INTEGERS - BASIC OPERATIONS AND THEIR PROPERTIES SUCH AS CLOSURE ETC.

Please study  Whole Numbers before Integers,
if you have not already done so.

There, we covered the basic operations such as
Addition, Subtraction, Multiplication and Division
and their properties such as
Closure, Commutativity, Associativity, Distributivity etc.

We have seen that closure property is not true
for Subtraction in the set of whole numbers,
with examples 3 - 7 = ? ∉ W; 20 - 23 = ? ∉ W;

i.e. when a bigger whole number is subtracted
from a smaller whole number, we do not get a whole number.

So, we need to have new type of numbers which
may represent the above differences.

Thus, corresponding to natural numbers 1, 2, 3, 4, 5, 6, 7, ....etc.,
we introduce new numbers denoted by -1, -2, -3, -4, -5, -6, -7, ....etc.
called minus one, minus two, minus three, minus four, minus five,
minus six, minus seven, etc., respectively such that :
1 + (-1) = 0, 2 + (-2) = 0, 3 + (-3) = 0 and so on.

The numbers -1, -2, -3, ....etc.are called opposites (or additive inverses)
of 1, 2, 3, ....etc respectively and vice versa.

Thus, we get a new set of numbers given by :
Z = {......., -4, -3, -2, -1, 0, 1, 2, 3, 4,...........}
These numbers are known as Integers.

We denote the set of Integers by Z.

The numbers 1, 2, 3, 4, 5, 6, 7, ....etc. are known as Positive Integers.

The numbers -1, -2, -3, -4, -5, -6, -7, ....etc.are known as Negative Integers.

0 is an integer which is neither positive nor negative.

The numbers 0, 1, 2, 3, 4, 5, 6, 7, ....etc. are known as Non-Negative Integers. or Whole Numbers, as we have already seen.

We may denote the positive integers by +1, +2, +3, +4, +5, +6, +7, ....etc.
Usually, we omit the "+" sign.

Use of Positive and Negative Integers in Daily Life

(i) The height of Mean Sea Level is taken as 0.
The heights above Mean sea Level are taken as Positive.
The heights below Mean sea Level are taken as Negative.

(ii) The freezing point of water is 0⁰C.
The temperatures above the freezing point are taken as Positive.
The temperatures below the freezing point are taken as Negative.

(iii) The Atmospheric Pressure is taken as 0.
The Pressures above Atmospheric Pressure are taken as Positive.
The Pressures below Atmospheric Pressure are taken as Negative.

(iv) Profits in a business are represented by Positive Numbers.
Losses in a business are represented by Negative Numbers.

(v) Deposits in a Bank Account are considered as Positive Numbers.
Withdrawls in a Bank Account are considered as Negative Numbers.

Comparison of Integers

From the knowledge of  Decimal Number System,
we can compare positive integers and know
which is bigger or which is smaller.

For example, 3 < 5; 9 < 12; 99 < 100; 189 < 197

In addition to that knowledge, the following points
will help in deciding which is bigger or smaller.

0 is less than every positive integer.

0 < 1 < 2 < 3 < 4 < 5 <..........etc.

0 is greater than every negative integer.

.......< -5 < -4 < -3 < -2 < -1 < 0

Every negative integer is less than every positive integer.

.......< -5 < -4 < -3 < -2 < -1 < 0 < 1 < 2 < 3 < 4 < 5 < ........

The greater is the integer, the lesser is its negative.

As 5 > 3, we have -5 < -3; As 11 > 9, we have -11 < -9.

Absolute value of an Integer

The absolute value of an integer "a" is the numerical value of "a" regardless of its sign. We denote it by | a |.

Thus, | -5 | = 5; | 5 | = 5; | 0 | = 0; | -19 | = 19; | 19 | = 19 etc.

So, we can say, absolute value | a | of an integer a is defined by
| a | = a if a is positive or zero.
| a | = -a if a is negative.

Operations on Integers

Addition of Integers

While adding, we have 3 rules depending
on the signs of the numbers being added.

Rule 1 : If the two integers being added are positive,
their sum is the sum of their numerical values.

Examples :
(i) (+ 3) + (+6) = +(3 + 6) = +9 = 9.
(ii) ( + 144) + (+36) = + (144 + 36) = +180 = 180.

Rule 2 : If the two integers being added are negative,
find the sum of their numerical values and
give negative sign to it, which is their sum.

Examples :
(i) (-8) + (-7) = -(8 + 7) = -15 = -15.
(ii) ( - 100) + (-299) = - (100 + 299) = -399 = -399.

Rule 3 : In the two integers being added, if one is positive and the other is negative, find the difference between their numerical values and give the sign of the integer with more numerical value, to it, which is their sum.

Examples :
(i) Consider the sum (+12) + (-9)
Here, one is positive and the other is nagative.
Numerical value of (+12) = 12.; Numerical value of (-9) = 9.
Their difference = 12 - 9 = 3
(+12) has more numerical value and its sign is +.
So, the sign to 3 is +.
Thus (+12) + (-9) = +(12 - 9) = +3 = 3.

