Relations Between Numbers

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 is One More Than 11

12 is One Less Than 13

12 is More Than 11,10,9 ……… and so on

12 is Less Than 13, 14, 15, 16, ……. And so on

12 Is a Factor of 24,36, 48 …. 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 a Multiple of 2,3, 4 and 6. 12 tokens can be arranged in arrays with rows being 2, 3, 4 & 6.

12 Is a Common Factor of 72 &48. Both 48 & 72 can be arranged in arrays with rows being 12.

12 Is a Common Multiple of 3 &4. 12 can be arranged in arrays with rows being 3 or 4.

12 Is the HCF of 36 &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 the LCM of 4 &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 & 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 & 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). Any power of an Even number would 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).

Any power of an Odd number would be Odd.

Exponential Relations between Odd & Even Numbers

What about exponential relations connection odd & even numbers? EvenEven, OddOdd, EvenOdd & OddEven?

These have to be derived by logical thinking.

Odd numbers & 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 31.9 on Number Patterns.

1 + 3 = 22,

1 + 3 + 5 = 32 etc

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

1 + 3 = 22

3 + 6 = 32

6 + 10 = 42

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

12 + 1 + 2 = 22

22 + 2 + 3 = 32

1002 + 100 + 101 = 1012

This is actually the algebraic relation (x + 1) 2 = x2 + (x) + (x+1)

Interesting Number Relations

111111111 X 111111111 = 12345678987654321

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.

Math Riddles

One of the best ways for students to learn about number properties and relations is to ask them to make simple riddles using them. They can take any number and try to form clues relating to the digits in the number. Students would enjoy constructing such riddles.

For example, if we take 28, we can provide the following clues - In unit’s place I have the least prime number. In tens place I have the cube of the digit in unit’s place.

Such riddles can also be seen as word problems involving number properties and relations.