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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