We have already studied about the numbers.when we are able to recoganize the numbers that was the first step towards understanding the science in nature. There are different types of numbers.
An Integers include all whole numbers and negative numbers. This means we create a set of whole numbers if we have negative numbers and whole numbers.
An integer is a number without an fractional part or decimal and includes negative and positive numbers, including zero.
A few examples of integers are: -5, 0, 1, 5, 8, 97, and 3043.
The set of integers that are represented as Z includes:
As we have already mentioned the three categories of integers, we can easily represent them on a number line based on zero, positive integers, and negative integers. Zero is the center of the integers on the number line. Negative integers lie on the left side. Positive integers lie on the right side. See the given image below.
Prime Numbers: The numbers other than 1 is called prime number if it is divisible only by 1 and itself.
Composite Numbers: A number, other than 1 which is not prime number is called composite number.
e.g. 4, 6, 8, 9, 10, 12,...etc. The first composite number is 4.Even Numbers: The number which is divisible by 2 is known as even number.
e.g .2, 4, 6, 8, 10,... etc. Even numbers are in the form of 2n. where n is whole numbers.Odd Numbers: The number is not divisible by 2 is known as an odd number.
e.g. 3, 9, 11, 17, 19,... etc. . Odd numbers are in the form of 2n+1 or 2n-1 . where n is whole numbers.Consecutive Numbers:A series of numbers in which each is greater than its predecessor by 1, is called consecutive numbers.
e.g. 1, 2, 3, or 3, 4, 5 or 101, 102, 103.Rational numbers: A number which can be written as p/q form where p and q are integers but q ≠ 0, are known as rational numbers.
e.g. 0, 1/2 ,3/4 , 8/9 ,13/15 are rational numbers.Integers: The set of numbers which consists of whole numbers and negative numbers is known as integers. It is denoted by I. Now we will study integers and trier properties in detail.
Basic operations performed on integers are:
Example: a+b = b+a
5 + 3 = 8 and 3 + 5 = 8
Example: (a + b) + c = a + (b + c)
(6 + 7) + 3 = 16 and 6 + (7+3) = 16
Example: a x (b+c) = (ab) + (ac)
3 x (6+4) = (3×6) + (3×4)
Example:
The additive inverse of -4 is 4, because -4 + 4 = 0.
The additive inverse of +4 is -4 as well.
The product of two integers of opposite sign is equal to the additive inverse of the product of their absolute values.
Product of Signs | Result | Example |
(+) × (+) | + | 2 × 3 = 6 |
(+) × (-) | - | 2 × (-3) = -6 |
(-) × (+) | - | (-2) × 3 = -6 |
(-) × (-) | + | (-2) × (-3) = 6 |
While subtracting two integers, always change the sign of the second number which is being subtracted, and follow the rules of addition.
For example,
(-7) – (+4) = (-7) + (-4) = -11
(+8) – (+2) = (+8) + (-2) = +6
Now,
(-16) ÷ (+4) = -4
(+6) ÷ (+2) = +3
Division of Signs | Result | Example |
(+) ÷ (+) | + | 12 ÷ 3 = 4 |
(+) ÷ (-) | - | 12 ÷ (-3) = -4 |
(-) ÷ (+) | - | (-12) ÷ 3 = -4 |
(-) ÷ (-) | + | (-12) ÷ (-3) = 4 |
The Properties of Integers are:
These properties of integers, when two integers are multiplied or added together, it results in an integer only. If both a and b are the integers, then:
Examples:
2 x 5 = 10 (is an integer)
2 + 5 = 7 (is an integer
If both a and b are two integers, then:
Examples:
3 x 8 = 8 x 3 = 24
3 + 8 = 8 + 3 = 11
If the a, b, and c are integers, then:
Examples:
2x(3×4) = (2×3)x4 = 24
2+(3+4) = (2+3)+4 = 9
If a, b and c is an integers, then a x (b + c) = a x b + a x c
Example: 2 x (5 + 1) = 2 x 5 + 3 x 1
LHS = 2 x (5 + 1) = 2 x 6 = 12
RHS = 2 x 5 + 2 x 1 = 10 + 2 = 12
Thus, LHS = RHS
If a is an integer, as per the additive inverse property of integers, then a+ (-a) = 0
Thus, -a is the additive inverse of integer a.
If a is an integer, as per the multiplicative inverse property of integers, then a x (1 / a) = 1
Therefore, 1 / a is the multiplicative inverse of integer a.
The identity elements of integers are:
Example: -100,-12,-1, 0, 2, 1000, 989 etc…
As a set, it can be represented by Z :
Z= {……-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7,……}
Q1. Solve the following:
Solution:
Q2. Calculate the following product of integers:
Solution:
Q3. Solve the following division of integers:
Solution:
Q1. Is 0 an integer ?
Ans. Yes, 0 is an integer. According to definition, integers are the numbers that include whole numbers and negative natural numbers.
Q2. What are negative numbers called?
Ans. The negative numbers ………. -6,-5, -4, -3, -2, -1 are called negative integers.
Q3. What numbers are not integers?
Ans. A number which is not a whole number, zero or negative whole number is defined as Non-Integer. It is any number that is not included in the integer set, which is expressed as { … -5,-4, -3, -2, -1, 0, 1, 2, 3, 4,5… }. Some of the examples of non-integers include decimals, fractions, and imaginary numbers.
Q4. Why integer is a rational number?
Ans. All integers are rational numbers because it is the set of numbers including all the positive counting numbers, zero and all negative counting numbers which count from negative infinity to positive infinity. The set doesn't include fractions and decimals.