# Properties Of Multiplication Of Integers

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## Closure under Multiplication

Drag and drop the products into the correct group.

What do you observe? Can you find a pair of integers whose product is not an integer?

This gives us an idea that the product of two integers is again an integer. So we can say that, integers are closed under multiplication.

In general,

**a × b is an integer, for all integers a and b**

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## Commutativity of Multiplication

We know that multiplication is commutative for whole numbers. Can we say, multiplication is also commutative for integers? Let's find out.

Verify if the below equalities are true or false and accordingly drag and drop them in the correct classification.

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What are your observations? The above examples suggest *multiplication is commutative for integers*.

In general, for any two integers a and b:

**a × b = b × a**

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## Multiplication by Zero

We know that any whole number when multiplied by zero gives zero.

Observe the following products of negative integers and zero. These are obtained from the patterns done earlier.

(–3) × 0 = 0

0 × (– 4) = 0

– 5 × 0 =

0 × (– 6) =

This shows that the product of a negative integer and zero is zero.

In general, for any integer a,

**a × 0 = 0 × a = 0**

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## Multiplicative Identity

We know that 1 is the multiplicative identity for whole numbers.

Let's find out if 1 is the multiplicative identity for integers as well.

Negative integers | Positive Integers |
---|---|

(–3) × 1 = –3 | 1 × 5 = 5 |

(– 4) × 1 = | 1 × 8 = |

1 × (–5) = | 3 × 1 = |

1 × (– 6) = | 7 × 1 = |

This shows that *1 is the multiplicative identity for integers as well.*

In general, for any integer *a* we have:

**a × 1 = 1 × a = a**

Now, what happens when we multiply any integer with –1? Complete the following table:

Product of Integers | Product of Integers |
---|---|

(–3) × (–1) = 3 | 3 × (–1) = –3 |

(6) × (–1) = | (–1) × (–25) = |

(–1) × 13 = | 18 × (–1) = |

What do you observe?

Can we say –1 is a multiplicative identity of integers?

0 is the additive identity whereas 1 is the multiplicative identity for integers. We get additive inverse of an integer *a* when we multiply (–1) to a, i.e.

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## Associativity for Multiplication

Earlier we saw the associative property for addition. Let's check if it holds for multiplication as well.

Take for example: [(–3) × (–2)] × 5 and (–3) × [(–2) × 5]

In the first case, (–3) and (–2) are grouped together and in the second, (–2) and 5 are grouped together.

- [(–3) × (–2)] × 5 =
× 5 = - Now, (–3) × [(–2) × 5] = (–3) ×
= - We get the same result of 30.
- Thus, associative property is valid for multiplication

Complete the given below products:

[(7) × (– 6)] × 4 =

7 × [(– 6) × 4] = 7 ×

Does the grouping of integers affect the product of integers?

In general, for any three integers a, b and c:

**(a × b) × c = a × (b × c)**

*Thus, like whole numbers, the product of three integers does not depend upon the grouping of integers and this is called the associative property for multiplication of integers.*

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## Distributive Property

We know:

16 × (10 + 2) = (16 × 10) + (16 × 2) [Distributivity of multiplication over addition]

Let us check if this is true for integers also.

Observe the following:

**(–2) × (3 + 5)**

- (–2) × (3 + 5) = –2 ×
= - We can re-write: (–2) × (3 + 5) = [(–2) × 3] + [(–2) × 5]
- Now, [(–2) × 3] + [(–2) × 5] =
+ = - Thus, distributive property has been verified.

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**(– 4) × [(–2) + 7]**

- (– 4) × [(–2) + 7] = (– 4) ×
= - We can re-write: (– 4) × [(–2) + 7] = [(– 4) × (–2)] + [(– 4) × 7]
- Now, [(– 4) × (–2)] + [(– 4) × 7]=
+ = - Thus, distributive property has been verified.

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**(– 8) × [(–2) + (–1)]**

- (– 8) × [(–2) + (-1)] = (– 8) ×
= - We can re-write: (– 8) × [(–2) + (–1)] = [(– 8) × (–2)] + [(– 8) × (–1)]
- Now, [(– 8) × (–2)] + [(– 8) × (–1)] =
+ = - Thus, distributive property has been verified.

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Can we say that the distributivity of multiplication over addition is true for integers also?

In general, for any integers a, b and c,

**a × (b + c) = a × b + a × c**

Now consider the following:

Can we say 4 × (3 – 8) = 4 × 3 – 4 × 8?

Let us check:

- 4 ×
= - Evaluating: 4 × 3 – 4 × 8 =
– = - Thus, proved.

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Let's check further:

Is ( –5) × [( – 4) – ( – 6)] = [( –5) × ( – 4)] – [ ( –5) × ( – 6)] ?

- ( –5) × [( – 4) – ( – 6)] = (–5) ×
= - Evaluating: [( –5) × ( – 4)] – [( –5) × ( – 6)] =
– = - Thus, proved.

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Now, check for ( –9) × [ 10 – ( –3)] and [( –9) × 10 ] – [ ( –9) × ( –3)]

- ( –9) × [ 10 – ( –3)] = (–9) ×
= - Evaluating: [( –9) × 10] – [( –9) × ( –3)] =
– = - Thus, proved.

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You will find that these are also equal.

In general, for any three integers a, b and c,

**a × (b – c) = a × b – a × c**