6+(-8) integer solving
Answers
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

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

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
1. What is 52 + 12?
Advance Example
Example 1:
1. What is the Additive inverse of 77? Which number we get if it is added with 55?
2. What is (–80) – 40?
3. What is (–200) – (–572)?
Real Life Examples
Singing Competition
Bank Membership Example
Distance Example
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 = 60
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) = -60
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) = -60
Simple Example
Example 1:
1. What is 35 × 7?
Example 2:
1. (-38) × (-2)
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
1. (-15) × [(-2) × (-2)× (-2)]
2. (-10) × [(-3) × (-3)]
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] = ples
1. Solve: [(– 3) × (–2)] + [(– 3) × 7]
2. Solve: (– 4) × [(–3) + (–1)]
3. Solve: [(–8) × 10] – [ ( –8) × ( –3)]
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
1. (–25) × 58
2. (–30) × (–98)
3. 54 × (– 4) + (–54) × 2
4. 70 × (–19) + (–1) × 70
5. (–17) × (–10) × 6
6. (–40) × (–2) × (–5) × 8
Integers: Word Problems
Classroom Test Example
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 (+).