Number System

Integers are a collection of natural numbers, zero and negative numbers.

Integers are denoted by ‘Z’

Integers can be represented by using a number line

In a number line, zero occupies the central position, separating positive and negative integers.

To the left of zero lies the negative integers and to the right of zero lies positive integers.

In a number line, integers are placed in an increasing order from left to right.

For example: -3, -2, -1, 0, 1, 2, 3 are in ascending order.

Moving along the left, integers are arranged in decreasing order

For example: Among -10, -9, -8, -7 etc, -7 > -8 > -9 > -10 is the order.

Among -65 and 65, 65 > -65 is the order.

1. Write the following integers in ascending order:

-5, 0, 2, 4, -6, 10,-1, -6, -5, -1, 0, 2, 4, 10 is the order.

2. If the integers –15, 12, –17, 5, –1, –5, 6 are marked on the number line then the integer on the extreme left is:-17

3. Write the given integers in descending order:

–27, 19, 0, 12, –4, –22, 47, 3, –9, –35, 47, 19, 12, 3, 0, -4, -9, -22, -27, -35

Addition of integers

Addition of integers can be done by using the number line.

To add (+5) and (–3), we start at five. Then, we move three steps backwards (since 3 has – sign) We reach +2, which is the answer

We can also do this by starting at -3 in the number line and move five steps forward. We reach +2, which is the answer

Similarly, to add (-6) and (-4), We can start at (-6) and move 4 steps backward (since four has negative sign). We can also start at (-4) and move six step backward to get the same answer.

We get -10, which is the answer.

To add (-6) and 4, we start at -6 and move four steps forward to reach (-2), which is the answer.

Examples

Add 10 and -15

Start at 10 on number line and move 15 steps backward. We get (-5) which is the answer.

Add (-7) and (-9)

Start at -7 and move 9 steps backward. We get (-16) which is the answer.

Rules for addition of integers

1. The integer without sign represents positive integer.

For eg: 9represents (+9), 27 represents (+27), etc.

2. When we add two integers of the same sign, the sum will also be an integer of the same sign.

Add (-70) and (-12)

To add (-70) and (-12), we add the two numbers 70 + 12 =82) and we put the sign (here, - sign). Hence, (-82) is the answer.

Add 103 and 39

103 + 39 = 142

3. When we add two integers of different sign, the sum will be the difference between the two integers and have the sign of the integer with greater value.

For eg: While adding (-7) and (+4) the answer is (-3) which is the difference between 7 and 4 with the sign of the greater number, 7.

Ie. Negative sign.

Examples

Add (-40) and 30

To add (-40) and 30, we find the difference between the two numbers (40 – 30 =10) and we put the sign of the greater number (here, - sign). Hence, -10 is the answer.

Add 60 and (–50)

To add 60 and (-50), we find the difference between the two number (60 – 50 =10) and we put the sign of the greater number ( here, + sign).

Hence, +10 is the answer.

Word problems

1. Sita saved Rs. 225.00 and she has spent Rs. 400 on credit basis for the purchase of stationery. Find her due amount?

The amount Sita has = 225 Rs

The amount spent for stationery on credit = 400 Rs

The due amount to be paid = 225 – 400 = – 175 Rs

Therefore, Sita has to pay Rs 175

Properties of Addition of integers

1. Closure property

The sum of two integers is an integer. This property is called closure property.

Since integers obeys closure property, we can say that integers are closed under addition.

For eg: The sum of -12 and +12 gives zero, which is also an integer.

Therefore, for any two integers a and b, (a + b) is also an integer.

2. Commutative property

The order in which we add two integers does not matter. This is called commutative property.

For eg: (-39) + 32 and 32 + (-39) gives the same answer, (-7)

(-8) + (-18) and (-18) + (-8) gives same answer, (-26)

Therefore, for any two integers, a and b, a + b = b + a

3. Associative property

Regrouping of integers does not change the value of their sum. This property is called associative property.

For eg: Take (-5), (-9) and 13. Now, let us group them in different way

[(-5) + (-9)] + 13 and (-5) + [(-9) + (13)]

[(-5) + (-9)] + 13 = [(-14)] + 13 = -1

(-5) + [(-9) + (13)] = (-5) + [(4)] = -1

Both of the cases give same answer. Hence, integers obey associative property.

