Page 64 - Maths Skills - 8

62 Maths

INTRODUCTION
Once a famous mathematician Prof. G.H. Hardy came to visit India’s great mathematical genius, S. Ramanujan
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 interesting. He said it is the smallest number that can be
expressed as a sum of two cubes in two different ways:
1729 = 1728 + 1 = 12 + 1 3
3
1729 = 1000 + 729 = 10 + 9 3
3
1729 is sometimes called the Hardy-Ramanujan Number.
Ramanujan 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.

HOW TO CUBE A NUMBER
To cube a number, just multiply it three times by itself. 2

Can you tell what is 2 cubed?
= 2 × 2 × 2 = 8 2 2

If a number x is cubed, i.e., multiplied by itself three times, the number obtained is written as x and we read it as
3
x raised to the power of 3. Therefore, if there exists a number C such that C = x , then C is called x-cubed or the
3
cube of x. In the above example, 8 is 2-cubed or 8 is the cube of 2. Similarly,
3 cubed = 3 = 3 × 3 × 3 = 27; 27 is the cube of 3
3
4 cubed = 4 = 4 × 4 × 4 = 64; 64 is the cube of 4
3
5 cubed = 5 = 5 × 5 × 5 = 125; 125 is the cube of 5
3
6 cubed = 6 = 6 × 6 × 6 = 216; 216 is the cube of 6
3
PERFECT CUBE
Any number which is a product of three identical numbers is called a perfect cube.

For example, 1 = 1 × 1 × 1 = 1, 2 = 2 × 2 × 2 = 8, 3 = 3 × 3 × 3 = 27, etc.
3
3
3
So 1, 8, 27, 64, 125, etc. are perfect cubes. Fact-o-meter
Procedure for checking if a given natural number is a perfect cube To find a natural number
Step 1: Express the given natural number as a product of its prime factors. whose cube is the given

Step 2: Group the factors in triplets of equal factors. number, take out one
Step 3: If no factor is left after step 2, then the given natural number is a factor from each triplet
perfect cube, otherwise not. and multiply them.

PROPERTIES OF CUBES OF NUMBERS
Property 1. The cube of an even natural number is even.
For example, (2) = 2 × 2 × 2 = 8, (4) = 4 × 4 × 4 = 64, etc.
3
3
Property 2. The cube of an odd natural number is odd.
For example, (3) = 3 × 3 × 3 = 27, (5) = 5 × 5 × 5 = 125, etc.
3
3
Hence, cube of an odd natural number is odd.