Mathematics

Positive and negative WHOLE numbers, including zero, are called INTEGERS.

FRACTIONS are parts of whole numbers. They may be formed by dividing one integer by another (for example, dividing the number ‘1′ by 2.

Fractions can be written down one of two ways. The first is to show two numbers divided by a line in the middle like this:

The fraction on the left here shows two thirds. So if you had a cake and split it into 3 (‘thirds’), then gave two pieces to someone, they would have two thirds of the cake.

The fraction on the right shows four sevenths. So again, if you had a cake, split it into 7, and gave four pieces to someone, they’d have four sevenths of the cake.

Fractions can also be written with a decimal point (called ‘DECIMAL NOTATION‘).

For example, two thirds can also be written :

Why is there a little dot over the final 6?

Take a calculator and do the sum 1 divided by 3.

You’ll see that the result (one third) is 0.3333333333…. and so on. You can’t keep writing threes (they are infinite/never ending) so we represent this number by putting a dot over the final 3, to show it is ongoing (it is called a RECURRING DECIMAL).

Of course, two of these thirds is 0.666666666….. (etc!) so instead of trying to write out all the sixes, we write a dot over the final six.

Rational Numbers

Fractions, and their decimal equivalents, are grouped together and named ‘RATIONAL NUMBERS’.  A rational number is one that either:

(a) results in a decimal that terminates

or

(b) results in a decimal that recurs.

Irrational Numbers

IRRATIONAL NUMBERS (somewhat logically) are those for which their decimal equivalent

(a) does not terminate

and

(b) does not recur.

These numbers cannot be created by dividing one integer by another.

Examples of irrational numbers:

Pi is probably the most famous example.

Pi is a mathematical constant equal to the value of the ratio of any circle’s circumference to its diameter. So, to explain, the value of pi is approximately 3.1415 (the numbers just keep going so we’re going to round this to a few places, which is commonly done).

A circle’s circumference is the measure of the outside of the circle. If you snapped a circle, straightened it out and measured it, you’d have the circumference.

The circle’s diameter (as can be seen on the picture above) is the measure from one side of the circle to the other.

The circle’s radius is simply half the diameter (i.e. the distance from the middle of the circle to the edge).

The number ‘pi’ is created by working out how many times the radius can be fit into the circumference. No matter how big your circle is, the circumference is always (approximately) 3.14159 times the length of the radius. But the value of pi is not really 3.14159 – the numbers on the end carry on infinitely and because they are not all the same (like 0.33333333…) they are said to be irrational. Look at the rule for an irrational number above – you can see that pi is an irrational number. It (a) does not terminate (it’s never ending and will never be written down fully) and (b) it does not recur (i.e. it isn’t 0.33333333333… or 0.666666666… – all the numbers on the end are different and not in a pattern).

Pi one of the most important mathematical and physical constants: many formulae from mathematics, science, and engineering involve the use of pi.

Why the funny letter?

The Greek letter used to represent pi was adopted for the number from the Greek word for perimeter “περίμετρος”, first by William Jones in 1707, and popularized by Leonhard Euler in 1737.

Real Numbers

You can represent numbers in a line, with 0 in the centre, and the positive numbers going one way (1, 2, 3 etc) and negative numbers going the other (-1, -2 etc). This is a number line (see below).

All the number we have looked at – integers, rational numbers, irrational numbers – can be placed somewhere on the number line. They are therefore called ‘real numbers’.

There are also (as you might expect) ‘imaginary numbers’!

For example, the square root of minus one is an imaginary number. Why? Well the equation is mind boggling, that’s why. If the square root of +1 is both +1 and -1, then what is the square root of -1? Don’t worry too much about getting your head round this just yet. Suffice to say that mathematicians solved this by creating the imaginary number, represented by another symbol, illustrated nicely by this cartoon (which also shows ‘pi’):

As you might imagine, some numbers involve both real and imaginary numbers. These are known as ‘complex numbers’. Here’s an example:

Number Patterns – The Fibonacci Sequence

Patterns of numbers are fascinating to mathematicians. The Fibonnacci Sequence is an example:

0. 1. 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…

This was first defined by Leonardo de Pisa, an Italian Mathematician, in 1202 (who was nicknamed Fibonacci).

Each term in the sequence after the first two is obtained by adding the previous two together.

So:

0+1 = 1 .. 1 + 1 = 2 .. 1 + 2 = 3 .. 2 + 3 = 5 etc etc

The interesting thing about this pattern is that it follows the spiral patterns seen in plant growth (see the centre of the flower below).

Summary of key terms on this page

Integer: A positive or negative whole number

Fraction: Part of whole number.  Formed by dividing one integer by another.

Decimal Notation: Fractions written with a decimal point e.g. 0.5 (half).

Recurring Decimal: A decimal in which after a certain point a particular digit or sequence of digits repeats itself indefinitely (e.g. 0.555555555555 etc)

Rational Numbers: A number which either: (a) results in a decimal that terminates or (b) results in a decimal that recurs.

Irrational Number: A number which has a resulting decimal that (a) does not terminate and (b) does not recur.

PI: The number of times a radius fits into the circumference. Approximately 3.14159.

Diameter: The distance across the circle.

Circumference: The distance all the way round the edge of the circle.

Radius: The distance from the middle of the circle to the edge.

Real Numbers: Numbers that can be put on the number line.

Imaginery Numbers: Numbers that don’t exist but represent something (like the square root of minus one).

Fibonacci Sequence: A pattern of numbers that spirals.