Integers

 Mind Maps

Class VII - maths: Integers
Q) Evaluate (-40)/5 ?

Q) Find the blank. 369 / ? = 369.

Q) Define the association property?

Q) Find the 16 *14 ?

Q) Define the Commutative property?

Q) Divide -98 by -14?

Q) The numbers 1, 2, 3, 4, 5, 6, 7, 8, ........, i.e. natural numbers, are called

Q) When two integers have the same sign, their product is the product of their absolute values with the _______ sign.

Q) Greatest negative integer is ?

Q) a×(b+c)=a × b +a × c is called?

Q) Verify (-40) * [18 + (-6) }=[ (-40) * 18] + [(-40) * (-6)]?

Q) (i) For any integer a, what is (-1)* a equal to? (ii) Determine the integer whose product with (-12) is (a) -24) ?(b) 39?

Q) Find the product, using suitable properties? (a) 24 * (-43) + (-54) * (56) (b) 15 * (-35) * (-4) * (-10) (c) (-17) *(-18) (d) (-42) *(154).

Q) Starting from (-1)* 5, write various product showing some pattern to show (-1)* (-1)=1?

Q) An elevator descends into a mine shaft at the rate of 6 m/mim. If the descent starts from 10 m above the ground , how long will it take to reach -360m?

Q) 0x (-5) is equal to ?

Q) If 16÷x is equal to ?

Q) The integer whose product with(-1) is 0,is?

Q) (-2) × 1 is equal to

Q) The integer whose product with(-1) is 1, is

Q) The temperature at 12 noon was 10° C above zero.If it decreases at the rate of 3° C per hour until midnight, at what time would the temperature be 8° C below Zero ? What would be the temperature at mid-night?

Q) Under addition and multiplication, integer show a property called distributive property. That is, a* ( b+c)= a* b+ a*c for any three integers a,b and c?

Q) A cement company earn a profit of $9 per bag of white cement sold and a loss of$ 3 per bag of yellow cement sold? (a) The company sell 3,000 bags of white cement and 5300 bags of yellow cement in a month. what is its profit or loss? (b) What is the number of white cement bags it must sell to have neither profit nor loss , if the number of yellow bags sold is 6,900 bags?

Q) The properties of commutativity, associativity under addition and multiplication ,and the distributive property to make our calculation easier?

Q) (a)We studied in the earlier class, about the representation of integers on the number line their addition and subtraction?
(b) Integer are a bigger collection of numbers which is formed by whole numbers and their negatives?

Q) If x is an integer different from 0, then x ÷ x =?

Q) Find the products of (-12) × (-13) × (-5) =?

Q) (-20)× (-5) is equal to?

Q) Opposite of the 50 km South is?

Content
• Integers
• Properties Of Addition And Subtraction Of Integers
• Multiplication Of Integers
• Properties Of Multiplication Of Integers
• Properties Of Division Of Integers
• Examples on Division of Integers

Associativity property
The multiplication of integers is associative, i.e., for any three integers a, b, c, we have
a * ( b * c) = (a * b) * c

Two positive integers integers a and b we can say
a xx (-b) =(-a) xx b = -(a xx b)

Multiplication Of Two Negative Integers
in general for any two positive integers a and b,
(- a) xx (-b) = a xx b

Closer Under Multiplication
" a x b is an integer, for all integers a and b"

commutativity of multiplication
a xx b = b xx a

Multiplication By Zero
a xx 0 = 0 xx a = 0
Multiplication of a negative integer and zero is zero

Distributivity of multiplication over addition property
The multiplication of integers is distributive over their addition.
That is, for any three integers a, b, c, we have
a * (b + c) =a * b + a * c
(b + c) * a = b * a + c * a

Existence of multiplicative identity property)
For every integer a, we have
a * 1 = a = 1 * a

Existence of multiplicative identity property)
For any integer, we have
a * 0 = 0 = 0 * a

Associativity For Multiplication
For any integer a, we have
a * (-1) = -a = (-1) * a

Formulae Of Division Of Integers

Integers

In integers we know that the numbers are
........... -6, -5, -4, - 3,- 2,- 1, 0, 1, 2, 3, 4, 5, 6, ...........
The numbers 1, 2, 3, 4, 5, 6, 7, 8, ........, i.e. natural numbers, are called positive integers
and the numbers - 1,- 2, - 3, - 4, - 5, - 6, -7, -8, ........, are called negative integers.
The number " 0 " is simply an integer. It is neither positive nor negative.

