# Rational Numbers

In mathematics, a rational number is a type of real number, which is represented in the form of p/q where q is not equal to zero. Any fractional number which doesn't have zero as its denominator is a rational number. Some examples of rational numbers are 15 / 14, 2 / 4, 6 / 7, and so on. The number "0" is also a rational number, as we can represent it in the form of 0 / 1, 0 / 2, and 0 /3. But, 1 / 0, 2 / 0, 3 / 0, etc., are not rational numbers, since the values that we get are infinite.

### What is a Rational Number?

The definition of a rational number in mathematics can be given as any number which can be represented in the form of p/q where q cannot be equal to zero. Also, we can say that any fraction fits under the category of rational numbers, where the denominator and numerator are integers and the denominator is not equal to zero. When the rational number (i.e., fraction) is divided, the result will be in decimal form, which can either be terminating decimal or a repeating decimal.

### How to identify rational numbers?

In order to identify a rational number, we can check the following conditions:

• It is represented in the form of p/q, where q≠0.
• The ratio p/q can be further simplified and represented in decimal form.

The set of rational numerals:

1. Include positive numbers, negative numbers, and zero.
2. Can be expressed as a fraction.

Examples of Rational Numbers:

p q P/q Rational
14 2 14 /2 =7 Rational
2 1000 2 /1000 = 0.002 Rational
60 10 60/10 = 6 Rational

### Different types of Rational Numbers

A number is called rational only if we can write it as a fraction, where both the numerator and denominator are integers and the denominator is a non-zero number.

• All rational numbers are included in the real numbers (R).
• Real numbers include the integers (Z).
• Integers involve natural numbers (N).

### The 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/ 48 is a rational number. But it can be more simplified to 1/ 4; common factors between the divisor and dividend are only one. So, we can say that the rational number 1 / 4 is in standard form.

### Positive and Negative Rational Numbers

As we have studied earlier rational number is represented in the form of p / q, where p and q are integers. Also, q should not be zero. There is both type of natural numbers positive and negative. If the rational number is positive, both p and q are positive integers. If the rational number takes the form -(p/q), then either p or q takes the negative value. It means that:

=> -(p / q) = (-p) / q = p / (-q)

Now, let's have a look at some examples of positive and negative rational numbers:

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

### The Arithmetic operations on Rational Numbers

In Maths, arithmetic operations are the basic operations we perform on integers. Let us discuss here how we can perform these operations on rational numbers, say p/q and s/t.

1. Addition: When we add p/q and s/t, we need to make the denominator the same or consider the least common multiple. Hence, we get (pt+qs)/qt.

Example: 1/ 4 + 3/4 = (1 +3) /4 = 4 /4.

2. Subtraction: Similarly, if we subtract p/q and s/t, then also, we need to make the denominator the same, first, and then do the subtraction.

Example: 1/ 4 - 3/4 = (1 - 3) /4 = -2 /4 = -1/2.

3. Multiplication: In the case of multiplication, while multiplying two rational numbers, the numerator and denominators of the rational numbers are multiplied, respectively. If p/q is multiplied by s/t, then we get (p×s)/(q×t).

Example: 13 /2 × 3/4 = (13 ×3)/(2×4) = 39 /8.

4. Division: If p/q is divided by s/t, then it is represented as: (p/q)÷(s/t) = p ×t/q× s

Example: 1/2 ÷ 3/ 8 = (1× 8)/(2×3) = 4/ 3 = 4 /3.

### Multiplicative Inverse of Rational Numbers

As we already know rational numbers are represented in the form of p / q, which is a fraction, definition of the multiplicative inverse is that it is the reciprocal of the given fraction.

For example: if 2 / 3 is a rational number, then the multiplicative inverse of the rational number 2 / 3 is 3 / 2 because as we will multiply both numbers, we will get 1. [(2 / 3) * (3 / 2) = 1]

### Properties of Rational Numbers

As we have the knowledge that rational number is a subset of real number, rational numbers will obey the properties of the real number system. Some of the important properties of rational numbers are as follows:

• If we do any operations on a rational number like addition multiplication, division, or subtraction, the result is always a rational number.
• There is no change in the rational number if we divide or multiply both the numerator and denominator with the same factor.
• If we add zero to a rational number, we get the same number itself.
• Rational numbers are closed under addition, subtraction, and multiplication.

### Rational Numbers and Irrational Numbers As the name itself suggests there is a difference between rational and Irrational Numbers. A fraction that has a non-zero denominator is called a rational number. The number 2 / 3 is a rational number because it is read as integer 2 divided by integer 3. All the numbers that are not rational are known as irrational numbers.

Rational Numbers Irrational Numbers
These are numbers that can be expressed as fractions of integers. Examples: 1/2, 0.75, -31/5, etc These are numbers that cannot be expressed as fractions of integers. Examples: √5, π, etc.
They are terminating decimals. They are NEVER terminating decimals that do not have an accurate value.
They can be non-terminating decimals with repetitive patterns of decimals or recurring decimals. Example: 1.414, 414, 414 ... has repeating patterns of decimals where 414 is repeating. They should be non-terminating decimals with NO repetitive patterns of decimals. Example: √5 = 2.236067977499789696409173.... has no repeating patterns of decimals
The set of rational numbers contains all-natural numbers, all whole numbers, and all integers. The set of irrational numbers is a separate set and it does NOT contain any of the other sets of numbers.

Rational Numbers can either be positive, negative, or zero. While specifying a negative rational number, the negative sign is either in front or with the numerator of the number, which is the standard mathematical notation. For example: we denote the negative of 5 / 2 as - 5 / 2.

An irrational number cannot be written as a simple fraction but can be represented with a decimal. It has endless non-repeating digits after the decimal point. Some of the common irrational numbers are:

Pi (π) = 3.142857...

Euler's Number (e) = 2.7182818284590452..

√2 = 1.414213...

### How to find Rational Numbers between two Rational Numbers?

There are infinite numbers of rational numbers between two rational numbers. The rational numbers between two rational numbers can be found easily using two different methods. Now, let us have a look at the two different methods.

Method 1:

Find out the equivalent fraction for the given rational numbers and find out the rational numbers in between them. Those numbers should be the required rational numbers.

Method 2:

Find out the mean value for the two given rational numbers. The mean value should be the required rational number. In order to find more rational numbers, repeat the same process with the old and the newly obtained rational numbers.

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