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# Numbers and the number system : Key Stage 4 : Mathematics Content Knowledge

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Key Stage 4
Numbers and the number system
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# 1. How confident are you that you can:

## Example

Natural numbers are the counting numbers 1, 2, 3, …

Integers are the set of positive and negative whole numbers and zero, e.g. –2, –1, 0, +1, +2.

A directed number is any positive or negative numbered point on a number line, e.g. –9, +4.6, 7

 1 2
, –345.

A rational number is any number that can be expressed in the form of a fraction, i.e. in the form

 a b
where a and b are whole numbers and b ≠ 0, e.g.

$2\frac{1} {2} = \frac{5} {2}, \ \ \ \ 1.3 = \frac{13} {10}, \ \ \ \ \sqrt{0.81} = 0.9 = \frac{9} {10}$

Rational numbers can always be written as terminating or recurring decimals.

Irrational numbers cannot be expressed as a fraction, i.e. in the form

 a b
where a and b are whole numbers and b ≠ 0. When expressed as decimals, irrational numbers are infinite non-recurring decimals, e.g. √3 and π.

Surds are irrational numbers expressed as roots of positive numbers or combinations of roots and real numbers; e.g.

$\sqrt[5]{13} , \ \ \ \ \sqrt{5} + 5, \ \ \ \ \frac{1} {\sqrt[3]{7} - \sqrt[2]{7}}$

The set of real numbers is the union of the sets of rational and irrational numbers.

## What this might look like in the classroom

Question 1

√10     1      3.5      −3       0
 1 3
π

From the list write down
1. a natural number
2. a negative integer
3. the largest rational number
4. the smallest irrational number
1. 1
2. −3
3. 3.5
4. √10

Question 2
For each of the numbers tick all of the boxes that apply

Natural number
Integer
Directed number
Rational number
Irrational number
Surd
Real number
2.5

-8

0

√5

π

 1 10

0.30

1010

3.142

2

Natural number
Integer
Directed number
Rational number
Irrational number
Surd
Real number
2.5

-8

0

√5

π

 1 10

0.3o

1010

3.142

2

## Taking this mathematics further

Explore the origins of zero..  When was it invented?

Find out about other irrational numbers of note; e.g.

1. Ø – phi, the golden ratio
2. e – sometimes known as Euler's number

Explore definitions of other types of number; e.g.

1.       complex
2.       imaginary
3.       transcendental

## Making connections

In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction

 p q
, where p and q are integers. The best known irrational numbers are π and √2.

There is a natural progression in the build up of the different types of numbers, and it can be linked to different types of equations that need to be solved (bear in mind that scientific and technological advances have in the past required ever more complicated equations to be solved).  Assume that we just have the counting numbers (or natural numbers) as a starting point.  Each subsequent equation requires a new type of number to be 'created'.

1. a + 2 = 17.  No problem here.  a must be 15.
2. a + 11 = 5.  This is a problem: the smallest number we know is 1. Never mind, let's invent negative numbers ... a = -6.  Along with the natural numbers we now have a set of integers
3. a × 3 = 12 so a = 4. But if a × 6 = 3 we are stuck again.  Now we need to invent fractions (or rational numbers, after 'ratio'). Conveniently enough, all the numbers so far can be written as fractions too.
4. x2 = 4 tells us that x = 2 (or x = -2), but x2 = 2 is once again pushing beyond our number system.  The solution cannot be written as a fraction (rational number), so we will call it an irrational number.
5. But what about x2 = -3? How can that be possible? We'll just have to imagine some more numbers – so let's call them imaginary numbers!  The ones that we had before must therefore be the real numbers as we can mark them on a number line.

If the idea of imaginary numbers seems hard to comprehend consider the fact that it wasn't until the time of the Industrial Revolution that mathematicians universally accepted negative numbers.  Before then some people had even called them absurd numbers!

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