RATIONAL NUMBERS

 Rational Numbers

Definition

• Rational  numbers are one very common type of number that we usually study after integers in math. These numbers are in the form of p/q, where p and q can be any integer and q ≠ 0.

Types of Rational Numbers

There are different types of rational numbers. We shouldn't assume that only fractions with integers are rational numbers. The different types of rational numbers are:

• Integers like -2, 0, 3 etc.
• Fractions whose numerators and denominators are integers like 3/7, -6/5, etc.
•      Terminating decimals like 0.35, 0.7116, 0.9768, e
• Non-terminating decimals with some repeating patterns (after the decimal point) such as 0.333..., 0.141414..., etc. These are popularly known as non-terminating repeating decimals

Smallest Rational Number

• Look at the chart given below to understand the difference between rational numbers and irrational numbers along with other types of numbers pictorially.

Standard Form of Rational Numbers

• The standard form of a rational number can be defined if it’s no common factors aside from one between the dividend and divisor and therefore the divisor is positive.
• For example, 12/36 is a rational number. But it can be simplified as 1/3; common factors between the divisor and dividend is only one. So we can say that rational number ⅓ is in standard form.

Positive and Negative Rational Number

Positive Rational Numbers	Negative Rational Numbers
If both the numerator and denominator are of the same signs.	If numerator and denominator are of opposite signs.
All are greater than 0	All are less than 0
Examples of positive rational numbers: 12/17, 9/11 and 3/5	Examples of negative rational numbers: -2/17, 9/-11 and -1/5.

Properties of rational numbers

• Since a rational number is a subset of the real number, the rational number will obey all the properties of the real number system. Some of the important properties of the rational numbers are as follows:

• The results are always a rational number if we multiply, add, or subtract any two rational numbers.
• A rational number remains the same if we divide or multiply both the numerator and denominator with the same factor.
• If we add zero to a rational number then we will get the same number itself.
• Rational numbers are closed under addition, subtraction, and multiplication.

Solved Examples

Example 1:

Identify each of the following as irrational or rational: ¾ , 90/12007, 12 and √5.

Solution:

Since a rational number is the one that can be expressed as a ratio. This indicates that it can be expressed as a fraction wherein both denominator and numerator are whole numbers.

• ¾ is a rational number as it can be expressed as a fraction. 3/4 = 0.75
• Fraction 90/12007 is rational.
• 12, also be written as 12/1. Again a rational number.
• Value of  √5 = 2.2360679775…….. It is a non-terminating value and hence cannot be written as a fraction. It is an irrational number.

Example 2:  

Identify whether mixed fraction, 11/2 is a rational number.

Solution: 

The Simplest form of 11/2 is 3/2

Numerator = 3, which is an integer

Denominator = 2, is an integer and not equal to zero.

So, yes, 3/2 is a rational number.

Fun Facts

• The ancient greek mathematician Pythagoras believed that all numbers were rational, but one of his students Hippasus proved (using geometry, it is thought) that you could not write the square root of 2 as a fraction, and so it was irrational.
• But followers of Pythagoras could not accept the existence of irrational numbers, and it is said that Hippasus was drowned at sea as a punishment from the gods!