

In a triangle number successive layers of tokens contain one less token, thus helping to form the shape of a triangle.

The triangle can be arranged either as an Isosceles Right-Angled triangle resting on one of its sides or as an Isosceles triangle resting on its base.

The first formation is easy for building the formations with tokens and exploring further.

The first few triangle numbers are

1, 3, 6, 10, 15, 21 …

All triangle numbers are the sum of Natural Numbers from 1 to a certain number. The 6th Triangle Number would be the sum of numbers from 1 to 6.

1st 2nd 3rd 4th 5th 6th 1 1 + 2 1 + 2 + 3 1 + 2 + 3 + 4 1+2+3+4+5 1+2+3+4+5+6

Relation Between Triangle Numbers & Square Numbers

The sum of any two consecutive triangle numbers is a square number.

We have shown above, pictorially, the addition of the 4th & 5th Triangle numbers.

Their total is 10 + 15 = 25 = 5^2

Another property of triangle numbers – the square of the number is equal to the sum of the cubes of the numbers which total to the triangle number.

1^3 +2^3 +3^3+4^3 = 1 + 8 + 27 + 64 = 100 = (〖1+2+3+4)〗^2

Representation of tens

Number 10 can be nicely shown as a bunch of grapes hanging down.