(ii) Consider the sum (- 136) + (+96)
Here, one is positive and the other is nagative.
Numerical value of (- 136) = 136.; Numerical value of (+96) = 96.
Their difference = 136 - 96 = 40
(- 136) has more numerical value and its sign is -.
So, the sign to 40 is -.
Thus (- 136) + (+96) = - (136 - 96) = -40 .

Properties of Addition of Integers

Closure Property of Addition :

Observe the following Examples :

(+ 3) + (+6) = 9 ∈ Z;
(-8) + (-7) = -(8 + 7) = -15 ∈ Z;
(+12) + (-9) = +(12 - 9) = +3 = 3 ∈ Z;
(- 136) + (+96) = - (136 - 96) = -40 ∈ Z;

What is seen to be true for these examples,
is true for any two elements of Z. So, we can say :

For any two elements a and b ∈ Z, (a + b) ∈ Z.
This is called the Closure Property of Addition for integers.

Commutative Property of Addition :

Look at the following Examples.

(+ 3) + (+6) = (+ 6) + (+3)
(-8) + (-7) = (-7) + (-8)
(+12) + (-9) = (-9) + (+12)
(- 136) + (+96) = (+96) + (- 136)

What is seen to be true for these examples,
is true for any two elements of Z. So, we can say :

For any two elements a and b ∈ Z, a + b = b + a
This is called the Commutative Property of Addition for integers.

Associative Property of Addition :

Look at the following Examples.

(15 + 27) + (-18) = 15 + {27 + (-18)} = 24;
(-3 + 6) + 1 = -3 + (6 + 1) = 4

What is seen to be true for in these examples,
is true for any three elements of Z. So, we can say :

For any three elements a, b and c ∈ Z, a + (b + c) = (a + b) + c
This is called the Associative Property of Addition for integers.

Additive Identity :

Look at the following Examples.

19 + 0 = 0 + 19 = 19 ; -1345 + 0 = 0 + (-1345) = -1345

What is seen to be true for these examples,
is true for any element of Z. So, we can say :

For every element a ∈ Z, a + 0 = 0 + a = a
0 is called the Additive Identity in the set Z.

Additive Inverse of an Integer :

We have, 1 + (-1) = 0, 2 + (-2) = 0, 3 + (-3) = 0 and so on.

What is seen to be true for these examples,
is true for any element of Z. So, we can say :

For every element a ∈ Z, we have a + (-a) = 0.
We call a and -a as the additive inverse of each other.

Subtraction of Integers

For any two integers a and b, we define : a - b = a + (-b) = a + (additive inverse of b).

Examples :
(i) Subtract (-7) from 8
8 - (-7) = 8 + (Additive inverse of -7) = 8 + 7 = 15.

(ii) Subtract (33) from (-67)
(-67) - (33) = -67 + (Additive inverse of 33) = -67 + (-33) = -100.

Properties of Subtraction of Integers

Closure Property is true for Subtraction in the set Z :

Observe the following Examples :

3 - 7 = -4 ∈ Z; 20 - 23 = -3 ∈ Z;

What is seen to be true for these examples,
is true for any two elements of Z. So, we can say :

For any two elements a and b ∈ Z, (a - b) ∈ Z.
This is called the Closure Property of Subtraction for integers.

Subtraction is not Commutative :

Observe the following Examples :

7 - 4 ≠ 4 - 7; 78 - 67 ≠ 67 - 78

So, we can say Subtraction is not Commutative.

Subtraction is not Commutative in the set Z.

Subtraction is not Associative :

Observe the following Examples :

(8 - 5) - 2 ≠ 8 - (5 - 2); (78 - 67) - 11 ≠ 78 - (67 - 11)

So, we can say Subtraction is not Associative.

Subtraction is not Associative in the set Z.

Multiplication of Integers

While multiplying, we have 2 rules depending
on the signs of the numbers being multiplied..

Rule 1 : If the two integers being multiplied have the same sign,
their product is the product of their numerical values.

Examples :
(i) (+ 3) x (+6) = +(3 x 6) = +18 = 18.
(ii) ( - 9) x (-16) = + (9 x 16) = +144 = 144.

Rule 2 : If the two integers being multiplied have opposite signs,
their product is the negative product of their numerical values.

Examples :
(i) (+ 3) x (-6) = -(3 x 6) = -18.
(ii) ( - 9) x (+16) = - (9 x 16) = -144.

Properties of Multiplication of Integers

Closure Property of Multiplication :

Observe the following Examples :

30 x 7 = 210 ∈ Z; -2 x 23 = -46 ∈ Z; -18 x -8 = 144 ∈ Z;

What is seen to be true for these examples,
is true for any two elements of Z. So, we can say :

For any two elements a and b ∈ Z, (a x b) ∈ Z.
This is called the Closure Property of Multiplication for integers.

Commutative Property of Multiplication :

Look at the following Examples.

10 x 18 = 18 x 10 = 180; -25 x 4 = 4 x -25 = -100; -9 x -16 = -16 x -9 = 144

What is seen to be true for in these examples,
is true for any two elements of Z. So, we can say :

For any two elements a and b ∈ Z, a x b = b x a
This is called the Commutative Property of Multiplication for integers.

