mathsrevision.net --> gcse
20th May 2013

## GCSE Maths

MathsRevision HOME

A-Level Home

GCSE Home

Revision World

Numbers

Decimals

Fractions

Directed Numbers

Number Sequences

Surds

Percentages

Standard Form

Ratios

Proportion

### Shape and Space

Angles

Circle Theorems

Loci

Shapes

Areas and Volumes

Constructions

Vectors

Transformations

### Statistics and Probability

Probability

Averages

Standard Deviation

Sampling

Cum. Freq. Graphs

Representing Data

Histograms

Travel Graphs

Graphs

### Algebra

Factorising

Algebraic Fractions

Solving Equations

Simultaneous Equations

Inequalities

Indices

Functions

### Trigonometry

Sin, Cos, Tan

Pythagoras

Sin and Cosine Formulae

Bearings

Intercept Theorem

Similar Triangles

Congruency

### Other

Coursework

Practice Questions

# Decimal Numbers

Any number can be written in "decimal form".

There are three different types of decimal number: exact, recurring and other decimals.

An exact or terminating decimal is one which does not go on forever, so you can write down all its digits. For example: 0.125

A recurring decimal is a decimal number which does go on forever, but where some of the digits are repeated over and over again. For example: 0.1252525252525252525... is a recurring decimal, where '25' is repeated forever.

Sometimes recurring decimals are written with a bar over the digits which are repeated, or with dots over the first and last digits that are repeated.

For example:

Other decimals are those which go on forever and don't have digits which repeat. For example pi = 3.141592653589793238462643...

### Relationship with Fractions

In decimal form, a rational number (fraction) is either an exact or a recurring decimal.

The reverse is also true: exact and recurring decimals can be written as fractions. For example, 0.175 =175/1000 = 7/40. Also, 0.2222222222... is rational since it is a recurring decimal = 2/9.

You can tell if a fraction will be an exact or a recurring decimal as follows: fractions with denominators that have only prime factors of 2 and 5 will be exact decimals. Others will be recurring decimals. This means that when you write the denominator of a fraction in its prime factor decomposition, if there are only 2's and 5's you will get an exact decimal.
For example, 1/8. The denominator is 8, which is 2 × 2 × 2. There are only 2's in the prime factor decomposition so the decimal will be exact (and it is 0.125).
On the other hand, 2/9 has denominator 9 = 3 × 3 and 3 isn't a 2 or a 5 so we have a recurring decimal (0.222222....).

### Converting a Recurring Decimal to a Fraction

We know that recurring decimals can be written as fractions. The trick is to use a little algebra.

#### Example

Convert 0.142857142857... into a fraction.

Let x = 0.142857142857...
We want to move the decimal point to the right, so that the first "block" of repeated digits appears before the decimal point. Remember that multiplying by 10 moves the decimal point 1 position to the right.

So in this example, we need to move the decimal point 6 places to the right (so we multiply both sides by 1 000 000):

1000000x = 142857.142857142857...

Now we can subtract our original number, x, from both sides to get rid of everything after the decimal point on the right:

1000000x - x = 142857

So 999999x = 142857

x = 142857/999999

= 1/7 (cancelling)

### Rounding Numbers

If the answer to a question was 0.00256023164, you would not usually write this down. Instead, you would 'round off' the answer to save space and time. There are two ways to do this: you can round off to a certain number of decimal places or a certain number of significant figures.

0.00256023164, rounded off to 5 decimal places (d.p.) is 0.00256 . You write down the 5 numbers after the decimal point. To round the number to 5 significant figures, you write down 5 numbers. However, you do not count any zeros at the beginning. So to 5 s.f. (significant figures), the number is 0.0025602 (5 numbers after the first non-zero number appears).

From what I have just said, if you rounded 4.909 to 2 decimal places, the answer would be 4.90 . However, the number is closer to 4.91 than 4.90, because the next number is a 9. Therefore, the rule is: if the number after the place you stop is 5 or above, you add one to the last number you write.
So 3.486 to 3s.f. is   3.49
0.0096 to 3d.p. is 0.010 (This is because you add 1 to the 9, making it 10. When rounding to a number of decimal places, always write any zeros at the end of the number).