# What is the difference between a rational number and an irrational number?

 Check out the subsets of the Real Numbers shown in the diagram at the right. Notice that the rational and irrational numbers are contained within the set of Real Numbers.  A rational number is a number that can be expressed as a fraction (ratio) in the form where p and q are integers and q is not zero.

Examples: A rational number can be expressed as a ratio

(fraction). When a rational number fraction is divided to form a decimal value,

it becomes a terminating or repeating decimal.  Some rational fractions may produce a large number of digits in their repeating patterns, which may exceed the size of the viewing screen on a calculator. The fraction 53/83 has a calculator display of 0.6385542169, which shows no repeating pattern, when in reality the pattern will repeat after 41 digits.

To convert a repeating decimal to a fraction: To show that the rational numbers are dense: (between any two rationals there is another rational)  An irrational number is a number that is NOT rational. It cannot be expressed as a fraction with integer values in the numerator and denominator.

Examples: When an irrational number is expressed in decimal form, it goes on forever without repeating.

 Regarding :
While it is popular to use 3.14 or to represent “pi”, these values are only estimates or approximations. Notice the differences in the decimal representations on the calculator screen at the right. Properties of Rational and Irrational Numbers:

Since rational and irrational numbers are subsets of the real numbers, they possess all of the properties assigned to the real number system.

## Difference between Rational and Irrational Numbers

Difference between rational and irrational numbers has been clearly explained in the picture given below. Example :

Rational : 1.2626262626……….(Repeated pattern is 26)

Irrational : 1.4142135623…………..(No repeated pattern)

• More clearly,
• A non terminating decimal which has repeated pattern is called as rational number.
• Because, the non terminating decimal which has repeated pattern can be converted into fraction.
• A non terminating decimal which does not have repeated pattern is called as irrational number.
• Because, the non terminating decimal which does not have repeated pattern can not be converted into fraction.
• To have better understanding on difference between rational and irrational numbers, let us come to know about rational numbers and irrational numbers more clearly.
• What are rational and irrational numbers ?

First let us come to know, what is rational number. Because, once we understand rational number, we can easily understand irrational number.

A rational number has to be in the form as given below.

### Rational Numbers So, any number in the form of fraction can be treated as rational number.

Examples of Rational Number :

5,   2.3,   0.02,   5/6

1. Because all these numbers can be written as fractions.
2. 5 = 5/1
3. 2.3 = 23/10
4. 0.02 = 2/100 = 1/50
5. 5/6 (This is already a fraction)

## Difference Between Rational and Irrational Numbers

In
Mathematics, we have to come across lots of numbers. In these numbers, there
come perfect squares, surds, terminating decimals, non-terminating decimals,
repeating decimals and non-repeating decimals etc. We usually divide these
numbers into two categories.

First category is known as rational numbers and
the second category is known as irrational numbers. No doubt, to understand the
difference between rational and irrational numbers is a difficult task for the
students.

Here, we will try to explain the difference between rational and
irrational numbers with the help of examples. In
Mathematics, rational numbers
are those numbers which are written in the form of p/q such that q≠0. The
condition for the rational numbers is that both p and q should belong to Z and
Z is a set of integers. The simplest examples of the rational numbers are given
below;

• ·
1/9
• ·
10  or 10/1
• The
irrational numbers are those numbers which are not written in the form of p/q.
The simplest examples of the irrational numbers are given below;
• ·
√3
• ·
3/0

Most of the
students are not able to understand the difference between the rational and
irrational numbers just with the help of their definitions. They require more
detail to understand the difference between rational and irrational numbers.
The key difference between them is given below;

All the
perfect squares are rational numbers and the perfect squares are those numbers
which are the squares of an integer. In other words, if we multiply an integer
with the same integer, we get a perfect square.

The examples of the perfect
squares are √ 4, √ 49, √ 324, √ 1089 and √ 1369. After taking the square roots
of these perfect squares, we get 2, 7, 18, 33 and 37 respectively. 2, 7, 18, 33
and 37 are all integers.

On the other hand, all the surds are the irrational
numbers and the surds are those numbers which are not the squares of an
integer. In other words, these are not the multiples of an integer with itself.
The examples of the surds are √2, √3 and √7.

After taking the square roots of
these surds, we get 1.41, 1.73 and 2.64 respectively. 1.41, 1.73 and 2.64 are
not integers.

All the
terminating decimals are rational numbers. Terminating decimals are those
decimals which have finite number of digits after the decimal point. For
example, 1.25, 2.34 and 6.94 are all rational numbers.

