What are Integers in Math

Integers are set of all negative and positive whole numbers. It also includes zero. In other words, integers are the set of whole numbers and their opposites - {...,-3,-2,-1,0,1,2,3,...}

The number line is used to represent integers. Let's first understand the number line.

Number line


Number Line

  These numbers are to the right of zero on the number line are called positive integers. They are +1, +2, +3…………
  These numbers are to the left of zero on the number line are called negative integers. They are -1, - 2, - 3…………
  The integer zeros neither positive nor negative and has no sign.

Note:

The number line goes on both directions.

On a number line when we have


Number Line

 We move to the right to add a positive integer.
  We move to the left to add a negative integer.
  We move to the left to subtract a positive integer.
  We move to the right to subtract a negative integer.


Adding of integers


  When two positive integers are added the result will be a positive integer.
Ex. 10 + 3 = 13
  When two negative integers are added the result will be a negative integer.
Ex. (-1) + (-3) = - 4
  When a positive integer and a negative integer are added, the result will be a negative or positive integer.
Ex. (-3) + 5 = 2 and 3+ (-5) = -2
We take their difference and place the sign of the bigger integer.


Additive inverse


Additive inverses are the opposite integer to the given integer.
Let us say the number 7.
Additive inverse of an integer 7 is (– 7) and additive inverse of (– 7) is 7.

Simple Example


 64

Explanation:

52+12
We have to add both positive numbers. We will get a positive integer
= 64


Advance Example

Example 1:
 -22

Explanation:

First we will write the additive inverse of 77 is -77.
-77 + 55
= -22


 -120

Explanation:

First add the integer then put minus sign.
= (–80) + (– 40)
= –120


 372

Explanation:

(–200) – (–572)
= –200 + 572
Subtract the integers.
= 372


Real Life Examples


In a singing competition, positive points are given for correct singing and negative points are given for incorrect singing. If Sophia got points in five rounds were 50, – 15, – 10, 75 and 40, what were her total points in all five rounds?
 Sophia got total 140 points in all 5 rounds.

Explanation :

We have to find total points in all five rounds.
Sophia got points in five rounds were 50, – 15, – 10, 75 and 40
= 50+ (– 15) + (– 10) + 75 + 40
= (50+ 75 + 40) + (– 15 ) + (– 10)
= 165 + (-25)
= 165 – 25
=140

Max deposited Rs 5,000 in his bank account. He withdraws Rs 3,150 from it for some purpose. Find the balance in Max's account after the withdrawal.

 The balance in Max's account after the withdrawal is Rs. 1850

Explanation :

Deposited amount is represented by a positive integer because it is adding money to the account.
Withdrawal of amount from the account is represented by a negative integer
Amount deposited= Rs 5,000
Amount withdrawn= Rs 3,150
The balance in Max's account= 5,000 + (-3150)
= 5,000 – 3150
= 1850

Jennifer moves for picnic from school which is far away from the school 55km north. Her house is at distance of 25km south along the same road from picnic place. If she returns back to the house after a picnic directly, how will you represent the distance that she has to travel everyday to reach to the school towards south?

 Jennifer has to travel 30km everyday to reach to the school towards south.

Explanation :

The distance towards north is represented by a positive integer and distance towards south is represented by a negative integer.
Picnic spot from school 55km north
Her house is at distance of 25km south
Distance from house to school is 55km + (-25 km)
55km + (-25 km)
= 55 - 25
=30


Multiplication of Integers


 Points to Remember

If Both Numbers are Positive or Negative


If any two positive integers a and b,
(-a ) × (-b) = a × b
If we take (-12) × ( -5)
  First multiply both the negative numbers as whole integers. (-12) × (-5)
  Then put positive sign before the product you obtained.
(-12) × (-5) = 12 × 5 = 70

Note:

 If both numbers are positive, the product is positive.
 If both numbers are negative, the product is positive.


