Rational numbers room one an extremely common kind of number that us usually study after integers in math. These numbers space in the form of p/q, whereby p and q have the right to be any integer and q ≠ 0. Many often world find the confusing to differentiate in between fractions and also rational numbers due to the fact that of the an easy structure of numbers, that is p/q form. Fountain are consisted of of totality numbers when rational numbers are made up of integers together their numerator and also denominator. Let's learn more about rational numbers in this lesson.

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1.What room Rational Numbers?
2.Types of rational Numbers
3.How to recognize Rational Numbers?
4.Arithmetic operations on reasonable Numbers
5. Irrational vs reasonable Numbers
6.FAQs on reasonable Numbers

What space Rational Numbers?

Do you recognize from where the word "Rational" originated? it is source from words "ratio". So, rational number are really well regarded the ratio principle of ratio.

Rational numbers Definition

A rational number is a number the is the the kind p/q wherein p and also q space integers and q is no equal come 0. The collection of rational numbers is denoted by Q.

In other words, If a number have the right to be expressed as a portion where both the numerator and also the denominator space integers, the number is a rational number.


Examples of rational Numbers

If a number have the right to be expressed as a fraction where both the numerator and also the denominator space integers, the number is a reasonable number. Some examples of rational numbers are:

Types of reasonable Numbers

There room different types of reasonable numbers. Us shouldn't i think that only fractions through integers space rational numbers. The different species of rational number are:

integers choose -2, 0, 3 etc.fractions who numerators and denominators are integers prefer 3/7, -6/5, etc.terminating decimals choose 0.35, 0.7116, 0.9768, etc.

How to determine Rational Numbers?

In each of the over cases, the number have the right to be expressed together a fraction of integers. Hence, every of this numbers is a reasonable number. To find whether a offered number is a rational number, us can check whether that matches with any of these conditions:

We deserve to represent the provided number as a fraction of integersWe the decimal expansion of the number is terminating or non-terminating repeating.


The given number has actually a collection of decimal 923076 which is repeating continuously.


Thus, the is a reasonable number.

Rational number in decimal form

Rational number can likewise be expressed in decimal form. Perform you know 1.1 is a rational number? Yes, that is due to the fact that 1.1 can be composed as 1.1= 11/10. Currently let's talk around non-terminating decimals such together 0.333..... Due to the fact that 0.333... Have the right to be created as 1/3, thus it is a reasonable number. Therefore, non-terminating decimals having repeated number after the decimal point are additionally rational numbers.

Is 0 a rational Number?

Yes, 0 is a reasonable number together it have the right to be written as a portion of integers favor 0/1, 0/-2,... Etc.


List of rational Numbers

From the above information, that is clear the there is one infinite variety of rational numbers. Hence, it is not possible to identify the list of rational numbers.

Smallest rational Number

Since we cannot identify the list of reasonable numbers, us cannot recognize the smallest rational number.

Points come Remember reasonable Numbers:

Rational numbers room NOT only fractions but any number that have the right to be expressed together fractions.Natural numbers, totality numbers, integers, fractions of integers, and also terminating decimals room rational numbers.Non-terminating decimals with repeating patterns of decimal are also rational numbers.If a fraction has a an adverse sign either to the molecule or come the denominator or in front of the fraction, the fraction is negative. I.e, -a/b = a/-b.

Arithmetic operations on reasonable Numbers

Rational numbers deserve to be added, subtracted, multiplied, or divided just like fractions. These are the four simple arithmetic operations perform on rational numbers.

Addition of reasonable numbersRational number subtractionRational numbers multiplicationDivision of rational numbers

Adding and Subtracting reasonable Numbers

The process of adding and subtracting rational numbers deserve to be excellent in the same way as fractions. To include or subtract any two rational numbers, we make your denominators the same and also then include the numerators.

Example : 1/2 - (-2/3)= 1/2 + 2/3 = 1/2 × 3/3 + 2/3 × 2/2 = 2/6 + 4/6 = 6/6 = 1

We deserve to learn much more about enhancement of fractions and also subtraction of fractions.

Multiplying and also Dividing reasonable Numbers

The procedure of multiplying and dividing reasonable numbers have the right to be done in the same method as fractions. Come multiply any two reasonable numbers, we multiply their numerators and also their platform separately and simplify the result fraction.

Example: 3/5 × -2/7 = (3 × -2)/(5 × 7)= -6/35

To divide any type of two fractions, we multiply the first fraction (which is dividend) through the mutual of the second portion (which is the divisor).

Example: 3/5 ÷ 2/7=3/5 × 7/2 = 21/10 or \(2\dfrac110\)

Irrational vs reasonable Numbers

The numbers which room NOT rational numbers are referred to as irrational numbers. The collection of irrational number is represented by Q´. The difference in between rational and irrational numbers space as follows:

Rational NumbersIrrational Numbers

These are numbers that deserve to be expressed together fractions of integers.

Examples: 0.75, -31/5, etc

These room numbers that CANNOT be expressed as fractions the integers.

Examples: √5, π, etc.

They have the right to be end decimals.They are never ever terminating decimals.

They can be non-terminating decimals with recurring patterns the decimals.

Example: 1.414, 414, 414 ... Has repeating trends of decimals whereby 414 is repeating.

They need to be non-terminating decimals with NO recurring patterns the decimals.

Example: √5 = 2.236067977499789696409173.... Has no repeating trends of decimals

The collection of rational numbers includes all-natural numbers, all whole numbers, and all integers.The collection of irrational number is a separate set and that does not contain any kind of of the various other sets the numbers.

Look at the graph given listed below to know the difference between rational numbers and also irrational numbers along with other species of number pictorially.


Example 2: uncover a rational number in between the following: 1/2 and also 2/3.

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We recognize that the typical of any type of two numbers lies in between the two numbers. Let's discover the median of the provided two reasonable numbers.

\( \beginaligned \dfrac \dfrac12+ \dfrac232 &= \dfrac\dfrac36+ \dfrac462\\<0.3cm> &= \dfrac \left(\dfrac76 \right)2\\<0.3cm> &= \dfrac \left(\dfrac76 \right) \left(\dfrac21 \right) \endaligned \)= 7/6 × 1/2= 7/12