Relations Between Numbers

< 8.3 Properties of Number Sets | Topic Index | 8.5 Number Theory >

Numbers are also connected to other numbers in many interesting ways. The study of the relations between numbers is supposed to have begun before the development of arithmetic. Today it has developed into a very sophisticated branch of mathematics called Theory of Numbers.

We will explore some simple examples in this chapter and the subsequent one.

Let us explore the relations of 12 with other numbers. Similar exercises can be done with other numbers.

All these relations can also be demonstrated visually using tokens.

12 isOne MoreThan 11

12 isOne Less Than13

12 isMore Than11,10,9 &hellip;&hellip;&hellip; and so on

12 isLess Than13, 14, 15, 16, &hellip;&hellip;. And so on

12 Is aFactor of24,36, 48 &hellip;. And so on. In an array with row being 12, the cumulative sum of the tokens in the rows starting with row 1 will be 24, 36, 48 etc.

12 Is aMultiple of2,3, 4 and 6. 12 tokens can be arranged in arrays with rows being 2, 3, 4 &amp; 6.

12 Is aCommon Factorof 72 &amp;48. Both 48 &amp; 72 can be arranged in arrays with rows being 12.

12 Is aCommon Multipleof 3 &amp;4. 12 can be arranged in arrays with rows being 3 or 4.

12 Is theHCF of36 &amp;48. 48 and 72 cannot be arranged in arrays with the same number of tokens in a row if the row contains more than 12 tokens.

12 Is theLCM of4 &amp;6.12 is the smallest number of tokens which can be arranged as arrays with rows either equal to 4 or 6.

Additive relations between Odd &amp; Even Numbers

Even + Even = Even, Even + 1 = Odd

Even + Odd = Odd

Odd + Odd = Even, Odd + 1 = Even

These relations can be visually represented with tokens

Multiplicative relations between Odd &amp; Even Numbers

These relations can also be visually represented with tokens.

Even X Even = Even (An array in which at least one of the sides is even will be even)

Even X Odd = Even (An array in which at least one of the sides is even will be even)

Odd X Odd = Odd (An Odd X Odd can be separated into an Odd X Even Array and a single row or column of Odd number of tokens. Hence it is Odd)

Odd numbers &amp; Square Numbers – The sum of all consecutive odd numbers from 1 upwards would be equal to the square of the number of terms. Please refer to Chapter 241 on Number Patterns.

1 + 3 = 2^2,

1 + 3 + 5 = 3^2 etc

Triangle numbers and Square numbers– the sum of any two consecutive Triangle numbers would be a square number.

1 + 3 = 2^2

3 + 6 = 3^2

6 + 10 = 4^2

Relation between a Square of any Number and Square of its successor

1^2 + 1 +  2 = 2^2

2^2 + 2 + 3 = 3^2

100^2 + 100 + 101 = 101^2

This is actually the algebraic relation (x + 1) ^2 = x^2 + (x) + (x+1)

Prime Factorization

One of the properties of any whole numbers is that every number can be uniquely expressed as a product of whole number powers of prime numbers.

24 = 2 X 2 X 2 X 3 – IT cannot be expressed as a product with any other set of prime numbers.

36 = 2 X 2 X 3 X 3

This is called the'''Fundamental Theorem of Arithmetic. '''

< 8.3 Properties of Number Sets | Topic Index | 8.5 Number Theory >