Rational and Irrational Numbers

• Rational Numbers can be written as a ratio that compares two numbers or quantities, giving a simple fraction or mixed fraction p/q. These include integers or decimals terminating (finite) or recurring (repeating patterns). The denominator ‘q’ is a natural number, i.e., non-zero.
• Irrational Numbers cannot be written as a ratio. In simple words, irrational numbers include numbers that cannot be simplified further (to fractions with natural numbers and integers.) The decimals of irrational numbers, if expanded, give neither finite nor recurring decimals. These include surds and unique numbers like π (pi). Surds are non-perfect squares or cubes that cannot be simplified and remove the square or cube roots. ‘pi’ is the most common irrational number.

Below is a Venn diagram showing the rational and irrational numbers in the number system.

We cannot list rational and irrational numbers since both have an infinite range. Some examples are given below.

Examples

Rational Numbers

Common examples of rational numbers are:

1. 6; it can be written as 6/1 where 6 and 1 are integers
2. 0.125; it can be written as 1/8 or 125/1000
3. √81; it can be simplified further to 9 or 9/1
4. 5.232323…, or 0.111; these are recurring decimals as they are repeated in patterns

Irrational Numbers

Common examples of irrational numbers are:

1. 1/0; denominator is zero
2. π; its value is 3.142, non-terminating and non-recurring
3. √99; its value is 9.94987.. and it cannot be simplified further

Rational Numbers vs Irrational Numbers

While discussing rational and irrational numbers, we need to compare them to find how terms mathematically differ from each other.

Below are the differences between rational and irrational numbers in a table.

Difference Between Rational and Irrational Numbers

Now, let us learn about identifying rational and irrational numbers through some examples.

Solved Examples

identify the rational and irrational numbers from the given set – 1.36591237, 5/8, 0.36, 0.19755683…, 0.7711, and 1/36

Solution:

As we know,
Rational numbers are finite or recurring and irrational numbers are infinite or non-recurring,
1.36591237…  ⇒   irrational
– 5/8  = 0.625 ⇒   rational
0.04 = 1/25 ⇒   rational
0.19755683… ⇒  irrational
– 0.7711 ⇒   rational
1/36 = 0.0277777777777778; ⇒   rational

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