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Mathematics Revision Guides – Rational and Irrational Numbers
Author: Mark Kudlowski
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Mathematics Revision Guides
Level: GCSE Higher Tier
RATIONAL AND IRRATIONAL NUMBERS
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Mark Kudlowski
1.2
11-01-2009
Mathematics Revision Guides – Rational and Irrational Numbers
Author: Mark Kudlowski
Page 2 of 3
RATIONAL AND IRRATIONAL NUMBERS
Review of number systems.
All number systems are infinite in terms of members.
The system of natural numbers, which are the ‘counting numbers’ learnt from childhood, begin with
1, 2, 3 and so forth. The set of natural numbers is denoted by the symbol
.
The next step up from the natural numbers is the set of all ‘whole’ numbers or integers.
This set contains all the natural numbers, plus zero and all the negative whole numbers starting with -1,
-2, -3 and so forth. The set of integers is denoted by the symbol
numbers).
(from German Zahlen =
m
We can then extend the number set to include all positive and negative fractions of the form n where
m and n are integers and n is not zero.
Because they represent quotients or ratios, they are termed rational numbers.
The set of rational numbers is denoted by the symbol
(for quotient).
This still leaves us with numbers such as and 2 , which cannot be expressed exactly as fractions.
The quantities are real enough - is the ratio of a circle’s circumference to its diameter, and a square
whose sides are one metre long has a diagonal of 2 metres. These numbers are irrational, and so we
extend our number set again to include them all.
The result is the set of real numbers, denoted by
.
A rational number is any number which can be written as a fraction whose top and bottom lines are
both integers. An integer is a special case of a fraction whose bottom line is 1.
When expressed as a decimal, a rational number may either terminate or go on for ever (recur).
Examples of rational numbers are 4, -2.6, 34 and 22
.
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Rational numbers therefore include all integers, fractions, terminating decimals and recurring decimals.
An irrational number also goes on for ever without giving an exact value, but there is no predictable
pattern, and the number cannot be expressed as a fraction.
Examples of irrational numbers are  (the ratio of a circle’s circumference to its diameter) and the
square roots of most positive integers, such as 2 .
Adding a rational number to an irrational number, or multiplying an irrational number by a rational
number will still result in an irrational one.
The fraction
22
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is often used as an approximation for , and its value is 3.142857142857....
Its decimal value does not terminate, but it is rational because it can be expressed as a fraction.
The decimal value of  begins 3.141592653589... but there is no repeating pattern and so  is
irrational.
Example (1): Give two rational numbers and two irrational numbers between 4 and 5.
Two rational numbers are 4.3 and
9
2
.
Two irrational numbers are  + 1 and
20 .
The square of 4 is 16 and the square of 5 is 25, therefore the square roots of all the integers in between
are irrational. In fact, square roots of non-square integers are a good source of irrational numbers.
Mathematics Revision Guides – Rational and Irrational Numbers
Author: Mark Kudlowski
Example (2): Give a rational number and an irrational number between and
Since = 3.141592... and
22
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22
7
.
= 3.142857... , one rational number between the two is 3.142.
Adding a rational number to an irrational one still leaves it irrational, so we can add 0.001 to  and get
3.142592....which is between and 22
.
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3.142 is a rational number between and
22
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, whilst  + 0.001 is an irrational one.
Example (3): Give two distinct irrational numbers whose product is a rational number.
2 is irrational, but it can be multiplied by
8 to give 16 , which is the rational number 4.
Example (4): Give two distinct irrational numbers whose sum is a rational number.
 is irrational, and so is 4-, but adding them gives 4which is 4 – a rational number.