Integers are numbers with a direction, one direction is positive, and the other is negative. Hence, integer is called a directed number that is preceded by a plus or minus sign.
‘Integer’ is a Latin word which means ‘whole’ or ‘intact’. Integers are whole numbers and their opposites (the negative whole numbers). Integers do not include fractions or decimals.
Integers include positive numbers, negative numbers and zero.
Positive numbers: These numbers are greater than zero. They can be written with or without a positive sign.
Negative numbers: These numbers are less than zero. They can be written with a negative sign.
Zero: 0 is a whole number. It is neither a positive number nor a negative number.
The number line below shows the clear distinction between integers, whole numbers and natural numbers.
All the natural and whole numbers are integers. But all the integers are not whole numbers.
Two integers are opposites if they are at the same distance from 0 in either the positive or negative direction on the number line.
On the number line, the actual distance of the integer from zero is called the absolute value.
Absolute value of a number is represented by two vertical lines.
|-7| = 7
and
|15| = 15
Comparing Integers
Addition of Integers
Addition of integers is finding the sum of two or more integers where the value of the sum depends on the integer being positive or negative.
Rules for adding integers:
When adding integers with the same signs, add the absolute values of integers, and give the same sign to the result. When we add two positive integers, their sum is a positive integer and when we add two negative integers, their sum is a negative integer.
(+3) + (6) = +9
and
(-5)+(-7) = -12
When adding Integers with different signs, find the difference of the absolute values of the numbers and then give the sign of the larger of integers to the result.
(-9) + (3) = -6
and
(+8) + (-2) = +6
Subtraction of Integers
Subtraction of integers is finding the difference between two or more integers where the value of the difference depends on the integer being positive or negative.
Rules for subtracting integers:
When subtracting two integers, change the sign of the second number which is being subtracted, and follow the rules of addition.
Multiplication of Integers
Multiplication of integers is finding the product of two or more integers where the value of the product depends on the integer being positive or negative. Multiplying integers is the same as multiplying whole numbers but the product has a sign.
Follow the below rules while multiplying two integers.
(-5) ✕ (-3) = 15
and
5 ✕ 3 = 15
(-5) ✕ 3 = -15
and
5 ✕ (-3) = -15
We can summarize the multiplication of two integers as shown below.
Division of Integers
Division of integers is finding equal groups or dividing an integer into a specific number of groups. Dividing integers is the same as dividing whole numbers but the quotient has a sign.
The rules to divide integers are the same as in multiplication of integers.
(-24) ÷ (-8) = 3
and
24 ÷ 8 = 3
(-24) ÷ 8 = -3
and
24 ÷ (-8) = -3
We can summarize the division of two integers as shown below.
Properties of Integers:
Closure property
The sum or difference or product of any two integers will always be an integer.
If a and b are any two integers, a + b, a − b and a x b will also be an integer.
The closure property does not hold for division of integers.
Commutative law
When adding or multiplying two numbers, the order of doing things doesn’t matter.
If a and b are any two integers, a + b = b + a and a x b = b x a
The commutative law does not hold for subtraction and division of integers.
Additive Identity
When any integer is added to zero it will give the same integer.
If ‘a’ is an integer, a + 0 = a.
Additive inverse
The addition operation between any integer and its negative value will give the result as zero (0).
If a is an integer, -a is additive inverse of a such that a – a = 0
Change the sign of the number and add it to the original number to get an answer equal to 0.
Associative law
When adding or multiplying three or more numbers, you can add any pair of numbers first.
If a, b and c are any three integers, a + (b + c) = (a + b) + c and a x (b x c) = (a x b) x c
The associative law does not hold for subtraction and division of integers.
Distributive law
When a sum or difference is being multiplied by a number, each number in the sum or difference can be multiplied by the number first then these products can be added or subtracted.
If a, b and c are any three integers, a x (b + c) = (a x b) + (a x c) and a x (b – c) = (a x b) – (a x c)
The distributive law does not hold for division of integers.
Identity property
When an integer is added to or subtracted by 0 it will give the integer itself as the result.
If a is an integer, a + 0 = a and a – 0 = a
0 is the additive identity of integers.
When an integer is multiplied or divided by 1 it will give the integer itself as the result.
If a is an integer, a x 1 = a and a ÷ 1 = a
0 is the multiplicative identity of integers.
Multiplicative Inverse
The multiplicative inverse of a number x is given by 1/x, such that when it is multiplied by its original number, it results in a value equal to 1.
If a is an integer, 1/a is the multiplicative inverse of a such that, a x 1/a = 1
The multiplication operation between any integer and its reciprocal will give the result as 1.
