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Any number can be written in "decimal form".

An exact decimal is a decimal number that does not go on forever.

A recurring decimal is a decimal number which does go on forever, but some of the digits are repeated over and over again. In decimal form, a rational number (fraction) is either an exact or a recurring decimal. For example 0.175 is rational since it is an exact decimal (=175/1000 = 7/40). Also, 0.345345345... is rational since it is a recurring decimal. However, 3.141592653589793238... is not rational because the number goes on forever and there is no pattern in the digits.

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: ### 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).

Copyright © Matthew Pinkney 2003

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