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Decimals/ Fractions

Decimals to Fractions

It is not hard to convert a decimal number with a finite number of digits to a fractional number. We can write each digit as a fraction, and add up those fractions.

Examples

Convert the given decimal number to a fractional equivalent:

Decimal	Fraction

Click Get New Example to convert a decimal number to a fractional equivalent:

Exercise

Convert the given decimal number to a fractional equivalent:

Repeating Decimals

We now provide a method for converting a decimal with repeated digits to a fractional number.

Note that

Let x be a one digit number. Then

Note the following pattern:

Let x, y, z and w be one digit numbers and let yx stand for a number where the ones place value is x and the 10s place value is y. Then it can be shown that

and the pattern continues.

If the first repeating digit starts after n zeros we simply divide the above by 10ⁿ. For example,

Here is an example demonstrating how to convert any decimal with repeated digits to a fractional number. Click on the question marks to see the example done step-by-step.

More Examples (opens in separate window)

The following examples show use of another method to solve any repeating fraction.

1. Choose 2 suitable powers of 10 as multipliers to write 2 equations so the decimal part is identical in both equations.
2. Subtract the 2 equations.
3. Solve the resulting equation, reducing the fraction if possible.

Example

Convert 0.4 to a fraction.

Convert 0.45 to a fraction.

Convert 0.0123 to a fraction.

Convert 0.83 to a fraction.

Exercise

Convert the given decimal number to a fraction :

Irrational Numbers

Only decimal numbers with a finite number of digits or repeated band of digits can be converted to fractional numbers and so are rational numbers.. Decimals that cannot be converted to fractions are called irrational. These are the decimals with an infinite number of digits where there is no repeated band. Thus 0.101001000100001... is irrational. It can be shown that the following numbers are irrational.

For any prime number p, the number is also irrational.

For some of these examples, a rational number that is close in value is sometimes used. For example many people will have used the fraction 22/7 for p.   22/7 = 3.142857 so is very close to the real value of p.

<< Fractions to Decimals | Decimals Index | Percentages >>
<< Fractions to Decimals | Fractions Index | Equivalent Fractions >>