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 nonzero 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