If One Number is Positive and the other Negative


If any two positive integers a and b,
(-a) × b = a × (-b) = - ( a×b)
If we take (12) × ( -5),
  First find the product of numbers. (12)×( -5)
  Then put minus sign before the product you obtained.
(12) × ( -5) = -(12 × 5) = -70

Note:

 If one number is positive and the other negative, the product is negative.
 If we have one number is negative and the other positive, the product is negative.
For example, (-12) × ( 5) = -(12 × 5) = -70


Simple Example


Example 1:
 245

Explanation:

First multiply both the positive numbers
35 × 7
= 245

Example 2:
 -76

Explanation:

Multiply both the negative numbers as whole integers and put then minus sign.
= - ( 38 × 2)
= -( 76)
= -76


Multiplication of Three or more Integers


We conclude here that if the number of negative integers that multiplied are even ( two, four, six) the product will be a positive integer but if the number of negative integers that multiplied are odd ( three, five) the product will be a negative integer.
For example,
(-3) × (-3) × (-3) = -(3 × 3 × 3) = -27 but (-3) × (-3) × (-3) × (-3)= (3 × 3 × 3 × 3)= 81

Simple Example


 120

Explanation:

(-15) × [(-2) × (-2) × (-2)]
= (-15) × - ( 2×2×2)
= (-15) × - 8
= 15 × 8
=120


 -90

Explanation:

(-10) × [(-3) × (-3)]
= (-10) × (3 × 3)
= (-10) × 9
= - (10 × 9)
= - 90


Properties of Multiplication of Integers

Commutativity of multiplication


For any two positive integers a and b,
a×b = b×a
For example, 2× (-5) = -10. And (-5) × 2 = -10. So that 2 × (-5) = (-5) × 2


Multiplication by Zero


For any integer a, a × 0 = 0 × a = 0
For example, (–2) × 0 = 0; 0 × (– 5) = 0; 7 × 0 =0


Note:

If any whole number multiplied by zero the product will be zero.


Multiplication by 1


For any integer a we have, a × 1 = 1 × a = a
For example, (–2) × (–1) = 2; 5 × (–1) = –5; 6 × 1=6


Multiplication of three integers


For any three integers a, b and c, (a × b) × c = a × (b × c)
If we say, [(–4) × (–2)] × 5 = 8 × 5 = 40
And, (–4) × [(–2) × 5] = (–4) × (–10) = 40
So that, [(–4) × (–2)] × 5= (–4) × [(–2) × 5]


Distributive Property


For any integers a, b and c, a × (b + c) = a × b + a × c
For example, (– 7) × [(–3) + (–2)] = (– 7) × (–5) = 35
And [(– 7) × (–3)] + [(– 7) × (–2)] = 21 + 14 = 35
So, (– 7) × [(–3) + (–2)] = [(– 7) × (–3)] + [(– 7) × (–2)]


Simple Example


 -25

Explanation:

(– 5) × [(–2) + 7]
= (– 5) × 5
= –25


Advance Examples


 -15

Explanation:

[(– 3) × (–2)] + [(– 3) × 7]
= (3×2) + (-21)
= 6 + (– 21)
= – (21– 6)
= – 15


 16

Explanation:

(– 4) × [(–3) + (–1)]
= (– 4) × [–(3+ 1)]
= (– 4) × (– 4)
= 16


 -104

Explanation:

[–(8× 10)] – [(8 × 3)]
= –80–24
= – (80 + 24)
= – 104


Calculations can be made easier using these properties.


For example, 18 × 12
We can write this as 18 × (10 + 2).
So that, 18 × 12
= 18 × (10 + 2)
= 18 × 10 + 18 × 2
= 180 + 36
= 216

Simple Example

 -1450

Explanation:

(–25) × 58
= (–25) × [60 – 2]
= (–25) × 60 – (–25) × 2
= (–1500) – (– 50)
= –1450


 2940

Explanation:

(–30) × (–98)
= (–30) × [(–100) + 2]
= (–30) × (–100) + (–30) × 2
= 3000 + (–60)
= 2940


 -324

Explanation:

