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## NUMBERS

In decimal number system, we use ten symbols \$0, 1, 2, 3, 4, 5, 6, 7, 8, 9\$ called digits, to represent any number.
Note: A group of figures, denoting a number is called numeral.

## Types of Numbers

### Natural Numbers:

Numbers which we use for counting the objects are known as natural numbers. It is denoted by ‘N’.
\$N=\{1,2,3,4 .... ..\}\$

### Whole Numbers:

All Natural Numbers together with zero form the set of all whole numbers. It is denoted by ‘W’.
\$W=\{0,1,2,3, .... ..\}\$

### Integers:

The set of numbers which consists of whole numbers and negative numbers is known as integers. It is denoted by Z.
\$Z=\{ .... .. -3,-2,-1,0,1,2,3, .... ..\}\$

### Positive Integers:

The set \$Z^{+} = \{1, 2, 3, 4, .... ..\}\$ is the set of all positive integers. It is clear that positive integers and natural numbers are synonyms.

### Negative Integers:

The set \$Z^{-} = \{-1, -2, -3, .... ..\}\$ is the setof all negative integers. Remember: “O” is neither positive nor negative.

### Non-Negative Integers:

The set \$\{0, 1, 2, 3, .... ..\}\$ is a set of non-negative integers.

### Non-Positive Integers:

The set \$\{0, -1, -2, -3, .... ..\}\$ is the set of non-positive integers.

### Even Numbers

The numbers which are divisible by 2 are called Even Numbers.
\$E = \{2, 4, 6, .... ..\}\$

### Odd Numbers:

The numbers which are not divisible by 2 are called Odd Numbers.
\$O = \{3,9,11,17,19, .... ..\}\$

## Properties of zero:

• \$0\$ is neither positive nor negative.
• \$0\$ is an even integer.
• \$0\$ is smaller than every positive number.
• \$0\$ is greater than every negative number.
• For any integer \$p\$; \$p x 0 = 0\$.
• For any integer \$p\$ (including 0):\$p + 0 = 0\$.
• For any positive integer \$p\$; \$0÷p;0/p\$ = undefined.
• For every integer \$p\$;\$p + 0\$ and \$p - 0 = p\$.
• If the product of two or more numbers is \$0\$, then at least one of them is \$0\$.

## Properties of one:

• For any number \$p\$; \$p x 1 = p\$ and \$p/1 = p\$.
• \$1\$ is the divisor of every integer.
• \$1\$ is an odd integer.
• \$1\$ is not a prime number, because prime numbers should be greater than \$1\$.
• \$1\$ is the smallest positive integer.
• For any number \$n\$: \$1^n = 1.\$

## Factors and Multiples:

A number which divides a given number exactly is called a factor of the given number.

Example 1: find the factors of (i) \$64\$ and (ii) \$75\$.

Solution: (i) \$64\$

\$= 1 x 64\$
\$= 2 x 32 \$
\$= 4 x l6 i\$
\$= 8 x 8\$
The factors of \$64\$ are \$1, 2, 4, 8, 16, 32\$ and \$64\$.

(ii) \$75\$
\$= 1 X 75\$
\$= 3 x 25\$
\$= 5 x 15\$
The factors of \$75\$ are \$1, 3, 5, 15,25\$ and \$75\$.

## Division Algorithm:

Let \$a\$ and \$b\$ be two given integers such that \$b ≠ 0\$. On dividing \$a\$ by \$b\$, let \$q\$ be the quotient and \$r\$ the remainder, then \$a = bq + r\$.

Clearly, \$0 < r < b\$
In general, we have
Dividend = (Divisor x Quotient) + Remainder

## Multiple of a Number:

A multiple of any natural number is a number obtained by multiplying that number by any natural number.

Example: find the multiples of:
(i) \$4\$ less than \$30\$ (ii) \$9\$ less than \$60\$ .

Solution: (i)

\$4 x 1 = 4\$
\$4 x 2 = 8\$
\$4 x 3 = 12\$
\$4 x 4 = 16\$
\$4 x 5 = 20\$
\$4 x 6 = 24\$
\$4 x 7 = 28\$ etc.
The multiple of \$4\$ less than \$30\$ are \$4, 8, 12, 16, 20, 24\$ and \$28\$.

(ii) \$9 x 1 = 9\$
\$9 x 2 = l8\$
\$9 x 3 = 27\$
\$9 x 4 = 36\$
\$9 x 5 = 45\$
\$9 x 6 = 54\$ etc.
The multiple of \$9\$ less than \$60\$ are \$9, 18, 27, 36, 45\$ and \$54\$.

## Divisible of a Number:

If a number divides a second number without leaving any remainder, then we say that the second number is divisible by the first number. For example, since the number \$2\$ divides \$14\$ without leaving any remainder, we say that \$14\$ is divisible by \$2\$.

›› Numbers