We know about the addition and subtraction of integers. In seventh grade maths under integers we will learn about multiplication and division of integers. We will also learn about various properties of these operations on integers.

In integers we will discuss on this in details and solve various types of examples on integers.

• The numbers ........, -6, -5, -4, -3,-1, 0, 1, 2, 3, 4, 5, 6, ........ etc. are integers.
• 1, 2, 3, 4, 5, 6, 7, ........ are positive integers and -1, -2, -3, -4, -5, -6, -7, ........ are negative integers.
• 0 is an integer which is neither positive nor negative.
• On an integer number line, all numbers to the right of 0 are positive integers and all numbers to the left of 0 are negative integers.
• 0 is less than every positive integer and greater than every negative integer.
• Every positive integer is greater than every negative integer.
• Two integers that are at the same distance from 0, but on opposite sides of it are called opposite numbers.
• The greater the number, the lesser is its opposite.
• The sum of an integer and its opposite is zero.
• The absolute value of an integer is the numerical value of the integer without regard to its sign. The absolute value of an integer x is denoted by |x| and is given by
• The sum of two integers of the same sign is an integer of the same sign whose absolute value is equal to the sum of the absolute values of the given integers.
• The sum of two integers of opposite signs is an integer whose absolute value is the difference of the absolute values of addend and whose sign is the sign of the addend having greater absolute value.
• To subtract an integer y from another integer x, we change the sign of y and add it to x.

Thus, x - y = x + (-y)

• All properties of operations on whole numbers are satisfied by these operations on integers.
• If m and n are two integers, then (m - n) is also an integer.
• -m and m are negative or additive inverses of each other.
• To find the product of two integers, we multiply their absolute values and give the result a plus sign if both the numbers have the same sign or a minus sign otherwise.
• To find the quotient of one integer divided by another non-zero integer, we divide their absolute values and give the result a plus sign if both the numbers have the same sign or a minus sign otherwise.
• All the properties applicable to whole numbers are applicable to integers in addition; the subtraction operation has the closure property.
• Any integer when multiplied or divided by 1 gives itself and when multiplied or divided by - 1 gives its opposite.
• When expression has different types of operations, some operations have to be performed before the others. That is, each operation has its own precedence. The order in which operations are performed is division, multiplication, addition and finally subtraction(DMAS) .
• Brackets are used in an expression when we want a set of operations to be performed before the others.
• While simplifying an expression containing brackets, the operations within the innermost set of brackets are performed first and then those brackets are removed followed by the ones immediately after them till all the brackets are removed.
• While simplifying arithmetic expressions involving various parentheses or brackets and operations, we use as PEMDAS rule or BODMAS rule. Both are same some followPEMDAS rule or some follow BODMAS rule.
• PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
• BODMAS stands for Brackets, Order, Division, Multiply, Addition, Subtraction.,
Properties Of Addition And Subtraction Of Integers

Sum of the two whole numbers is again a whole number.

example
19 + 34 =53
which is again a whole number..
This property is known as the Closure Property For Addition Of The Whole Numbers
example
37 + (-37) = 0
27 + 33 = 60
- 20 + 0 = -20..
addition of integers gives integers, we say Integers Are Closed Under Addition
Generally..,for any two integers x and y , x+y is an integer.

Closure Under Subtraction

example
statement
(1) 3 - 5 = -2
observation
integer
statement
(2) 15 - (-11)...?
observation
note If x and y are two integers then x-y is also an integer.

