Menu Navigation

  • Cell: +92(308)611-2254

ARITHMETIC

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 $2$.