A Rational Number is represented as a fraction. For example, x / y is a rational number. Here, the upper term of the fraction is called a numerator that is ‘x’, and the lower term of the fraction is called a denominator that is ‘y’. Both terms numerator (x) and denominator (y) must be integers. Integer numbers are both positive and negative numbers like -4, -3, -2, -1, 1, 2, 3, 4…etc… The most important thing in the rational numbers is the denominator of the rational number cannot be equal to zero.
Rational Numbers are 2/3, 4/5, 6/7, 8/9, 10/15, 20/4, 125/100, ….etc… Check out the complete concept of Rational Numbers in Terminating and Non-Terminating Decimals below.
Rational Numbers in Decimal Fractions
By simplifying the rational numbers, we will get the result in the form of decimal fractions. We have two types of decimal fractions. They are
- Terminating Numbers
- Non – Terminating Numbers
When we converted the rational numbers into a decimal fraction, we will get either finite numbers of digits or infinite numbers of digits after the decimal point. If we get the decimal fraction with the finite number of digits, then it is called Terminating Numbers. If we get the decimal fraction with the infinite number of digits after the decimal point, then it is called Non – Terminating Numbers.
Terminating Number
Example for Terminating Numbers are 1.25, 0.68, 1.234, 2.456, 3. 4567, 5.687, 6.24, 8. 46, ….etc….The below examples are in the form of rational numbers and we need to convert that numbers into the form of decimal numbers.
(1) x / y = 100 / 25 = 0.4
(2) x / y = 644 / 8 = 8.5
(3) x / y = 5 / 4 = 1.25
Non – Terminating Number
Example for Non – Terminating Numbers are 1.23333, 2.566666, 5.8678888, 3.467777, 4.6899999,…..etc… The below-mentioned x / y fraction indicates the rational numbers and by simplifying it, we will get the decimal numbers.
(1) x / y = 256 / 6 =42.66666…
(2) x / y = 10 / 3 = 3.33333…
(3) x / y = 20 / 9 = 2.222222…
Note: If a rational number (≠ integer) can be expressed in the form p/(2^n × 5^m) where p ∈ Z, n ∈ W, and m ∈ W then the rational number will become a terminating decimal. If not, the rational number becomes the Non – Terminating Numbers.
Examples of Repeating and Non-Repeating Decimals
1. Find out the conversion of rational numbers to terminating decimal fractions?
(i) 1/4 is a rational fraction of form p/q. When this rational fraction is converted to decimal it becomes 0.25, which is a terminating decimal fraction.
(ii) 1/8 is a rational fraction of form p/q. When this rational fraction is converted to decimal fraction it becomes 0.125, which is also an example of a terminating decimal fraction.
(iii) 4/40 is a rational fraction of form p/q. When this rational fraction is converted to decimal fraction it becomes 0.1, which is an example of a terminating decimal fraction.
2. Find out the conversion of rational numbers to nonterminating decimal fractions.
(i) 1/11 is a rational fraction of form p/q. When we convert this rational fraction into a decimal, it becomes 0.090909… which is a non-terminating decimal.
(ii) 1/13 is a rational fraction of form p/q. When we convert this rational fraction into a decimal, it becomes 0.0769230769230… which is a non-terminating decimal.
(iii) 2/3 is a rational fraction of form p/q. When this is converted to a decimal number it becomes 0.66666667… which is a non-terminating decimal fraction.
Irrational Numbers
You may see different types of numbers such as real numbers, whole numbers, rational numbers, etc. Now, let us check out the irrational numbers. Irrational numbers are also real numbers that are represented as a simple fraction. There is no repeating or no terminate pattern available in Irrational Numbers. the numbers which do not consist of exact square roots of integers treats as Irrational Numbers. Also, the Irrational Number is pi and that is equal to the value of 3.14.
Solved Problems on Rational and Irrational Numbers
Addition
Add the two rational numbers. For example x / y = 1 / 2 and p / q = 2 / 6
To add the two rational numbers, we need to find out the LCM of the denominators.
That is, LCM of 2 and 6 is 6.
x / y + p / q = 1 / 2 + 2 / 6 = (3 + 2) / 6 = 5 / 6.
Multiplication
Multiply the two rational numbers such as 2 / 3 and 5 / 6.
x / y = 2 / 3 and p / q = 5 / 6
x / y X p / q = 2 / 3 x 5 / 6
(xX p) / (y X q) = (2 x 5) / 3 x 6)
px / yq = 10 / 18 = 5 / 9.
Subtraction
Subtract the two rational numbers. Here, 5 / 8 and 12 / 5 are rational numbers.
x / y = 5 / 8 and p / q = 12 / 5
x / y – p / q = 5 / 8 – 12 / 5
For subtraction, we need to find out the LCM of denominator values.
LCM of 8 and 5 is 40.
5 / 8 – 12 / 5 = [(5 x 5) – (12 x 8)] / 40 = (25 – 96) / 40 = -71 / 40.
Division
To divide the two rational numbers, we need to cross multiply the terms. For example, x / y and p / q are two rational numbers.
(x / y) ÷( p / q) = (2 / 5) ÷ ( 3 / 7)
Cross Multiply the first fraction numerator with second fraction denominator and vice versa.
xq / py = (2 x 7) / (5 x 3 )
xq / py= 14 / 15.
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Rational Numbers
A rational number is represented as a fraction, where the upper term of the fraction is called the numerator and the lower term is called the denominator. Both the numerator and denominator must be integers, and the denominator cannot be equal to zero.
Rational Numbers in Decimal Fractions
When rational numbers are simplified, they can be expressed as decimal fractions. There are two types of decimal fractions: terminating numbers and non-terminating numbers. Terminating numbers have a finite number of digits after the decimal point, while non-terminating numbers have an infinite number of digits after the decimal point.
Terminating and Non-Terminating Numbers
- Terminating Numbers: Examples of terminating numbers include 1.25, 0.68, and 1.234. Rational numbers such as 100/25, 644/8, and 5/4 can be converted into terminating decimal fractions.
- Non-Terminating Numbers: Examples of non-terminating numbers include 1.23333, 2.566666, and 5.8678888. Rational numbers such as 256/6, 10/3, and 20/9 can be converted into non-terminating decimal fractions.
Repeating and Non-Repeating Decimals
Rational numbers can be converted into terminating or non-terminating decimal fractions. For example, 1/4 becomes 0.25, a terminating decimal fraction, while 1/11 becomes 0.090909..., a non-terminating decimal.
Irrational Numbers
Irrational numbers are also real numbers but cannot be represented as simple fractions. They do not have a repeating or terminating pattern. An example of an irrational number is π, which is approximately equal to 3.14.
Operations on Rational Numbers
- Addition: To add two rational numbers, find the least common multiple (LCM) of the denominators and then add the fractions.
- Multiplication: Multiply the numerators together and the denominators together to get the result.
- Subtraction: Find the LCM of the denominators and then subtract the fractions.
- Division: Cross multiply the terms to divide the rational numbers.
These concepts provide a foundational understanding of rational and irrational numbers, their representation, and operations involving rational numbers. If you have further questions or would like to explore additional topics, feel free to ask!
