Natural number: Counting numbers are known as natural numbers. Thus, 1,2,3,4,5,6,7,………,etc are all natural numbers.
Whole numbers: All natural numbers together 0 form the set of all whole numbers. Clearly, every natural number is a whole number. And, 0 is the only whole number that is not a natural number.
Integers: All natural numbers, 0 and negatives of natural numbers form the set of all integers. Thus,….,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ….,etc., are all integers.
Clearly, every natural number is an integer, and every whole number is an integer.
Number Line: Let us draw a line XY which extends in both the directions, as indicated by the arrowheads in the diagram below.
Let us take any point O on this line. Let this point represent the integer 0 (zero). Now, taking a fixed length, called unit length, set off equal distances to the right as well to the left of O.
X<..|…|…|…|…|…O…|…|…|…|…|......>Y
-5 -4 -3 -2 -1 0 1 2 3 4 5
On the right-hand-side of O, the points at distances of 1 unit, 2 unit, 3 unit, 4 unit, 5 unit. etc.,from O denotes respectively the integers 1, 2, 3, 4, 5, etc.
Similarly, on the left -hand-side of O, the points at distances of -1unit , -2 unit, -3 unit, -4 unit, -5 unit etc,.
Since the line XY can be extended endlessly on both sides of O, it follows that we can represent each and every integer by some point on this line.
For instance, starting from O and moving to its right, we get a point after 10000 units that represents the integer 10000. Similarly, starting from O and moving to its left, a point after 750 unit represents the integer -750
Thus line is known as a number line.
We have studied about natural number or counting numbers (123….) and whole numbers (0123….). Whole numbers are really useful but they have certain limitations as explained below. The search for a number system to overcome the limitations of natural number led to rational number.
Closure Property
In any number system if application of any of the four fundamental operations (namely, addition, substraction, multiplication and division) on any two numbers results in a numbers results in a number of the same number system, then the number system is said to be closed for that operation or the closure property is said to be satisfied.
You also know that addition or multiplication of two whole numbers gives a sum or product that is also a whole number or a natural number. Hence, the closure property is said to be true for whole numbers for addition and multiplication. Your can also say that whole numbers are closed for addition and multiplication.
Limitation Of Whole Numbers
When you subtract two whole numbers, the result need not always be a whole number. For example, the subtraction of two whole numbers 9 and 17, as shown below, gives -8 as result which is not a whole number
9- 17 = -8
As whole numbers are not closed under subtraction, this created the need for a more extensive number system. Then integers were thought of as a new and more extensive number system to overcome the limitations of whole numbers. Integers, as you know, consist of all whole numbers and also negative numbers like -1, -2, -3, -4, -5…………..
When you subtract tow integers, the result is always an integer. Therefore, the number system of integers satisfied the closure property for subtraction.
Limitation Of Integers
The number system of integers too has a limitation, it does not satisfy the closure property of division. When you divide two integers, the result or quotient need not always an integer. For example,
-5÷25 = - 1/5, which is not an integer
Therefore, integers are closed for addition, subtraction and multiplication but not closed for division. This again created the need for a much more extensive number system. Then rational numbers were thought of as a number system that would overcome the limitation of integers.
RATIONAL NUMBERS
The numbers which can be written in the form of p/q, where p and q are integers and q ≠ 0.
We know that every natural number is a whole number and every whole number is an integer. However, the converse is not true, that is, every integer is not a whole number nor is every whole number a natural number.
What about the relationship between an integer and a rational number? Is every integer a rational number? The answer is yes, because every integer can be written in the form of p/1 {p/q, where q=1}. For example, 25 =25/1, 15 =15/1, -27=-27/1 and soon.
We have seen that every integer is a rational number, but every rational number is not an integer. Hence, it follows that all properties that are true for integers will also be true for rational numbers, but the converse need not be true.
Numerator and Denominator
In the rational number p/q, p is called the numerator and q as the
denominator.
If the numerator and denominator have the same sing, the rational number is said to be positive. If they have different signs, the rational number is said to be negative. Observe the following examples of positive and negative rational numbers.
