Rational and Irrational numbers

Rational numbers: - Rational numbers can be represented in the p/q form, where p and q are integers, and q is not equal to 0. p can be a positive, negative, or even a zero integer.

Example – 2/3, 5/6

Irrational numbers: - In simple terms, the numbers which are not rational are called irrational numbers. Strictly speaking, the numbers represented in decimal but not in the fractional form are called irrational numbers. It is because they have an endless set of repeating or non-repeating terms after the decimal point.

Example – The value of the square root of 2 = 1.41421356537…

The given Venn diagram shows the relation between rational and irrational numbers.

Properties of rational and irrational numbers

1. The sum of two rational numbers is also a rational number.
   
   Example: 1/2 + 1/3 = (3+2)/6 = 5/6
2. The product of two rational numbers is a rational number.
   
   Example: 1/2 x 1/3 = 1/6
3. The sum of two irrational numbers is not always irrational.
   
   Example: √2+√2 = 2√2, which is an irrational number
   
   2+2√5+(-2√5) = 2, which is a rational number
4. The product of two irrational numbers is not always irrational.
   
   Example: √2 x √3 = √6, which an irrational number
   
   √2 x √2 = √4 = 2, which is a rational number

Example

Find four rational numbers between 2/5 and ½.

Solution

To find rational numbers between a given set of numbers, let us make the denominator equal first.

Taking LCM of 5 and 2, we get 10.

Making the fraction equal as – 2/5 x 2/2 = 4/10

And, ½ x 5/5 = 5/10

Since the numbers 4 and 5 are adjacent numbers, we need to make the denominator bigger enough to find four rational numbers.

Therefore, 2/5 x 10/10 = 20/50

And, ½ x 25/25 = 25/50

Now, we can easily find four rational numbers between 20/50 and 25/50. Four rational numbers are – 21/50, 22/50, 23/50 and 24/50.

Some common properties of irrational numbers

1. Irrational numbers are made up of non-terminating and non-recurring decimals.
2. These are real numbers only.
3. For any two irrational numbers, their least common multiple (LCM) may or may not exist.

Some common properties of irrational numbers

1. The square root of 2 – Hippassus accidentally introduced the value of the square root of 2 is irrational. He was not able to write the value in fraction form. Hence, he termed the value as irrational.
2. Π – Pi is a commonly known irrational number whose value goes up to 22 trillion digits. It took 105 days and 24 hard disk drives to derive the value of pi.
3. Euler’s number e – The Swiss mathematician introduced the value e as 2.718281828459045… as an irrational value. e is commonly known as Napier’s number and is mostly used in trigonometry and logarithms.

Difference between rational and irrational numbers