Integers - MathsTips.com

Integers refer to as the positive and negative numbers. Integers are -4, -3, -2, -1, 0, 1, 2, 3, 4 and so on. All natural numbers, negative’s of natural numbers and 0, together form the set $Z$ or $I$ of all integers.

Thus $Z=\left\{....-3, -2, -1, 0, 1, 2, 3,....\right\}$

$Z^{+}=\left\{1, 2, 3, ....\right\}$ is the set of all positive integers.
$Z^{-}=\left\{-1, -2, -3, ....\right\}$ is the set of all negative integers.
Integers are also known as directed numbers.

Four Fundamental Operations on Integers

1. Addition of integers:

a. When the integers have like signs, i.e., when both the integers to be added are either positive or are negative, we add their absolute values and assign the same sign to the sum.

Example: Add:

1. 57 and 112

The sum of their absolute values =57+112 = +169

2. -32 and -83

The sum of their absolute values = 32+83 =115

Therefore (-32) + (-83)= -115

b. When the integers have unlike signs i.e. , one is positive and the other is negative, we determine the difference of their absolute values and assign the sign of integer of greater absolute value.

Example: Add:

1. 358 and -564

Since, $\lvert 358\rvert\ =358$ and $\lvert-564\rvert\ =564$

Difference of their absolute values =564 – 358=206

Therefore (358)+ (-564) = -206

Properties of addition of integers:

The sum of any two integers is always an integer.
For any two integers $a$ and $b$ , $a+b= b+a$
For any three integers $a$ , $b$ and $c$ , $a+(b+c) = (a+b)+c$
For any integer $a$ , $a+0=0+a = a$
For each integer $a$ there exists another integer $(-a)$ such that $a+(-a)=(-a)+a=0$ . The integer $latex(-a) $ is called the opposite or negative or additive inverse of the integer $a$

2. Subtraction of Integers

We change the sign of the integer to be subtracted and then add.

Example: Subtract :

1. -5 from 8

8 – (-5) =+8+5 =+13

2. -5 from -8

(-8)- (-5) = -8+5 = -3

Properties of subtraction of integers:

The difference of any two integers is always an integer
For any two integers $a$ and $b$ ; $a-b$ is not equal to $b-a$
For any three integers $a$ , $b$ , $c$ , $a-(b-c) = (a-b)-c$
For any integer $a$ ; $(a-0) \neq 0-a$ and $a-0=a$ .
If $a, b, c$ are integers and $a> b$ , then $(a-c)>(b-c)$

To evaluate an expression containing various integers with plus and minus signs:

Example: (+18) + (-12) – (+6) –(-9)

Solution: +18 -12 -6+9 = +27 – 18 (Adding all integers with plus sign together and with minus sign separately together)

=+9 or simply 9.

3. Multiplication of Integers

1. When the integers have same signs

The multiplication of two integers, both positive or both negative, is always

a positive integer equal to the product of their absolute values .

2. When the integers have unlike signs :

The multiplication of a positive and a negative integer is always negative.

Example:

(-4) x (-7) = 4 x 7 =28
(-30) x (5) = -(30 x 5) = -150

Properties of multiplication of integers:

Product of two integers is always an integer.
$a \times b= b \times a$
$a \times (b \times c) = (a \times b) \times c$
$a \times 0= 0 \times a$
$a \times 1 = 1 \times a = a$
$a \times (b+c) = a \times b + a \times c$
$a \times (b-c) = a \times b -a \times c$ and so on..

Important Reesults:

1. $(-a_{}1) \times (-a_{2}) \times (-a_{3}) \times .... \times (-a_{n})$

$=-(a_{}1 \times a_{2} \times a_{3} \times .... \times a_{n})$ , when $n$ is odd.

$=(a_{}1 \times a_{2} \times a_{3} \times .... \times a_{n})$ when $n$ is even.

2. $(-a) \times (-a) \times (-a) \times .... n \: times$

$=-a^n$ , when $n$ is odd.

$=a^n$ , when $n$ is even.

3. $(-1) \times (-1) \times (-1) \times .... n \: times$

$=-1$ , when n $n$ is odd.

$=1$ , when $n$ is even.

4. Division of Integers

If both the integers have like signs the sign of division is always positive.
If both the integers have unlike signs the division is always negative.

