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CUBES AND CUBE ROOTS 109

CHAPTER
7
Cubes and Cube Roots

7.1 Introduction

This is a story about one of India’s great mathematical geniuses, S. Ramanujan. Once
another famous mathematician Prof. G.H. Hardy came to visit him in a taxi whose number
was 1729. While talking to Ramanujan, Hardy described this number
“a dull number”. Ramanujan quickly pointed out that 1729 was indeed Hardy – Ramanujan
interesting. He said it is the smallest number that can be expressed Number
as a sum of two cubes in two different ways: 1729 is the smallest Hardy–
1729 = 1728 + 1 = 12 + 1 3 Ramanujan Number. There
3
are an infinitely many such
1729 = 1000 + 729 = 10 + 9 3 numbers. Few are 4104
3
1729 has since been known as the Hardy – Ramanujan Number, (2, 16; 9, 15), 13832 (18, 20;
even though this feature of 1729 was known more than 300 years 2, 24), Check it with the
before Ramanujan. numbers given in the brackets.
How did Ramanujan know this? Well, he loved numbers. All
through his life, he experimented with numbers. He probably found
numbers that were expressed as the sum of two squares and sum of
two cubes also.
There are many other interesting patterns of cubes. Let us learn about cubes, cube
roots and many other interesting facts related to them.

Figures which have
7.2 Cubes 3-dimensions are known as
You know that the word ‘cube’ is used in geometry. A cube is solid figures.
a solid figure which has all its sides equal. How many cubes of
side 1 cm will make a cube of side 2 cm?
How many cubes of side 1 cm will make a cube of side 3 cm?
Consider the numbers 1, 8, 27, ...

These are called perfect cubes or cube numbers. Can you say why
they are named so? Each of them is obtained when a number is multiplied by
itself three times.