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Examples of irrational numbers are π (the ratio of a circle's circumference to its diameter) and the square roots of most positive integers, ...
Typology: Schemes and Mind Maps
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Author: Mark Kudlowski
Author: Mark Kudlowski
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 (from German Zahlen = numbers).
We can then extend the number set to include all positive and negative fractions of the form n m 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, 43 and 227.
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 227 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 29.
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.
