Rational Numbers

Rational Numbers are a type of real numbers. Go through the article to learn in detail about what is a rational number, the types of rational numbers, properties of rational numbers explained. In addition to these, you will learn the standard form of a rational number, how to identify a rational number, the difference between a rational number and an irrational number. Check out solved examples on rational numbers available and master the concept.

Also, See: Rational Numbers in Terminating and Non-Terminating Decimals

List of Rational Numbers Topics

What is a Rational Number?

In Mathematics, a Rational Number is defined as any number that can be denoted in the form of a/b where b≠0. In other words, we can say that a fraction is said to be a rational number if both the numerator and denominator are integers and the denominator is not equal to zero. On dividing a rational number we get the result in decimal form and it can be either a terminating or a repeating decimal.

Examples of Rational Numbers are 1/3, 4/5, 6/7, etc.

How to Identify Rational Numbers?

You need to check the below-listed conditions to determine if a number is a rational number or not. They are outlined for your reference

  • Rational Number is represented in the form of a/b where b≠0
  • We can simplify and represent the ratio a/b in decimal form.

Standard Form of Rational Numbers

The Standard Form of a Rational Number is defined as the number if it has no common factors between the dividend and divisor. For Example, 12/24 reduced in its simplest form is 1/2. Common factors between the divisor and dividend are only one thus it is said to be in its simplest form.

Positive and Negative Rational Numbers

A Rational number is in the form of p/q, in which p and q are integers and q should be non-zero integers. Rational Numbers can be either positive or negative. If a rational number is positive, then both p and q are positive integers. If a rational number takes the form of -(p/q), then either p or q takes the negative value. It means that

-(p/q) = (-p)/q = p/(-q).

Positive Rational Numbers Negative Rational Numbers
In the Case of Positive Rational Numbers, both the numerator and denominator will be of the same signs. In Negative Rational Numbers, numerator and denominator will be of opposite signs.
All are greater than 0 All are less than 0
Examples of positive rational numbers: 13/14, 9/10 and 3/4 Examples of negative rational numbers: -1/12, 7/-13 and -3/6.

Arithmetic Operations on Rational Numbers

Arithmetic Operations are the basic operations we perform on Integers. Below we will discuss how to perform these arithmetic operations on rational numbers.

Addition: While adding two rational numbers say a/b, c/d we need to make the denominators same. Thus we get (ad+bc)/bd. For Example 1/3+4/5 added gives (1*5+4*3)/15 = 17/15

Subtraction: In the case of subtracting rational numbers a/b, c/d we need to make the denominators the same and then subtract. For Example, 1/3-1/2 subtracted gives (6- 2)/6 i.e. 4/6 or 2/3

Multiplication: In the Case of Rational Numbers Multiplication we need to multiply the numerators and denominators respectively. If a/b * c/d is multiplied we have ac/bd. For Example 3/4*5/3 = (3*5)/(4*3) = 15/12

Division: If a/b÷c/d then we can denote the rational numbers division as (a/b)÷(c/d) = ad/bc

Multiplicative Inverse of Rational Numbers

Multiplicative Inverse of a Rational Numbers is nothing but the reciprocal of the given fraction. For Example, if 3/4 is a rational number then the multiplicative inverse of the given rational number is 1/(3/4) = 4/3

Rational Numbers Properties

As we all know Rational Numbers are a set of real numbers it obeys all the properties of real numbers. We have outlined a few of the important properties of rational numbers for your reference. They are as such

  • Whenever you add, subtract, multiply or divide a rational number with another rational number a result is always a rational number.
  • Rational Numbers will not change if we multiply or divide both the numerator and denominator with the same factor.
  • If we add zero to a rational number we will get the same number itself.
  • Rational Numbers are closed under addition, subtraction and multiplication.

Rational Numbers Vs Irrational Numbers

Fractions having non-zero denominators is called rational number. Irrational Numbers can’t be written in simple fractions instead we can represent them using decimals. Irrational Numbers have endless non-repeating digits next to the decimal point.

How to Find the Rational Numbers between Two Rational Numbers?

There are n number of rational numbers between 2 rational numbers. We can find rational numbers between two rational numbers using 2 different methods. The two methods are listed as such

Method 1:
Firstly, find the equivalent fraction for the given rational numbers and find out the rational numbers between them. The numbers obtained are the required rational numbers.

Method 2:
Obtain the mean value of the given two rational numbers. Mean value is the required rational number. To obtain even more rational numbers repeat the process with old and newly obtained rational numbers.

Solved Examples on Rational Numbers

Example 1.
Determine whether 1.25 is a rational number or not?
Solution:
Since the given number 1.25 is in the decimal format we need to change it to fraction form.  if the denominator of the fraction is non zero then the number is a rational number.
1.25=125/100
=5/4
Since denominator 4 is non zero given decimal number 1.25 is a rational number.

Example 2.
Identify whether the mixed fraction 1 4/2 is a rational number or not?
Solution:
Simplest Form of a Rational Number = 1 4/2
=6/2
Since denominator 2 is non-zero 6/2 is a rational number.

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