Decimals, repeating decimals and rounding

Place value after the decimal point

Decimals extend place value to the right of the decimal point: tenths, hundredths, thousandths and so on. In \(6.384\):

You read it as "six and three hundred eighty-four thousandths". The word "and" marks the decimal point.

Zeros at the end of a decimal don't change its value: \(0.7 = 0.70 = 0.700\). That's useful for comparing.

Decimals to fractions

A terminating decimal converts directly using its place value. Read it aloud and write what you hear, then simplify.

Fractions to decimals

Divide the numerator by the denominator.

• \(\frac{5}{8} = 5 \div 8 = 0.625\). The division ends: this is a terminating decimal.
• \(\frac{4}{11} = 4 \div 11 = 0.363636\dots\) The digits \(36\) repeat forever: this is a repeating decimal.

We mark the repeating block with a bar: \(0.363636\dots = 0.\overline{36}\). The bar covers only the part that repeats, so \(0.8333\dots = 0.8\overline{3}\).

That's because our place values are powers of \(10 = 2 \cdot 5\). A denominator like \(40 = 2^3 \cdot 5\) can be scaled up to \(1{,}000\), but \(12 = 2^2 \cdot 3\) can never become a power of \(10\).

Repeating decimals to fractions

Every repeating decimal is a fraction in disguise. Here's an algebra trick to find it.

The pattern: a one-digit repeating block goes over \(9\), a two-digit block over \(99\), and so on. So \(0.\overline{2} = \frac{2}{9}\) and \(0.\overline{13} = \frac{13}{99}\).

Rounding

To round to a given place, look at the digit just to its right:

• \(4\) or less: leave the rounding digit alone and drop the rest;
• \(5\) or more: increase the rounding digit by \(1\).

\(7.2849\) rounded to the nearest hundredth is \(7.28\). Rounded to the nearest tenth it's \(7.3\). Money is usually rounded to the nearest hundredth, or cent.

Practice