#### Property 2

Commutativity property

For any two integer's m and n, we have

m * n = n * m

That is, multiplication of integers is commutative.

For example:

• 7 * (-3) = -(7 * 3) = -21 and (-3) * 7 = -(3 * 7) = -21
Therefore, 7 * (-3) = (-3) * 7

• (-5) * (-8) = 5 * 8 = 40 and (-8) * (-5) = 8 * 5 = 40
Therefore, (-5) * (-8) = (-8) * (-5).

#### Property 3

Associativity property

The multiplication of integers is associative, i.e., for any three integers a, b, c, we have
a * ( b * c) = (a * b) * c

For example:

• (-3) * {4 * (-5)} = (-3) * (-20) = 3 * 20 = 60
and, {(-3) * 4} * (-5) = (-12) * (-5) = 12 * 5 = 60
Therefore, (- 3) * {4 * (-5)} = {(-3) * 4} * (-5)

• (-2) * {(-3) * (-5)} = (-2) * 15 = -(2 * 15)= -30
and, {(-2) * (-3)} * (-5) = 6 * (-5) = -(6 * 5) = -30
Therefore, (- 2) * {(-3) * (-5)} = {(-2) * (-3)} * (-5)

Adding zero to a number leaves it unchanged:
a + 0 = 0 + a = a
Identity
An equation that is true no matter what values are chosen.
Example:
a/2 = a xx 0.5 is true no matter what value is chosen for "a"

Multiplication Of Integers

#### Multiplication Of a positive and a negative integer

Multiplication Of whole numbers is repeated addition.
Example
7 + 7 + 7 +7 = 4 ⨯ 7 =28

we have from the following number line, ( 2 ) + ( 2 ) + ( 2 ) + ( 2 ) = 8

but we can also write
( 2 ) + ( 2 ) + ( 2 ) + ( 2 ) = 4 ⨯ 2
4 ⨯ 2 = 8

two positive integers a and b we can say

a ⨯ ( -b ) =( -a ) ⨯ b = -( a ⨯ b )

#### Multiplication Of Two Negative Integers

in general for any two positive integers a and b,

( - a ) ⨯ ( - b ) = a ⨯ b

(1) Find - 4 ⨯ ( - 2 ⨯ - 4 ) = ?
- 2 ⨯ - 4 = 8
- 4 ⨯ 8 = - 32 { 4 ⨯ ( 2 ⨯ 4 ) = 4 ⨯ 8 }
=> - 4 ⨯ ( - 2 ⨯ - 4 ) = - 32

#### Product Of Three Or More Negative Integers

Properties Of Multiplication Of Integers

#### Closure Under Multiplication

" a ⨯ b is an integer, for all integers a and b"
ex:

Statement Inference
( - 10 ) ⨯ ( - 3 ) = 30 product is an integer
( - 22 ) ⨯ ( 5 ) = - 110 product is an integer
( - 11 ) ⨯ ( 5 ) = ? ?

#### commutativity of multiplication

a ⨯ b = b ⨯ a

multiplication is commutative for whole numbers.

Statement1 Statement2 Inference
( - 7 ) ⨯ ( - 3 ) = 21 ( - 3 ) ⨯ ( - 7 ) = 21 product is an integer
( - 8 ) ⨯ ( 5 ) = - 40 ( - 5 ) ⨯ ( 8 ) = - 40 product is an integer
( - 0 ) ⨯ ( 5 ) = 0 ( - 5 ) ⨯ ( 0 ) = ? ?

#### Multiplication By Zero

a ⨯ 0 = 0 ⨯ a = 0
Multiplication of a negative integer and zero is zero
example
0 ⨯ 8 = 0
( - 1) ⨯ 0 = 0
( - 11 ) ⨯ 9 ⨯ 7 ⨯ 0 = ?