Associative Property of Multiplication :

Look at the following Examples.

(4 x -2) x 3 = 4 x (-2 x 3) = -24;
(-16 x -9) x 2 = -16 x (-9 x 2) = 288

What is seen to be true for in these examples,
is true for any three elements of Z. So, we can say :

For any three elements a, b and c ∈ Z, a x (b x c) = (a x b) x c
This is called the Associative Property of Multiplication for integers.

Multiplicative Identity :

Look at the following Examples.

14 x 1 = 1 x 14 = 14 ; -9875 x 1 = 1 x -9875 = -9875

What is seen to be true for these examples,
is true for any element of Z. So, we can say :

For every element a ∈ Z, a x 1 = 1 x a = a
1 is called the Multiplicative Identity in the set Z.

Zero Property of Multiplication :

Look at the following Examples.

-234 x 0 = 0 x -234 = 0 ; 5647 x 0 = 0 x 5647 = 0

What is seen to be true for these examples,
is true for any element of Z. So, we can say :

For every element a ∈ Z, a x 0 = 0 x a = 0
This is called the Zero Property of Multiplication in the set Z.

Also,

For any two elements a and b ∈ Z,
If a x b = 0, then either a = 0 or b = 0 or a = b = 0

Distributive Property of Multiplication over Addition :

Look at the following Examples.

5 x (-4 + 3) = 5 x -4 + 5 x 3 = -20 + 15 = -5 same as 5 x -1;
-2 x (-7 + 8) = -2 x -7 + (-2) x 8 = 14 - 16 = -2 same as -2 x 1;

What is seen to be true for these examples,
is true for any three elements of Z. So, we can say :

For any three elements a, b and c ∈ Z, a x (b + c) = (a x b) + (a x c)
This is called the Distributive Property of Multiplication over Addition.

Division of Integers

While dividing, we have 2 rules depending
on the signs of the dividend and the divisor.

Rule 1 : If the dividend and the divisor have the same sign,
their quotient is the division of their numerical values.

Examples :
(i) (+ 6) ÷ (+2) = +(6 ÷ 2) = +3 = 3.
(ii) ( - 96) ÷ (-16) = + (96 ÷ 16) = +6= 6.

Rule 2 : If the dividend and the divisor have opposite signs,
their quotient is the negative division of their numerical values.

Examples :
(i) (+ 6) ÷ (-3) = -(6 ÷ 3) = -2.
(ii) ( - 96) ÷ (+16) = - (96 ÷ 16) = -6

Properties of Division of Integers

Closure Property is not true for Division :

Observe the following Examples :

8 ÷ 3 = ? ∉ Z; 36 ÷ -5 = ? ∉ Z;

Thus For some elements of Z,
division does not give elements ∈ Z. So, we can say

The Closure Property is not true for Division in the set Z.

Division is not Commutative :

Observe the following Examples :

8 ÷ 4 ≠ 4 ÷ 8; 144 ÷ -9 ≠ -9 ÷ 144

So, we can say Division is not Commutative.

Division is not Commutative in the set Z.

Division is not Associative :

Observe the following Examples :

(8 ÷ 4) ÷ 2 ≠ 8 ÷ (4 ÷ 2);
(54 ÷ -6) ÷ 3 ≠ 54 ÷ (-6 ÷ 3)

So, we can say Division is not Associative.

Division is not Associative in the set Z.

Division with 1 of any number gives that number :

Look at the following Examples.

-19 ÷ 1 = 19 ; 9776 ÷ 1 = 9776

What is seen to be true for in these examples,
is true for any element of Z. So, we can say :

For every element a ∈ Z, a ÷ 1 = a

Division is inverse operation to Multiplication :

Look at the following Examples.

144 ÷ 18 = 8 and 8 x 18 = 144;
81 ÷ -3 = -27 and -27 x -3 = 81.

What is seen to be true for these examples,
is true for any element of Z. So, we can say :

For any three elements a, b(≠0) and c ∈ Z, a ÷ b = c means a = b x c

Division by zero is not defined :

-9 ÷ 0 = ?
Is it possible to assign any whole number to this quotient ?
Suppose, if possible, -9 ÷ 0 = a where a is an integer.
Since division is the inverse process of multiplication, we have
a x 0 = -9.
But, any whole number x 0 = 0
[see zero property of multiplication above.]
So a x 0 = 0 and not -9.
Thus there is no number which when
multiplied by zero gives a non zero number.

Therefore -9 ÷ 0 can not be equal to any integer.

Thus division by zero is not defined.

Division by zero is not defined.

What about 0 ÷ 0 ?

Let 0 ÷ 0 = b
Then 0 = b x 0.
This is satisfied for all values of b.
So 0 ÷ 0 has innumerable Answers.
It can not be any one umber.

So, we can say

0 ÷ 0 is indeterminate.

Division of zero with any number gives zero :

Look at the following Examples.

0 ÷ -34 = 34 ; 0 ÷ 5617 = 0

What is seen to be true for these examples,
is true for any element of Z. So, we can say :

For every element a ∈ Z, 0 ÷ a = 0