Types of Numbers

Integers 7th Math

Types of Numbers

Numbers can be divided into three types. These are Natural Numbers, Whole Numbers and Integers.

Natural Numbers:

Counting Numbers are called NATURAL NUMBERS.

Thus, 1, 2, 3, 4, 5, . . . . . . . . are all natural numbers

Whole Numbers

All counting numbers together with zero (0) are called WHOLE NUMBERS.

Therefore, 0, 1, 2, 3, 4, 5, . . . . . . . . are all whole numbers

Integers

All natural numbers, zero (0) and negative of counting numbers together forms INTEGERS.

Therefore, . . . . . –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, . . . . . . . . are all Integers.

Types of Integers

Integers further can be divided into three types: Positive Integers, Negative Integers and Zero.

Positive Integers

Numbers with positive signs are called POSITIVE INTEGERS.

1, 2, 3, 4, 5, . . . . . . . . are all called positive Integers.

An integer with no positive (+) or negative (–) signs before it are considered positive integers.

For convenience, no sign is given before a positive integer. That is numbers with no signs are considered as positive integers. Although a positive sign (+) is given to a number for positive integers as required.

Negative Integers

Number having negative signs (–) before them are called NEGATIVE INTEGERS.

Thus, –1, –2, –3, –4, –5, . . . . . . . . are all Negative Integers.

Zero

Zero (0) is an integer, which is neither negative (–) nor positive.

Operations on Integers

Multiplication or Product of Integers

Finding the Multiplication or Product of two Positive Integers

To find the multiplication or product of two integers, simply multiplied them. Resulting product is a positive integer.

Example: (1) 15 × 10

Solution:

15 × 10 = 150

Here, given integers 15 and 10 are positive integers, and their product 150 is also a positive integer.

Example: (2) 8 × 7

Solution:

8 × 7 = 56

Here, given integers 8 and 7 are positive integers, and their product 56 is also a positive integer.

Example: (3) 22 × 9

Solution:

22 × 9 = 1098

Here, given integers 22 and 9 are positive integers, and their product 1098 is also a positive integer.

Example: (4) Multiply 16 and 12

Solution:

16 × 12 = 192 Answer

Example: (5) Multiply 8 and 9

Solution:

8 × 7 = 56 Answer

Example: (6) Multiply 9 and 11

Solution:

9 × 11 = 99 Answer

Thus, when two positive integers are multiplied, their product is also a positive integer.

Multiplication of two integers with unlike signs

To find the product of two integers, one positive (+) and one negative (–) do the following:

(a) Simply multiply them without taking their sign into consideration

(b) Put a negative sign before their product.

This, means `(-)xx(+) = -`

And, `(+)xx(-)=-`

This means product of one positive (+) and one negative (–) is always a negative (–) integer.

Example: (1) 10 × –2

Solution:

Simply Multiply the given integers ignoring their sign

10 × 2 = 20

And put a negative sign (–) before their product

That is, their product, 20 becomes –20.

Thus, 10 × –2 = –20

Example: (2) 12 × –8

Solution:

Simply Multiply the given integers ignoring their sign

12 × 8 = 96

And put a negative sign (–) before their product

That is, their product, 96 becomes –96

Thus, 12 × –8 = –96

Example: (3) –6 × 9

Solution:

Simply Multiply the given integers ignoring their sign

6 × 9 = 54

And put a negative sign (–) before their product

That is, their product, 54 becomes –54

Thus, –6 × 9 = –54

Example: (4) –16 × 4

Solution:

Simply Multiply the given integers ignoring their sign

16 × 4 = 64

And put a negative sign (–) before their product

That is, their product, 64 becomes –64

Thus, –16 × 4 = –64

Multiplication of two Negative Integers

Product of two negative integers is a positive integer.

To multiply two negative integers, multiply them ignoring their signs, and put a positive sign before their product.

Example: (1) – 10 × –5

Solution:

Simply multiply the given integers ignoring their signs

i.e. 10 × 5 = 50

And put a positive sign before their product

That is 50 will remain +50.

Thus, – 10 × –5 = +50

Since, for convenience, no sign is given before a positive integer,

Thus, – 10 × –5 = 50

Example: (2) –8 × –7

Solution:

Simply multiply the given integers ignoring their signs

i.e. 8 × 7 = 56

And put a positive sign before their product,

That is, product 56 of given two negative integers will remain 56 or +56

Thus, –8 × –7 = 56

Example: (3) –16 × –10

Solution:

–16 × –10 = 160 Answer

Example: (4) –12 × –12

Solution:

–12 × –12 = 144 Answer

Example: (5) –2 × –12

Solution:

–2 × –12 = 24 Answer

Example: (6) –22 × –11

Solution:

–22 × –11 = 242 Answer

Summary of Multiplication of Integers:

(1) To find the multiplication (product) of two integers with unlike signs (one positive and other one negative), multiply them regardless of their sign, and assign a negative sign to their product.

This means product of two integers having unlike signs, is always a negative integer.

(2) To find the multiplication (product) of two integers having same sign (i.e. both of the integers are either positive or negative), multiply them regardless of their sign, and assign a positive sign to their product.

This means product of two integers having like signs is always a positive integer.