54 × (– 4) + (–54) × 2
=54 × (– 4) + (–54) × 2
= 54 × (– 4) + 54 × (–2)
= 54 × [(– 4) + (–2)]
= 54 × [(–6)]
= –324


 -1400

Explanation:

70 × (–19) + (–1) × 70
= 70 × (–19) +70× (–1)
= 70 × [(– 19) + (–1)]
= 70 × [(– 20)]
= –1400


 1020

Explanation:

(–17) × (–10) × 6
= [(–17) × (–10)] × 6
= 170 × 6
= 1020


 -3200

Explanation:

(–40) × (–2) × (–5) × 8
= – 40 × (–2 × –5) × 8
= [–40 × 10] × 8
= – 400× 8
= – 3200


Integers: Word Problems

There is a class test given to students which has all 10 questions of 20 marks. Every correct answer can give 2 marks and (–1) marks are given for every incorrect answer.
1. Elizabeth writes only 5 answers correct. What will be her score?
2. Jesson writes all 10 answers correct. What will be her score?
 Elizabeth’s total score = 5 And Jesson’s total score = 20

Explanation :

1. Elizabeth writes only 5 answers correct.
Marks given for one correct answer = 2
Marks given for one incorrect answer = – 1
Elizabeth writes 5 answers correct= 5×2= 10
So that, Elizabeth writes 5 answers incorrect= 5×– 1= – 5
Therefore, Elizabeth’s total score = 10 + (–5) = 5

2. Jesson writes all 10 answers correct.
Marks given for one correct answer = 2
Marks given for one incorrect answer = – 1
Jesson writes all 10 answers correct= 10×2= 20
Therefore, Jesson’s total score = 20


Division of a integer


For any two positive integers a and b
a ÷ (–b) = (– a) ÷ b where b ? 0
For example,
56 ÷ (–8) = –7 and 40 ÷ (–10) = –4
As well as 56 ÷ (–7) = –8 and 40 ÷ (–4) = –10


For any two positive integers a and b
(– a) ÷ (– b) = a ÷ b where b ? 0
For example, (–56) ÷ (–8) = 7 and (–40) ÷ (–10) = 4
As well as (–56) ÷ (–7) = –8 and (–40) ÷ (–4) = 10

Note:

  •  When we divide a positive integer by a negative integer, we first divide them as whole numbers and then put a minus sign (–) before the quotient.
  •  When we divide a negative integer by a negative integer, we first divide them as whole numbers and then put a positive sign (+).

Division by Zero


Any integer divided by zero is not defined but zero divided by an integer other than zero is equal to zero.
For example,
0 ÷ a = 0 for a 0

Any integer divided by 1 gives the same integer.
For any integer a, a ÷ 1 = a
For example,
(– 7) ÷ 1 = (– 7), (–12) ÷ 1 = –12, (13) ÷ 1 = 13

Simple Example


 -5

Explanation:

(14) × (–5) – (–75) ÷ (5) + 50
= (14) × (–5) – (–75) ÷ (5) + 50
= (–70) – (–75) ÷ (5) + 50
= (–70) – (–15) + 50
= (–70) +15+ 50
= (–55) + 50
= –5


Integers : Word Problems


In a test (+10) marks are given for every correct answer and (–5) marks are given for every incorrect answer. Monika attempted 10 questions and scored 50 marks. How many incorrect answers had they attempted?
 10

Explanation :

Marks given for one correct answer = 10
Marks given for one correct answer = 10
Monika attempted 10 questions = 10 × 10 = 100
But she score = 50
Marks obtained for incorrect answers = 50 – 100 = – 50
Marks given for one incorrect answer = (–5)
Therefore, number of incorrect answers = (– 50) ÷ (–5) = 10


 Points to Remember

Quick Tips

  •  When multiply the integers the product sign depends on the number of negative integers that multiplied are even or odd.
  •  When add or subtract the integers the answer sign depends on the negative integers that are bigger or smaller.

Think

  •  Can you write a pair of integers whose sum is negative number?
  •  Multiply 7 negative integers and 13 positive integers? What would be the sign of the product?





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