### Distributivity of multiplication over addition property

The multiplication of integers is distributive over their addition. That is, for any three integers a, b, c, we have

a * (b + c) =a * b + a * c
(b + c) * a = b * a + c * a

For example:

(i) (-3) * {(-5) + 2} = (-3) * (-3) = 3 * 3 = 9
and, (-3) * (-5) + (-3) * 2 = (3 * 5 ) -( 3 * 2 ) = 15 - 6 = 9
Therefore, (-3) * {(-5) + 2 } = ( -3) * (-5) + (-3) * 2.

(ii) (-4) * {(-2) + (-3)} = (-4) * (-5) = 4 * 5 = 20
and, (-4) * (-2) + (-4) * (-3) = (4 * 2) + (4 * 3) = 8 + 12 = 20
Therefore, (-4) * {-2) + (-3)} = (-4) * (-2) + (-4) * (-3).

Note: A direct consequence of the distributivity of multiplication over addition is
a * (b - c) =a * b - a * c

#### Property 5 (Existence of multiplicative identity property)

For every integer a, we have
a * 1 = a = 1 * a

The integer 1 is called the multiplicative identity for integers.

#### Property 6 (Existence of multiplicative identity property)

For any integer, we have

a * 0 = 0 = 0 * a

For example:

• m * 0 = 0
• 0 * y = 0

### Associativity For Multiplication

For any integer a, we have

a * (-1) = -a = (-1) * a

Note:

(i) We know that -a is additive inverse or opposite of a. Thus, to find the opposite of inverse or negative of an integer, we multiply the integer by -1.

(ii) Since multiplication of integers is associative. Therefore, for any three integers a, b, c, we have

(a * b) * c = a * (b * c)

example

In what follows, we will write a * b * c for the equal products (a * b) * c and a * (b * c).

(iii) Since multiplication of integers is both commutative and associative. Therefore, in a product of three or more integers even if we rearrange the integers the product will not change.

(iv) When the number of negative integers in a product is odd, the product is negative.

(v) When the number of negative integers in a product is even, the product is positive.

#### Property 8

If x, y, z are integers, such that x > y, then

• x * z > y * z, if z is positive
• x * z < y * z , if z is negative.

These are the properties of multiplication of integers needed to follow while solving the multiplication of integers.

### Examples on Multiplication of Integers

Examples on multiplication of integers on different types of questions on integers are discussed here step by step.

1. Find the products of (-10) * 11

Solution:

• (-10) * 11
= - (10 * 11)
= - 110

2. Find the products of 5 * (-7)

Solution:

• 5 * (-7)
= - (5 * 7)
= - 35

3. Find the products of (-115) * 8

Solution:

• (-115) * 8
= - (115 * 8)
= - 920

4. Find the products of 9 * (-3) * (-6)

Solution:

• 9 * (- 3) * (- 6)
= - {(9 * (-3)} * (- 6)
= -(9 * 3) * (-6)
= - 27 * (-6)
= 27 * 6 = 162

5. Find the products of (-12) * (-13) * (-5)

Solution:

• (- 12) * (- 13) * (-5)
= {(-12) * (-13)} * (-5)
= (12 * 13 ) * (-5)
= 156 * (-5)
= -(156 * 5)
= -780

6. Evaluate the products of (-1) * (-2) * (-3) * (-4) * (-5)

Solution:

Since the number of negative integers in the product their product is odd. Therefore their product is negative.

• (-1) * (-2) * (-3) * (-4) * (-5)
= -(1*2*3*4*5)
= -(2*3*4*5) [Since, 1 * 2 = 2]
=-(6 * 4 * 5 ) [Since, 2 * 3 = 6 ]
= -(24 * 5 ) = -120 [Since, 6 * 4 = 24 ]

7. Evaluate the products of (3) * (-6) * (-9) * (-12)

Solution:

Since the number of negative integers in the given product is even. Therefore, their product is positive.