On the other hand,
non-terminating decimals are those numbers which have infinite number of digits
after the decimal point. For example, 1.235434…, 3.4444… and 6.909090… are
all non-terminating decimals.

Non-terminating decimals can be rational or
irrational. These are explained in the next point.

All the
repeating decimals are the rational numbers and the repeating decimals are
those decimals whose digits repeat over and over again. The examples of the
repeating decimals are .33333333, .222222 and .555555.

On the other hand, all
the non-repeating decimals are the irrational numbers and the non-repeating
decimals are those digits which don’t repeat over and over again. The examples
of the non-repeating decimals are .0435623, .

3426452 and .908612.

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## Irrational Numbers

An Irrational Number is a real number that cannot be written as a simple fraction.

Irrational means not Rational

Let's look at what makes a number rational or irrational …

### Rational Numbers

A Rational Number can be written as a Ratio of two integers (ie a simple fraction).

Example: 1.5 is rational, because it can be written as the ratio 3/2

Example: 7 is rational, because it can be written as the ratio 7/1

Example 0.333… (3 repeating) is also rational, because it can be written as the ratio 1/3

But some numbers cannot be written as a ratio of two integers …

…they are called Irrational Numbers.

π = 3.1415926535897932384626433832795… (and more)

We cannot write down a simple fraction that equals Pi.

The popular approximation of 22/7 = 3.1428571428571… is close but not accurate.

Another clue is that the decimal goes on forever without repeating.

### Cannot Be Written as a Fraction

It is irrational because it cannot be written as a ratio (or fraction),
not because it is crazy!

So we can tell if it is Rational or Irrational by trying to write the number as a simple fraction.

9.5 = 192

So it is a rational number (and so is not irrational)

Here are some more examples:

Number   As a Fraction   Rational or
Irrational?
1.75 74 Rational
.001 11000 Rational
√2
(square root of 2)
? Irrational !

Let's look at the square root of 2 more closely. When we draw a square of size “1”, what is the distance across the diagonal?

The answer is the square root of 2, which is 1.4142135623730950…(etc)

## Identifying Rational and Irrational Numbers

### Learning Outcomes

• Identify rational numbers from a list of numbers
• Identify irrational numbers from a list of numbers

In this chapter, we’ll make sure your skills are firmly set.

We’ll take another look at the kinds of numbers we have worked with in all previous chapters. We’ll work with properties of numbers that will help you improve your number sense.

And we’ll practice using them in ways that we’ll use when we solve equations and complete other procedures in algebra.

We have already described numbers as counting numbers, whole numbers, and integers. Do you remember what the difference is among these types of numbers?

 counting numbers [latex]1,2,3,4dots [/latex] whole numbers [latex]0,1,2,3,4dots[/latex] integers [latex]dots -3,-2,-1,0,1,2,3,4dots [/latex]

### Rational Numbers

What type of numbers would you get if you started with all the integers and then included all the fractions? The numbers you would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.

A rational number is a number that can be written in the form [latex]frac{p}{q}[/latex], where [latex]p[/latex] and [latex]q[/latex] are integers and [latex]q
e o[/latex].

• All fractions, both positive and negative, are rational numbers. A few examples are
• [latex]frac{4}{5},-frac{7}{8},frac{13}{4}, ext{and}-frac{20}{3}[/latex]
• Each numerator and each denominator is an integer.

We need to look at all the numbers we have used so far and verify that they are rational. The definition of rational numbers tells us that all fractions are rational.

We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational.
Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers.

An easy way to do this is to write it as a fraction with denominator one.

[latex]3=frac{3}{1}-8=frac{-8}{1}0=frac{0}{1}[/latex]

Since any integer can be written as the ratio of two integers, all integers are rational numbers. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational.

What about decimals? Are they rational? Let’s look at a few to see if we can write each of them as the ratio of two integers. We’ve already seen that integers are rational numbers. The integer [latex]-8[/latex] could be written as the decimal [latex]-8.0[/latex]. So, clearly, some decimals are rational.

Think about the decimal [latex]7.3[/latex]. Can we write it as a ratio of two integers? Because [latex]7.3[/latex] means [latex]7frac{3}{10}[/latex], we can write it as an improper fraction, [latex]frac{73}{10}[/latex]. So [latex]7.3[/latex] is the ratio of the integers [latex]73[/latex] and [latex]10[/latex]. It is a rational number.

In general, any decimal that ends after a number of digits such as [latex]7.3[/latex] or [latex]-1.2684[/latex] is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.