Positive rational number: 6/7, -5/-9, 11/20, -3/-5
Negative rational numbers: -11/13, 2/-5, 7/-15, 6/-13
Lowest Form Of A Rational Number
A rational number p/q is said to be in the lowest or simplest form if p and q have no common factor other than 1. For example, rational numbers 7/8 and 5/9 are in the lowest form but 6/8 and 15/20 are not in the lowest form.
To reduce a rational number p/q to its lowest form, you have to divide its numerator p and denominator q by the HCF of p and q.
For example, to reduce the rational number 15/20 to its lowest form, first find the HCF of 15 and 20 –which is 5 – and then divide both the numerator and denominator by 5.
15÷5/20÷5= 3/4
3/4 is the lowest form of the rational number 15/20.
Standard Form Of A Rational Number
A rational number is said to be in the standard form if it is in the lowest form and its denominator is positive.
For example, the rational numbers 21/16, -7/9, -4/5 are in standard form. But the rational numbers 4/-5, 12/16, -8/12 are not in standard form.
Equivalent Rational Numbers
Two rational numbers P/q and r/s are said to be equal if and only if
p × s = r × q
It means
[Numerator of the first rational number × Denominator of the second rational number] = [Numerator of the second rational number × Denominator of the first rational number]
There is another easier method to find out whether the two rational numbers are qual. Rewrite both the rational numbers in their standard form. If they have the same standard form, they are equal, otherwise, they are not equal.
Example 1: Express 3/4 as a rational number with numerator
a. -15 b. 36
Solution: a. As 3 ×-5 = -15,
We need to multiply both the numerator and denominator of 3/4 by -5.
3 × -5/4 × -5 =-15/-20
b. As 3 × 12 = 36
We need to multiply both the numerator and denominator of 3/4 by 12.
3 × 12/4 × 12 = 36/48.
Addition And Subtraction Of Rational Numbers
1. Rational Numbers with the Same Denominator: If two rational numbers have the same denominator, then their sum or difference is a rational number. The numerator is the sum or difference of the numerators of given rational numbers and denominator is the same that of the given rational numbers. For example,
Sum of 5/13 + 7/13 = 5+7/13 = 12/13
Difference of 9/11-5/11 = 9-5/11 = 4/11
2. Rational Numbers with different Denominators: Rewrite the given rational numbers with positive denominators. Find the LCM of the denominators and again rewrite the given rational numbers as equivalent rational numbers having this LCM as denominators. Then add or subtract as mentioned above.
Example 2: Add 3/4 + 5/6
Solution: LCM of 4 and 6 is 12.
Rewriting given rational numbers as
3 × 3/4 ×3 = 9/12
and 5 × 2/6 ×2 = 10/12
Now, 9/12 +10/12 = 19/12
Multiplication Of Rational Numbers
The product of two numbers a/b and c/d is a rational number whose numerator is the product of the numerator s (a × c) of the given rational numbers and whose denominator is the product of the denominators (b × d) of the given rational numbers 5/8 and 7/9, we get
a /b × c/d = ac/bd
For example, multiplying the two rational numbers 5/8 and 7/9, we get
5/8 × 7/9 = 35/72
Division Of Rational Numbers
Dividing a rational number c/d by another rational number a/b is the same as multiplying c/d by the reciprocal of a/b. The reciprocal of a/b is b/a that is obtained by interchanging the numerator and denominator of a/b. Therefore, dividing c/d by a/b , we get
c/d ÷ a/b = c/d × b/a
For example, dividing the rational number -15/28 by another rational number 25/42, we get
-15/28 ÷ 25/42 = -15/28 × 42/25
= -9/10
Example 3: Rewrite -7/8 such that its denominator is equal to
a. 48 b. 128
Solution: a. We need to find an equivalent rational number of -7/8 such that it’s denominator is 48. So let us suppose that nominator of that rational number be x such that
-7/8 = x/48
Cross multiplying, we get
8 × x = -7 ×48
x = -7 × 48/8
= -7 × 6
= -42
b. Similarly we need to find y such that
-7/8 = y/128
Cross-multiplying, we get
8y = -7 × 128
Y = -7 × 128/8
= -7 × 16
= -112
Therefore, the required equivalent fractions are a. -42/48 b. -112/128.
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