• (-3) * (-6) * (-9) * (-12)
= (3 * 6 * 9 * 12)
= (18 * 9 * 12) [Since, 3 * 6 =18]
= (162 * 12) [Since, 18 * 9 =162]
= 1944

8. Find the value of 15625 * (-2) + (-15625) * 98

Solution:

• 15625 * (-2) + (-15625) * 98
= (-15625) * 2 + (-15625) * 98
[Since, 15625 * (-2) = -(15625 * 2) = (-15625 * 2)]
= (-15625) * (2 + 98) [Using: a * b + a * c = a * (b + c)]
= (-15625) * 100
= -(15625 * 100)
= -1562500

9. Find the value of 18946 * 99 - (-18946)

Solution:

• 18946 * 99 - (-18946)
= 18946 * 99 + 18946
= 18946 * 99 + 18946 * 1 [Since, 18946 = 18946*1]
= 18946 * (99 + 1) [Using: a * b + a * c = a * (b + c)]
= 18946 * 100
=1894600

10. Find the value of 1569 * 887 -569 * 887

Solution:

• 1569 * 887 – 569 * 887
= (1569 - 569) * 887 [Since, b * a - c * a = (b - c) * a]
= 1000 * 887
= 887000
• Students can practice these examples on multiplication of integers which are discussed here step by step.

Multiplicative Identity

The Multiplicative Identity Property. For a property with such a long name, it's really a simple math law.
The multiplicative identity property states that any time you multiply a number by 1, the result, or product, is that original number.

To write out this property using variables, we can say that

n * 1 = n.

It doesn't matter if n equals one, one million or 3.566879.
The property always hold true.
Therefore:
2 * 1 = 2
56 * 1 = 56
100,000,000,000 * 1 = 100,000,000,000
57,687.758943768579875986754890 * 1 = 57,687.758943768579875986754890

Associativity For Multiplication

The Associative Property of Multiplication
Similar examples can illustrate how the associative property works for multiplication.
Example 3:
(3 * 5 ) * 6 = 3 * (5 * 6)

15 x 6 = 3 x 30
90 = 90
L.H.S. = R.H.S.
Now which side of the equation is easier for you? Most often, it is 5 * 6 on the right side.

Associativity For Multiplication

Formula
( a x b ) x c = a x ( b x c )

Example 4:
2 * (18 * 10) = (2 * 18) * 10
Here the left side is written differently,
yet you can still see how the associative property makes the multiplication on the left side easier.
2 x 180 = 36 x 10
360 = 360
L.H.S. = R.H.S.

Distributive Property

In general, this term refers to the distributive property of multiplication which states that the. Definition:
The distributive property lets you multiply a sum by multiplying each addend separately and then add the products.

Formula
a x ( b + c ) = ( a x b ) + ( a x c)
a x ( b - c ) = ( a x b ) - ( a x c )

Example:

Properties Of Division Of Integers

Statement1 Inference Statement2 Inference
( - 7 ) / ( - 3 ) = 7/3 Result is an integer ( - 9 ) / ( - 3 ) = 3 Result is an integer
( - 8 ) / ( 5 ) = - 8/5 Result is an integer ( - 6 ) / ( 2 ) = - 3 Result is an integer
( - 0 ) / ( 7 ) = ? ? ( - 9 ) / ( 14 ) = ? ?

#### Rules For Dividing Integers

a / 1 = a

Division of integers is discussed here. Division of whole numbers is an inverse process of multiplication.

Dividing 20 by 4 means finding an integer which when multiplied with 4 gives us 20, such an integer is 5.

Therefore, we write as 20 ÷ 4 = 5 or, 20/4 = 5

Similarly, dividing 45 by -9 means, finding an integer which when multiplied with -9 gives 45, such an integer is -5.

Therefore, we write 45 ÷ (-9) = -5 or, 45/-9 = -5

Dividing (-28) by (-4) means, what integer should be multiplied with (-4) to get (-28), such an integer is 7.

