Rational and Irrational Number

Topic Overview

A rational number is part of a whole expressed as a fraction, decimal or a percentage.

A number is rational if we can write it as a fraction where the top number of the fraction and bottom number are both whole numbers.

The term rational is derived from the word 'ratio' because the rational numbers are figures which can be written in the ratio form.

Every whole number, including negative numbers and zero, is a rational number. This is because every whole number ‘n’ can be written in the form n/1

For example, 3 = 3/1 and therefore 3 is a rational number.

Numbers such as 3/8 and -4/9 are also rational because their numerators and denominators are both whole numbers.

Recurring decimals such as 0.26262626…, all integers and all finite decimals, such as 0.241, are also rational numbers.

Alternatively, an irrational number is any number that is not rational. It is a number that cannot be written as a ratio of two integers (or cannot be expressed as a fraction).

For example, the square root of 2 is an irrational number because it cannot be written as a ratio of two integers.

The square root of 2 is not a number of arithmetic: no whole number, fraction, or decimal has a square of 2. Irrational numbers are square roots of non-perfect squares. Only the square roots of square numbers are rational.

Similarly Pi (π) is an irrational number because it cannot be expressed as a fraction of two whole numbers and it has no accurate decimal equivalent.

Pi is an unending, never repeating decimal, or an irrational number. The value of Pi is actually 3.14159265358979323… There is no pattern to the decimals and you cannot write down a simple fraction that equals Pi.

Euler's Number (e) is another famous irrational number. Like Pi, Euler's Number has been calculated to many decimal places without any pattern showing. The value of e is 2.7182818284590452353… and keeps going much like the value of Pi.

The golden ratio (whose symbol is the Greek letter "phi") is also an irrational number. It is a special number approximately equal to 1.618 but again its value is never ending: 1.61803398874989484820...

Key Concepts

In the new linear GCSE Maths paper, you will be required to recognise certain properties of rational and irrational numbers. According to the Edexcel Revision Checklist for the linear GCSE Maths paper, you will be required to:

  • Recognise and understand calculations which involve surds
  • Use surds in exact calculations
  • Recognise the difference between rational and irrational numbers

Listed below are a series of summaries and worked examples to help you solidify your knowledge about rational and irrational numbers.

Worked Examples

1 - Recognising Surds
A surd is a square root which cannot be reduced to a whole number.

For example,

\[\sqrt 4 = 2\]

is not a surd, because the answer is a whole number.

Alternatively

\[\sqrt 5 \]

is a surd because the answer is not a whole number.

You could use a calculator to find that

\[\sqrt 5 = 2.236067977...\]

but instead of this we often leave our answers in the square root form, as a surd.

2 - Simplifying Surds
During your exam, you will be asked to simplify expressions which include surds. In order to correctly simplify surds, you must adhere to the following principles:

\[\sqrt {ab} = \sqrt a *\sqrt b \]
\[\sqrt a *\sqrt a = a\]

Example
(a) - Simplify

\[\sqrt {27} \]

Solution
(a) - The surd √27 can be written as:

\[\sqrt {27} = \sqrt 9 *\sqrt 3 \]
\[\sqrt 9 = 3\]

Therefore,

\[\sqrt {27} = 3\sqrt 3 \]

Example
(b) - Simplify

\[\sqrt {12} \sqrt 3 \]

Solution
(b) -

\[\sqrt {12} \sqrt 3 = \sqrt {12} *\sqrt 3 = \sqrt {(12*3)} = \sqrt {36} \]
\[\sqrt {36} = 6\]

Therefore,

\[\sqrt {12} \sqrt 3 = 6\]

Example
(c) - Simplify

\[\frac{{\sqrt {45} }}{{\sqrt 5 }}\]

Solution
(c) -

\[\frac{{\sqrt {45} }}{{\sqrt 5 }} = \sqrt {45/5} = \sqrt 9 = 3\]

Therefore,

\[\frac{{\sqrt {45} }}{{\sqrt 5 }} = 3\]

3 - Adding and Subtracting Surds
In order to add and subtract surds, the numbers which are being square rooted (or cube rooted) must be the same.

Example
(a) - Simplify

\[\sqrt {12} + \sqrt {27} \]

Solution
(b) - The numbers which are being square rooted must be the same, so it is necessary to find a common multiple of 12 and 27

\[\sqrt {12} = \sqrt {(3*4)} = \sqrt 3 *\sqrt 4 = 2*\sqrt 3 \]

Similarly,

\[\sqrt {27} = \sqrt {(9*3)} = \sqrt 9 *\sqrt 3 = 3*\sqrt 3 \]

Therefore,

\[\sqrt {12} + \sqrt {27} = 2\sqrt 3 + 3\sqrt 3 = 5\sqrt 3 \]

By making the numbers which are being square rooted the same, you can easily add and subtract surds.

Example
(a) - Simplify

\[\sqrt {90} - \sqrt {45} \]

Solution
(a) -

\[\sqrt {90} = \sqrt {(16*5)} = \sqrt {16} *\sqrt 5 = 4\sqrt 5 \]
\[\sqrt {45} = \sqrt {(9*5)} = \sqrt 9 *\sqrt 5 = 3\sqrt 5 \]

Therefore,

\[\sqrt {90} - \sqrt {45} = 4\sqrt 5 - 3\sqrt 5 = \sqrt 5 \]

4 - Rationalising Surds
The term 'rationalising an expression' simply means removing any surds from the denominators of fractions. This process of simplifying fractions with surds in the denominator often involves rationalising the expression.

Example
(a) - Simplify

\[\frac{{\sqrt 8 }}{{\sqrt 6 }}\]

Solution
(a) -

\[\frac{{\sqrt 8 *\sqrt 6 }}{{\sqrt 6 *\sqrt 6 }}\]
\[\frac{{(\sqrt {48} )}}{6}\]
\[\frac{{\sqrt {16*3} }}{6}\]
\[\frac{{4\sqrt 3 }}{6}\]
\[\frac{{2\sqrt 3 }}{3}\]

Exam Tips

  1. Memorise the general principles of surds as mentioned in the guide above
  2. Remember that a rational number is part of a whole expressed as a fraction, decimal or a percentage. A number is rational if we can write it as a fraction where the top number of the fraction and bottom number are both whole numbers
  3. Remember that an irrational number is any number which is not rational, such as Pi or e
  4. Write down every stage of your working out in order to score maximum method marks

Topic Summary

When solving problems related to surds, rational and irrational numbers, it is extremely important that you clearly demonstrate your working out and double check your method. Otherwise, a small mistake could cost you vital marks. However, with care and attention, you can be experienced at recognising rational and irrational number in order to solve related mathematical problems.

Related Topics

  • Arithmetic with Fractions
  • Indices
  • Problem Solving with Decimals and Percentages
  • Accuracy of Measurement Problems
  • Arithmetic with Positive and Negative Integers
  • Standard Form
  • Combinations