A rational number is a form of real number that is written as d/c where c is not equivalent to zero; in simple terms, a rational number is any fraction having non-zero denominators. 1/3, 12/7, 113/6 are some instances of rational numbers. The number "0" is also rational as it can be represented in various ways, including 0/1, 0/99, 0/54677 and more. However, 1/0, 54/0, 999/0, etc., are irrational since they offer us unlimited value, that is the value of infinity. NCERT Class 9 Maths Chapter 1 Number System talks extensively about real numbers and their types including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Let us dive deep into the concepts and ideas of rational numbers in this article.
All the rational numbers are real numbers, and the converse is also true; only real numbers can be rational numbers. We can say that a fractional integer that has a non-zero denominator is considered a rational number. For instance, 8 is a rational number because it can be written as 8/1, or 16/2, or 32/4.
There are two types of rational numbers: Positive rational numbers and negative rational numbers. The positive rational numbers lie to the right side of 0 on the number line and start from 1. They extend to positive infinity and must contain both the numerator and denominator as positive numbers. For example, 1, 10.2214, 33/2, 999/6, 2341, 2334546, etc. They can be decimal as well as fractional in form. The negative rational number lies to the left of 0 on the number line and ranges from -1 to (-infinity). Like positive rational numbers, they are also in decimal and fractional form, and either the denominator or numerator must be negative. For example, -3, -5.67, -21/4, -33435, etc. 0 is neither positive nor negative, it is a neutral rational number.
All the basic arithmetic operations can be carried on rational numbers with ease. Let us discuss all the arithmetic operations one at a time.
Addition: To add two rational numbers, we first need to make the denominators equal. This is done by determining the Least Common Multiple and then multiplying the numerators with suitable numbers. Example: 3/2 + 5/4, the Lcm of both denominators is 4. Multiplying 3 by 2 and 5 by 1 and keeping the LCM as the denominator, we get (6 + 5)/4 = 11/4
Subtraction: Similarly, to subtract two rational numbers, we first need to make the denominators equal by finding the Least Common Multiple and then multiplying the numerators with suitable numbers, eventually finding the difference. Example: 3/4 - 5/8, the Lcm of both denominators is 8. Multiplying 3 by 2 and 5 by 1 and keeping the LCM as the denominator, we get (6 - 5)/8 = 1/8
Multiplication: Two rational numbers are multiplied by finding the product of respective numerators and denominators. Example: 2/11 * 4/5 = (2 * 4)/ (11 * 5) = 8/55
Division: Two rational numbers are divided by finding the product of the first number and the reciprocal of the second number. Example: 3/12 ÷ 7/3 = 3/12 * 3/7 = (3 * 3)/ (12 * 7) = 9/84 = 3/28