Therefore, (-28) ÷ (-4) = 7 or, -28/-4 = 7

Formulae Of Division Of Integers

#### Definitions of the following terms used in division:

Dividend:- The number to be divided is called dividend.
Divisor:- The number which divides is called the divisor.
Quotient:- The result of division is called the quotient.

When dividend is negative and divisor is negative, the quotient is positive. When the dividend is negative and the divisor is positive, the quotient is negative.

#### Rule 1

The quotient of two integers either both positive or negative is a positive integer equal to the quotient of the corresponding absolute values of the integers.

(i) The quotient of two positive integers is positive. Here, we divide the numerical value of the dividend by the numerical value of the divisor.
For example: (+ 9) ÷ (+ 3) = + 3

(ii) The quotient of two negative integers is positive. Here, we divide the numerical value of the dividend by the numerical value of the divisor and assign (+) sign to the quotient obtained.
For example: (- 9) ÷ (- 3) = + 3

Thus, for dividing two integers with like signs, we divide their values and give plus sign to the quotient.

#### Rule 2

1. The quotient of a positive and a negative integer is a negative integer and its absolute value is equal to the quotient of the corresponding absolute values of the integers.

For example: (+ 16) ÷ (- 4) = - 4

Thus, for dividing integers with unlike signs, we divide their values and give minus sign to the quotient.

Properties of Division of Integers

#### The following properties of division of integers are:

• If x and y are integers, then x ÷ y is not necessarily an integer. For example; 16 ÷ 3, -17 ÷ 5 are not integers.

• If x is an integer different from 0, then x ÷ x = 1.

• For every integer x, we have x ÷ 1= x.

• If x is a non-zero integer, then 0 ÷ x = 0.

• If x is an integer, then x ÷ 0 is not meaningful.

• If x, y, z are non-zero integers, then (x ÷ y) ÷ z ≠ x ÷ (y ÷ z), unless z = 1.

• If x, y, z are integers, then
• x > y ⇒ x ÷ z > y ÷ z, if z is positive.
• x > y ⇒ x ÷ z < y ÷ z, if z is negative.

Examples on Division of Integers

Examples on division of integers on different types of questions on integers are discussed here step by step.

1. Divide 91 by 7

Solution:

• 91 ÷ 7
= 91/7
=13

2. Divide -117 by 13

Solution:

• -117 ÷ 13
= -117/13
= -9

3. Divide -98 by -14

Solution:

• -98 ÷ (-14)
= -98/-14
= 98/14
= 7

4. Divide 324 by -27

Solution:

• 324 ÷ (-27)
= - 324/27
= -12

5. Find the quotient of (-1728) ÷ 12

Solution:

• (-1728) ÷ 12
= - 1728/12
= -144

6. (-15625) ÷ (-125)

Solution:

• (-15625) ÷ (-125)
= (-15625/-125)
= (15625/125)
= 125

7. 50000 ÷ (-100)

Solution:

• 50000 ÷ (-100)
= 50000/-100
= -50000/100
=-500

8. Find the value of [32 + 2 ⨯ 17 + (-6)] ÷ 15

Solution:

• [32 + 2 ⨯ 17 + (-6)] ÷ 15
= [32 + 34 + (-6)] ÷ 15
= (66 - 6) ÷ 15
= 60 ÷ 15
= 60/15
= 4

9. Find the value of ||-17| + 17| ÷ ||-25| - 42|

Solution:

• ||-17| + 17| ÷ ||-25| - 42|
= |17 + 17| ÷ |25 - 42|
= |34| ÷ |-17|
= 34 ÷ 17
= 34/17
= 2

10. Simplify: {36 ÷ (-9)} ÷ {(-24) ÷ 6}

Solution:

• {36 ÷ (-9)} ÷ {(-24) ÷ 6}
= {36/-9} ÷ {-24/6}
= - (36/9) ÷ - (24/6)
= (-4) ÷ (-4)
= -4/-4
= 4/4
=1
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