# Rational Numbers vs. Irrational Numbers: What's the Difference?

Edited by Huma Saeed || By Sumera Saeed || Updated on October 17, 2023
Rational numbers are numbers that can be expressed as a fraction of two integers, while irrational numbers cannot be written as such a fraction.

## Key Differences

In mathematics, numbers are classified into various types to understand their properties and characteristics better. Rational numbers and irrational numbers represent two distinct categories within the realm of real numbers. Rational numbers can always be expressed as a fraction of two integers, meaning they have a numerator and a denominator where the denominator isn't zero.
Sumera Saeed
Oct 17, 2023
On the contrary, irrational numbers are those numbers which cannot be written as a simple fraction or ratio of two integers. Their decimal representation goes on forever without repeating. For instance, while the number 0.5 is rational because it can be represented as 1/2, the number π (pi) is irrational because it cannot be exactly expressed as a fraction.
Sumera Saeed
Oct 17, 2023
Another way to distinguish between rational numbers and irrational numbers is by looking at their decimal expansions. A rational number will always have a decimal expansion that either terminates (like 0.25) or eventually repeats (like 1/3 = 0.333...). In contrast, the decimal expansion of an irrational number neither terminates nor repeats.
Sumera Saeed
Oct 17, 2023
The existence of irrational numbers was a significant realization in ancient mathematics. Pythagoras's followers initially believed that all numbers were rational until they discovered the irrationality of the square root of 2. This discovery challenged their foundational beliefs about numbers and ratios.
Sumera Saeed
Oct 17, 2023
Both rational numbers and irrational numbers are vital for various mathematical and real-world applications. While rational numbers are more intuitive and frequently used in everyday calculations, irrational numbers, like π and e, play crucial roles in advanced mathematics and sciences.
Sumera Saeed
Oct 17, 2023

## Comparison Chart

### Definition

Can be expressed as a fraction
Cannot be expressed as a simple fraction
Sumera Saeed
Oct 17, 2023

### Decimal Expansion

Terminates or eventually repeats
Neither terminates nor repeats
Sumera Saeed
Oct 17, 2023

1/2, -3, 0.75
π, √2, e
Sumera Saeed
Oct 17, 2023

### Presence in Real Number Line

Everywhere on the number line
Fill the gaps between rational numbers
Harlon Moss
Oct 17, 2023

### Historical Discovery

Always known and used
Sumera Saeed
Oct 17, 2023

## Rational Numbers and Irrational Numbers Definitions

#### Rational Numbers

Numbers that can be placed exactly on the number line using fractions.
Every point on the number line corresponds to a rational number or an irrational number.
Sumera Saeed
Oct 17, 2023

#### Irrational Numbers

Mathematical constants and root values often not expressible as fractions.
The golden ratio, often denoted by φ, is another example of an irrational number.
Harlon Moss
Oct 17, 2023

#### Rational Numbers

Numbers with decimal expansions that terminate or eventually repeat.
0.33... is a rational number because its decimal expansion repeats.
Huma Saeed
Oct 17, 2023

#### Irrational Numbers

Numbers that fill the gaps on the number line between rational numbers.
Between any two rational numbers, there are infinitely many irrational numbers.
Sumera Saeed
Oct 17, 2023

#### Rational Numbers

Numbers that can be written as p/q where p and q are integers and q ≠ 0.
7 is a rational number because it can be expressed as 7/1.
Janet White
Oct 17, 2023

#### Irrational Numbers

Numbers with non-repeating and non-terminating decimal expansions.
√2 is an irrational number because its decimal representation goes on forever without repeating.
Sumera Saeed
Oct 17, 2023

#### Rational Numbers

Numbers that can be expressed as a ratio of two integers.
5/2 and -4 are both examples of rational numbers.
Sumera Saeed
Oct 17, 2023

#### Irrational Numbers

Numbers that cannot be written as a simple fraction.
The number π is an irrational number as it can't be expressed as a fraction.
Huma Saeed
Oct 17, 2023

#### Rational Numbers

The set of numbers that includes integers, finite decimals, and repeating decimals.
The number 0.125 is a rational number because it is a finite decimal.
Harlon Moss
Oct 17, 2023

#### Irrational Numbers

Numbers that cannot be expressed as the ratio of two integers.
The number e, the base of natural logarithms, is irrational.
Sumera Saeed
Oct 17, 2023

## FAQs

#### Are all fractions rational numbers?

Yes, all fractions with non-zero denominators are rational numbers.
Sumera Saeed
Oct 17, 2023

#### Can irrational numbers be approximated by rational numbers?

Yes, irrational numbers can be closely approximated by rational numbers.
Huma Saeed
Oct 17, 2023

#### Do rational and irrational numbers together form the set of real numbers?

Yes, the set of real numbers consists of both rational and irrational numbers.
Sumera Saeed
Oct 17, 2023

#### Are percentages rational numbers?

Yes, percentages can be expressed as fractions and hence are rational.
Janet White
Oct 17, 2023

#### Is the number π (pi) rational or irrational?

π (pi) is an irrational number.
Janet White
Oct 17, 2023

#### Can the square root of a prime number be rational?

No, the square root of a prime number is always irrational.
Sara Rehman
Oct 17, 2023

#### Is zero a rational number?

Yes, zero is a rational number because it can be expressed as 0/1.
Sumera Saeed
Oct 17, 2023

#### Can a number be both rational and irrational?

No, a number cannot be both rational and irrational at the same time.
Harlon Moss
Oct 17, 2023

#### Do irrational numbers have exact values?

Yes, though they can't be expressed as fractions, they have exact values, like √2 or π.
Sumera Saeed
Oct 17, 2023

#### Are there any operations that can turn rational numbers into irrational ones?

Yes, for example, taking the square root of a non-perfect square, like 2, results in an irrational number.
Harlon Moss
Oct 17, 2023

#### Which is more common, rational numbers or irrational numbers?

Both are infinitely abundant, but there are more irrational numbers between any two rational numbers.
Sumera Saeed
Oct 17, 2023

#### Are repeating decimals always rational numbers?

Yes, repeating decimals are always rational numbers.
Sara Rehman
Oct 17, 2023

#### Are all rational numbers whole numbers?

No, not all rational numbers are whole numbers; they can also be fractions.
Harlon Moss
Oct 17, 2023

#### Are all integers also rational numbers?

Yes, all integers are rational numbers because they can be written as a ratio of two integers.
Aimie Carlson
Oct 17, 2023

#### Are all decimals rational numbers?

No, only decimals that terminate or repeat are rational. Others are irrational.
Janet White
Oct 17, 2023

#### Is the number 2.101010101... rational or irrational?

It's rational because its decimal expansion eventually repeats.
Sumera Saeed
Oct 17, 2023

#### Is the value of e (base of the natural logarithm) rational?

No, e is an irrational number.
Sumera Saeed
Oct 17, 2023

#### Can irrational numbers be converted into rational form?

No, irrational numbers cannot be exactly expressed as a fraction of two integers.
Sara Rehman
Oct 17, 2023

#### Why are irrational numbers called "irrational"?

They are called "irrational" because they cannot be expressed as a "ratio" of two integers.
Janet White
Oct 17, 2023

#### Can an irrational number be negative?

Yes, irrational numbers can be both positive and negative.
Sumera Saeed
Oct 17, 2023