Decimal Numbers Worksheet Page 2
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2
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4o
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D ecim al N um bers - 3
E SO
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: 3.2014014014 · · · = 3.2014 = 3.2 ˙ 01 ˙ 4
Irrational numbers are those which go on forever and don’t have digits which repeat. For example:
√
2 = 1.4142135 . . . , π = 3.14159265 . . .
All rational numbers (fractions) can be expressed as either terminating decimals or recurring
decimals, dividing the numerator by the denominator.
7
8
5
For example:
= 0, 175;
= 0.72727272 . . . ;
= 0.4166666 . . .
40
11
12
Conversely, all terminating and recurring decimals can be expressed as fractions.
However, irrational numbers cannot be expressed as fractions, they are not rational. All rational and
irrational numbers form the set of real numbers , which is represented by the letter R.
How to convert a terminating decimal into a fraction:
• Write the decimal as a fraction with denominator 10, 100, 1000,. . . , according to the number
of decimal places.
45
For example: 0.45 =
100
45
9
• Simplify the fraction to its lowest terms:
0.45 =
=
100
20
How to convert a recurring decimal into a fraction:
Look at these examples:
5.454545. . .
1. Let x = 5.454545 . . .
(A)
2. Multiply by 100 (because there are two recurring figures; if there were three recurring figures,
you would multiply by 1000):
100x = 545.454545 . . .
(B)
3. Subtract B A:
100x = 545.454545 . . .
x =
5.454545 . . .
99x = 540
540
60
4. Divide by 99 and simplify: x =
=
99
11
2.5636363. . .
1. Let x = 2.5636363 . . .
2. Multiply by 10 (because there is one figure between the whole part and the recurring figures;
if there were two figures between the whole part and the recurring figures, you would multiply
by 100):
10x = 25.636363 . . .
(A)
D pto. M atemáticas. IES Jovellanos. 2011
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