Write each as the ratio of two integers:

1. [latex]-15[/latex]

## Rational and Irrational Numbers

A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers.

• The number 8 is a rational number because it can be written as the fraction 8/1.
• Likewise, 3/4 is a rational number because it can be written as a fraction.
• Even a big, clunky fraction like 7,324,908/56,003,492 is rational, simply because it can be written as a fraction.

Every whole number is a rational number, because any whole number can be written as a fraction. For example, 4 can be written as 4/1, 65 can be written as 65/1, and 3,867 can be written as 3,867/1.

### Ratio and Rates: A Video

Watch this video to better understand the relationship between two numbers—a ratio—and a particular kind of ratio involving time, which is called a rate.

### Irrational Numbers

All numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction.

An irrational number has endless non-repeating digits to the right of the decimal point. Here are some irrational numbers:

π = 3.141592… = 1.414213…

Although irrational numbers are not often used in daily life, they do exist on the number line. In fact, between 0 and 1 on the number line, there are an infinite number of irrational numbers!

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## Difference between Rational & Irrational Numbers with Examples

Mathematics involves numbers, proofs, theorems, which are applied to study the topic such as quantity, space, structure and change. Just think of your date of birth, doesn’t it involve the use of Mathematics?

We use Mathematics in our everyday life, such as in purchasing goods, the money we have etc. Numbers are involved in our day-to-day life.

In the classification of numbers, rational and irrational numbers play a lot of confusion to many students. So here we provide a detailed explanation to these numbers which is easy to remember. Let us discuss the difference between the rational and irrational number in detail.

### Key Differences Between Rational and Irrational Numbers

 S.No Rational Numbers Irrational Numbers 1 Numbers that can be expressed as a ratio of two number (p/q form) are termed as a rational number. Numbers that cannot be expressed as a ratio of two numbers are termed as an irrational number. 2 Rational Number includes numbers, which are finite or are recurring in nature. These consist of numbers, which are non-terminating and non-repeating in nature. 3 Rational Numbers includes perfect squares such as 4, 9, 16, 25, and so on Irrational Numbers includes surds such as 2, 3, 5, 7 and so on. 4 Both the numerator and denominator are whole numbers, in which the denominator is not equal to zero. Irrational numbers cannot be written in fractional form. 5 Example: 3/2 = 1.5, 1/ 6 =0.1666.. Example: √5, √11

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## Difference Between Rational and Irrational Numbers (With Table)

Rational vs Irrational Numbers

The construct of rational and irrational numbers is in vogue from a very long time ago. The difference between a rational number and an irrational one is that a rational number can be stated in the form of a fraction such as a/b where both “a” and “b” are integers whereas an irrational number cannot be written in such way.

However, the above is not the only difference. A comparison between both the terms on certain parameters can shed light on subtle aspects:

Parameter of ComparisonRational NumberIrrational Number
Definition A number which can be expressed as a ratio of two integers A number which cannot be written as a ratio of two integers, that means “no ratio”
Examples Perfect squares such as 4, 9, 16, 25, and so on Surds such as √5, √11
How Expressed In both fraction and decimal form Only in decimal form
Whether can be written in fraction? Yes No
Relation with integers All integers are rational numbers Example:6 = 6/1, so 6 is a rational number. Irrational numbers cannot be integers
Pattern of Decimals a) Can be recurring or repeating decimal Example: 1.26262626 (Repeated pattern is 26) b) Is finite in nature i.e. a terminating decimal Example:1.25 (decimals are finite) a) Non-repeating in nature Example: 1.4142135623……. (No repeated pattern) b) Is non-terminating i.e. has non-repeating (endless) digits to the right of decimal point Example: 0.4141141114….(decimals are non-terminating)
Relation with numerator and denominator The numerator and denominator are integers and denominator is not zero. Example:1/9 Numerator and denominator cannot be integers. Example:3/0 – Fraction with denominator zero

A rational number (derived from the word “ratio”) is stated as a ratio of two numbers/integers. That is, it can be stated as a fraction. Example: a/b. Basically “a” is divisible by “b” and “b” is not “0”.

Common examples of rational numbers include 1/2, 1, 0.68, -6, 5.67, √4 etc. Think, for example, the number 4 which can be stated as a ratio of two numbers i.e. 4 and 1 or a ratio of 4/1. Similarly, 4/8 can be stated as a fraction and hence constitute a rational number.

A rational number can be simplified. The decimal expansion of a rational number terminates after a finite number of digits.

Also, a rational number portrays a continual finite sequence of digits again and again (i.e. repeated pattern). The set of all rational numbers is possible to count.