Precisely, Rules of multiplication of two integers

(a) `(+) xx (-) = -`

(b) `(-) xx( +) = -`

(c) `(+) xx (+) = +`

(d) `(-) xx (-) = +`

Properties of Multiplication of Integers

Closure Property of Multiplication

The product of two integers is always an integer.

Example: (1) 5 × 4 = 20

Here, 5 and 4 are integers and their product 20 is also an integer.

Example: (2) 6 × 10 = 60

Here, 6 and 10 are integers and their product 60 is also an integer.

Example: (3) 12 × 7 = 84

Here, 12 and 7 are integers and their product 84 is also an integer.

Commutative Law for Multiplication

The Multiplication is commutative.

Commutative Law for Multiplication says,

If, `a` and `b` are two integers,

Then, `a xx b = b xx a`

Example: (1) 11 × 8 = 88 And, 8 × 11 = 88

That is, 11 × 8 = 8 × 11

Or, 11 × 8 = 8 × 11 = 88

Example: (2) 4 × 12 = 48 And, 12 × 4 = 48

That is, 4 × 12 = 12 × 4

Or, 4 × 12 = 12 × 4 = 48

Example: (3) –5 × 6 = 30 And 6 × (–5) = 30

Thus, –5 × 6 = 6 × (–5)

Or, –5 × 6 = 6 × (–5) = 30

Example: (4) –10 × (–11) = 110 And –11 × (–10) = 110

Thus, –10 × (–11) = –11 × (–10)

Or, –10 × (–11) = –11 × (–10) = 110

Associative Law for Multiplication

Multiplications are associative.

Associative Law says,

If, there are three integers `a, b` and `c`

Thus, `(axxb)xxc = axx(bxxc)`

Example: (1) (2 × 3) × 4 = 24 And 2 × (3 × 4) = 24

Thus, (2 × 3) × 4 = 2 × (3 × 4)

Or, Thus, (2 × 3) × 4 = 2 × (3 × 4) = 24

Example: (2) `{3xx(-2)}xx(-2) = 12`

And `3xx{(-2)xx(-2)}=12`

Thus, `{3xx(-2)}xx(-2)` `= 3xx{(-2)xx(-2)}`

Or, Thus, `{3xx(-2)}xx(-2)` `= 3xx{(-2)xx(-2)}=12`

Example: (3) `{-4xx(-2)}xx(-3) = -24` And `-4xx{-2xx(-3)} = 24`

Thus, `{-4xx(-2)}xx(-3)` ` = -4xx{-2xx(-3)} `

Or, `{-4xx(-2)}xx(-3)` ` = -4xx{-2xx(-3)} = 24`

Distributive Law of Multiplication over Addition

Multiplications are distributive over addition.

Distributive Law of Multiplication over Addition says,

If `a, b` and `c` are three integers,

Therefore, `axx(b+c) = (axxb)+(axxc)`

Example: (1) Let, there are three integers 2, 3 and 5

Therefore, 2 × (3+5) = 16 And (2 × 3) + (2 × 5) = 16

Thus, 2 × (3+5) = (2 × 3) + (2 × 5)

Or, 2 × (3+5) = (2 × 3) + (2 × 5) = 16

Example : (2) Let there are three integers, –3, –6 and –7

Therefore, –3 × {–6 + (–7)} = –39

And, {(–3) × ( –6)} + {(–3) × (–7)} =39

Thus, –3 × {–6 + (–7)}

= {(–3) × ( –6)} + {(–3) × (–7)}

And, –3 × {–6 + (–7)}

= {(–3) × ( –6)} + {(–3) × (–7)} = 39

Existence of Multiplicative Identity

For every integer `a`, we have `(axx1) = (1xxa) =a`

Thus, 1 is called the multiplicative identity for integers.

Example: (1) 8 × 1 = 8 And, 1 × 8 = 8

Or, 8 × 1 = 1 × 8 = 8

Thus, 1 is multiplicative identity of 8

Example: (2) –3 × 1 = –3 And 1 × (–) 3 = –3

Thus, –3 × 1 = 1 × (–) 3 = –3

Thus, 1 is the multiplicative identity for –3

Thus, 1 is the multiplicative identity for integers.

Existence of Multiplicative Inverse

Multiplicative inverse is the reciprocal of a non zero integer and which yields 1 (one) when multiplied with the given integer.

Thus, let `a` is an non zero integer

Therefore, reciprocal of `a` is equal to `1/a`

Thus, `axx1/a = 1/a xx a = 1`

Example: (1) Multiplicative inverse of 2 is `1/2`

When, 2 is multiplied by `1/2` it gives 1

i.e. `2 xx 1/2 = 1`

Thus, multiplicative inverse of `2` is the reciprocal of `2` i.e. `1/2`

Example: (2) Multiplicative inverse of –5 is `1/(-5)`

Thus, `-5 xx 1/(-5) = 1`

Thus, multiplicative inverse of –5 is reciprocal of –5, i.e. `1/(-5)`

Property of Zero

An integer gives zero (0) when multiplied by zero (0)

That is, Let `a` is an integer,

Therefore, `(axx0) = (0xxa) = 0`

Example: (1) 5 × 0 = 0 And 0 × 5 = 0

Thus, 5 × 0 = 0 × 5 = 0

Example: (2) –8 × 0 = 0 And 0 × (–8)=0

Thus, –8 × 0 = 0 × (–8)=0

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