Therefore, for any three integers a, b, and c, a + ( b +c ) = ( a + b) + c

4. Additive identity

Adding zero to any integer gives the same integer. Due to this property, zero is called the identity with respect to addition or ‘additive identity’.

For eg: 0 + 99 = 99, 0 + (-27) = -27, 0+ (–85) = (–85)

Therefore, for any integer a, a + 0 = 0 + a = a

5. Additive inverse

The pair of opposite integers, whose sum is zero is called additive inverse.

In a number line, additive inverses are equidistant from zero.

For example: (-17) and +17, (-5) and 5, (8) and (-8) are additive inverses of each other.

Therefore, for any integer a, –a is the additive inverse.

A+ (–a) = (–a) +a = 0

Subtraction of integers

Subtraction of integers can be done using the number line.

For subtraction, we move backward in a number line. But, when two (-) signs come next to each other, we move forward.

To subtract (+4) from (+7), we start at 7 and move four steps backward to reach (+3), which is the answer.

Subtract (6) from (-9)

(-9) - (6), we start at (-9) and move six steps backward to reach (-15), which is the answer.

Subtract (-5) from (-8)

(-8) – (-5). We start at (-8) and move five steps forward ( we move forward when two negative signs come next to each other). We reach (-3), which is the answer.

Subtract (-7) from (8)

8 – (-7) We start at 8 and move 7 steps forward, to reach 15, which is the answer

Rules for subtraction of integers

1. To subtract a negative integer from a negative integer, we add the additive inverse of the integer to be subtracted.

For eg: Subtract (-3) from 7

7 – (-3)

Add the additive inverse of (-3), ie +3

= 7 +3

= 10

Find the values of: (–6)–(–2), 26 – (+10), 35 + (–7)

(a) -6 – (-2) = -6 +2

= -4

(b) 26 – (+10) = 26 -10

= 16

= 35 – 7

=28

Note

Every subtraction statement has a corresponding addition statement.

For example, 8 – 5 = 3 is a subtraction statement. This can be seen as the addition statement 3 + 5 = 8

In the same way, (-8) – (-5) = (-3) is a subtraction statement which can be written as the addition statement (-8) = (-3) + (-5)

Properties of Subtraction of integers

1. Closure property

The difference of two integers is an integer. This property is called closure property.

Since integers obey closure property, we can say that integers are closed under subtraction.

For eg: The difference between (-7) and (-2) gives (-5), which is also an integer.

Therefore, for any two integers, a and b; a–b is also an integer.

The commutative and associative properties do not hold for subtraction.

Multiplication of Integers

Multiplication of negative integers is also repeated addition just like positive integers or whole numbers.

For eg: [(-5) + (-5) + (-5)] can be represented as (-5) × (-3)

The following number line shows this

The product of two negative integers is always a positive integer.

For eg: (-8) × (-6) = 48, (-9) ×− (-9) = 81

The product of a positive and negative integer is a negative integer

For eg: (-5) × (7) = (-35), (4) × (-5) = (-20)

Which of the following is incorrect?

a. -55 × -22 × -33 < 0 b. (-1521) × 2511 > 0

A is correct as the product of two negative numbers is positive. The

Product of that positive and negative number is a negative (less than 0)

b is incorrect as the product of a positive and negative number is always negative (less than 0)

Properties of multiplication of integers

1. Closure property

The product of two integers is an integer. Therefore, integers are closed under multiplication.

For eg: (-9) × 5 = (-45), which is also an integer

(-6) × (-6) = (-36), which is also an integer

Therefore, for any two integers a and b; a × b is also an integer.

2. Commutative property

Changing the order of multiplication does not change the value of the product.

Thus, multiplication of integers is said to be commutative.

For eg: 21 × (-5) = (-105), (-5) × 21 = (-105)

(-8) × (-9) = 72, (-9) × (-8) = 72

In both cases, we get same answer, even though order is changed.

Therefore, for any two integers a and b, a × b = b × a

3. Associative property

Re grouping of integers does not change the value of their product. This property is called associative property.

For eg: [(-5) ×7] × (-18) and (-5) × [7× (-18)] are equal

[(-5) ×7] × (-18) = [-35] × (-18) = 630

(-5) × [7× (-18)] = (-5) × [126] = 630

Therefore, for any three integers a, b, c, (a × b) ×c = a× (b × c)

4. Multiplicative identity

Multiplying one with any integer gives the same integer. Therefore, 1 is called multiplicative identity.

