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Shape and Space
Statistics and Probability
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.
Other decimals are those which go on forever and don't have digits which repeat. For example pi = 3.141592653589793238462643...
Relationship with FractionsIn 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.
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.
Convert 0.142857142857... into a fraction.
Let x = 0.142857142857...
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)
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).