Properties of division of integers:

If $a$ and $b$ are any two integers then $\dfrac{a}{b}$ is not necessarily an integer
If $a$ is a non zero integer then $\dfrac{a}{a}=1$
If $a$ is an integer we have $a \div 1=a$
For any non zero integer $a$ , $\dfrac{0}{a}=0$ but $\dfrac{a}{0}$ is not defined.
If $a, b, c$ are integers then $(a \div b) \div c \neq a \div (b \div c)$
If $a, b, c$ are integers and $a>b$ , then: i) $(a \div c) > (b \div c)$ , if $c$ is positive. ii) $latex (a \div c) < (b \div c) $, if $c$ is negative.

Simplification of Expressions Involving Integers:

In order to find the value of a given expression involving integers we use the ‘BODMAS rule’ that depicts the correct sequence in which the operations are to be executed. In this rule ‘B’ stands for ‘Bracket’, ‘O’ stands for ‘of’, ‘D’ for ‘Division’, ‘M’ for ‘Multiplication’, ‘A’ for ‘Addition’ and ‘S’ for ‘Subtraction’. Thus, in simplifying an expression first of all the brackets must be removed, strictly in the following order:

Parenthesis or common brackets or first bracket-()
Braces or curly brackets or second bracket-{}
Rectangular or Box Brackets or third bracket-[]

After removing the brackets, we must use the other operations strictly following ‘BODMAS rule’.

Vinculum or bar: Certain expresiions contain a Bar or Vinculum over a part of them. In such cases, we simplify the expression under the vinculum before applying the ‘BODMAS rule’.

Example: $63-[25-{6 \div 3 \: of \: (7+\overline{8-12})}]$

$63-[25-\left\{6 \div 3 \: of \: (7+\overline{8-13})\right\}]$

$=63-[25-\left\{6 \div 3 \: of \: (7-5)\right\}]$

$=63-[25-\left\{6 \div 3 \: of \: 2\right\}]$

$=63-[25-\left\{6 \div 6\right\}]$

$63-[25-1]=63-24=39$

Introduction to Number Line

A number line is shown below:

The line continues to be of infinite length at both the ends and so the numbers are also upto infinity on both the sides.
To the left of zero are negative and to it’s right are positive numbers.
For any two numbers on the number line , the one which is on the right of the other is greater . And the number which is on the left is smaller.
If $a$ is greater than $b$ ; then $(-a)$ is smaller than $(-b)$ .
If $a$ is smaller than $b$ ; then $(-a)$ is greater than $(-b)$ .
Every negative number is smaller than zero
Every positive number is greater than zero
Every positive number is greater than every negative number.
0 is neither positive nor negative.

Representation of Integers on the Number Line:

We draw a line. We mark a point on it and label it as 0. We set off steps on both sides of the point representing 0. We label the points lying on the right of zero as 1, 2, 3, 4, 5… and those lying on the left of 0 as -1, -2, -3, -4, -5… as shown in the following figure:

The arrowheads on both sides of the number line indicate the continuation of integers indefinitely on each side. On the right, the integers keep increasing by unity for every step we take away from the point 0. On the left, the integers keep decreasing by unity for every step we take away from the point 0.

The numbers to the left of zero are the negative integers while those to the right of zero are the positive integers. The numbers lying on either side of the point 0 are opposites of each other.

Exercise

Add: i) 2138 and -1562 ii) -75 and -56 iii) 394 and -492
Subtract : i) 346 from 293 ii) -350 from 200 iii) 63 from -63
Multiply : i) 0 and -297 ii) -12 and 47 iii) -8 and -7
Divide : i) 0 by 8 ii) 1728 by -12 iii) -729 by – 81
Fill in the blanks :
1. $(-17)$ – __________ $= -4$
2. $(-98) \div$ ___________ $= 1$
3. ________ $\div 97 = 0$
4. $-12 \times$ __________ $-36$
5. $(-8)+$ __________ $= -4$
Simplify:
1. $24-[16-\left\{19-(14-\overline{8-4})\right\}]$
2. $47-[29-\left\{6 \div 3-(6-9 \div 3) \div 3\right\}]$
3. $53-[28-\left\{16 \div (37+3 \times 2 -35)\right\}]$
4. $[67-(-3)\left\{16-(56-33)\right\}] \div [3+4 \times \left\{5+(-3) \times (-2)\right\}-24]$
The sum of two integers is (-41). If one of the integers is 185, find the other.
From (-38) subtract the sum of 39 and (-66).
Subtract the sum of 273 and (-512) from the sum of (-486) and 597.
A shopkeeper bought a pen for Rs 75, a book for Rs 240 and a pencil box for rs 46. He sold the pen for Rs 68, book for Rs 250 and the pencil box for Rs 52. What was his gain or loss?
A man starts from his home and drives 48Km to the north and then 72Km to the south. How far is he from his home finally?