For eg: 1× (-7) = (-7) ×1 = (-7), 57 ×1 = 1× 57 = 57

Therefore, for any integer a, a×1 = 1×a =a.

Note

The product of odd number of negative integers is negative.

Eg: (-3) × (-4) × (-6) = (-72), (-2) × (-4) × (-6) = (-48)

(-1) × (-3) × (-5) × (-7) × (-9) = (-105)

The product of even number of negative integers is positive.

For eg: (-6) × (-7) = 42, (-1) × (-3) × (-5) × (-7) = 105

(-1) × (-1) × (-1) × (-1) = +1

Generally, if negative integers are multiplied even number of times, the product is a positive integer, whereas, if negative integers are multiplied odd number of times, the product is a negative integer.

Distributive property of multiplication over addition

For integers, multiplication is distributive over addition.

For eg: (-2) × (4+5) = [(-2) ×4] + [(-2) ×5]

LHS = (-2) × (4+5) RHS = [(-2) ×4] + [(-2) ×5]

= (-2) × (9) = [-8] + [-10]

= -18 = -18

LHS = RHS. Therefore, multiplication distributes over addition for integers.

Therefore, for any three integers a, b, and c , a × (b+c) = (a × b) + (a×c)

Word problems

1. During summer, the level of the water in a pond decreases by 2 inches every week due to evaporation. What is the change in the level of the water over a period of 6 weeks?

Change in level of water in one week = -2 inches

The change in level of water after six weeks = (-2) ×6 = -12 inches

Therefore, the level of water decreases by 12 inches after 6 weeks

Division of integers

The rules for division of integers is similar to that of multiplication

To divide integers with same sign, divide as usual and put the sign.

The division of two integers with the same sign gives a positive integer

For eg: 72 ÷ 9 = 8, (-60) ÷ (-12) = 5

To divide integers with opposite sign, divide as usual and put negative sign.

The division of two integers with opposite signs gives a negative integer

For eg: (-81) ÷3 = (-27), 50 ÷ (-5) = (-10)

Properties of division

Integers are not closed under division.

For eg: (-7) ÷5 is a fraction and is not an integer. Similarly, 9 ÷ (-5) is not an integer.

Integers are not commutative under division.

For eg: (-14) ÷ 7 = (-2), but 7 ÷ (-14) ≠ (-2) similarly, 15 ÷ 5 ≠ 5÷ 15

Integers are not associative under division.

For eg: [(-12 ÷2)] ÷3 and (-12) ÷ [(2÷3)] are not equal.

Similarly, [9÷ (-3)] ÷ 3 and 9÷ [(-3) ÷3] are not equal.

There is no identity element for integers under division

For eg: Take the case of 1. 7÷1 = 7, but 1÷7 ≠ 7

Note

Each multiplication statement has two division statements in whole numbers.

For eg: The multiplication statement 8×9 = 72 has 72÷ 8 = 9 and 72÷ 9 =8 as division statements.

An integer divided by zero is not defined, but zero divided by any number is zero.

For eg: 1÷ 0 is not defined, but 0 ÷1 = 0

1. An elevator descends into a mine shaft at the rate of 5 m/min. If the descent starts from 15 m above the ground level, how long will it take to reach - 250 m?

Rate of descent of mine shaft = 5 m/min

Starting position of the shaft = 15 m above the ground

Final position of the shaft = 250 m below the ground

To reach -250 m, the shaft has to descend 15 m below to reach ground

Level, and then descend 250 m below ground level.

Therefore, total distance = 250 + 15 = 265 m

Time taken =

= = 53

Therefore, the shaft takes 53 minutes to reach (-250) m.

Summary

Integers are a collection of natural numbers, zero and negative numbers.

The number line gives a visual representation of the set of all integers with positive integers to the right of zero and negative integers to the left of zero.

The sum of the two positive integers is positive and two negative integers is negative

The sum of a positive and a negative integer is the difference of the two numbers in value and has the sign of the greater integer.

The addition of integers has the closure, commutative and associative properties

The product of two positive integers and two negative integers are positive

The product of two integers with opposite signs is negative

The multiplication of integers has the closure, commutative and associative properties

The integer 0 is the additive identity for integers.

The integer 1 is the multiplicative identity for integers