# Decimals

Decimals use strings of digits to represent numbers, rather than using fractions. Number shown by calculators are usually written as decimals.

$$10$$, $$1.234$$, $$-0.5$$, $$3.1416$$ are written as decimals.

Numbers are divided into two parts using a dot called the decimal point.

The number on the left of the decimal point is the integer part; the number on the right is the fractional part.

The easiest way to find the decimal corresponding to a number is to use a calculator. $$ \frac{1304}{125}= \Tblue{10}\Torange{.}\Tgreen{432}$$ The integer part is $$\Tblue{10}$$; the fractional part is $$\Tgreen{0.432}$$

Fraction | $$\frac{1}{2}$$ | $$\frac{1}{3}$$ | $$\frac{1}{4}$$ | $$\frac{1}{5}$$ | $$\frac{1}{6}$$ | $$\frac{1}{7}$$ |
---|---|---|---|---|---|---|

Decimal | $$0.5$$ | $$0.33\dots$$ | $$0.25$$ | $$0.2$$ | $$0.16\dots$$ | $$0.14\dots$$ |

The decimals with the $$\dots$$ are truncated, because they should have an infinite number of digits. $$ \frac{1}{7} = 0.\Tred{142857}\Tblue{142857}\Tgreen{142857}\dots $$ Some fractions have decimals that **stop** while others have a pattern that **repeats infinitely**.

**Comparing** numbers is simpler in decimal notation than in fractional notation.

- Write the numbers in decimal notation.
- Compare the integer parts.
- If they are equal, compare the decimals from left to right.

**Adding** or **subtracting** two numbers is simpler in decimal notation than in fractional notation.

- Write the numbers so that they have the same number of decimal places (putting 0 on the right if needed).
- Add the numbers in the usual manner, ignoring the decimal point (from right to left).
- Put back the decimal point at the same position from the right.

**Multiplication** and **division** are complex. We normally use a calculator for long multiplication and division methods.

Every number has a unique decimal representation.

- If the prime factorisation of the denominator of a
**rational number**only has $$2$$s and $$5$$s, the decimal has finite length. It is called a terminating decimal. $$ \frac{1}{2} = 0.\Tblue{5},\quad \frac{1}{5} = 0.\Tblue{2},\quad \frac{1}{8} = 0.\Tblue{125}, \quad\frac{231}{125} = 1.\Tblue{848}. $$ - All other
**rational numbers**have recurring decimals. This means that they have an infinite decimal pattern that repeats itself. The recurring part is often written with a dot or a bar on top. \begin{align*} \frac{1}{3} &= 0.\Tgreen{3}\Tblue{3}\Tviolet{3}\dots = 0.\dot{\Tgreen{3}},\quad \frac{1}{7} = 0.\overline{\Tgreen{142857}},\\ &\frac{231}{162} = 1.4\Tgreen{259}\Tblue{259}\Tviolet{259}\ldots = 1.4\overline{\Tgreen{259}} \end{align*} -
**Irrational numbers**have infinite**non-recurring decimals**. $$ \sqrt{2}= 1.\Torange{414\;213\;562\dots}, \quad \pi = 3.\Torange{141\;592\;653\dots}$$

Using a calculator, you can write a fraction as a decimal $$\frac{2}{25} = \Tred{0.08},\quad \frac{121}{12} = \Tred{10.0833\dot{3}} $$ How can you go **from the decimal to the fraction**?

- If the decimal is
**terminating**, multiply the number by ten to the power of the number of decimals. You get an integer that you can divide back and simplify. $$ \Tred{0.08} \times\Tblue{10} = \Torange{8},\quad \Tred{0.08} = \frac{\Torange{8}}{\Tblue{10}} = \frac{2}{25}$$ - If the decimal is
**recurring**, multiply the number by a power of ten to have the recurring part begin at the digit. Multiply by ten to the power of the recurring sequence and subtract. You have an integer. You can then work out the fraction and simplify. \begin{align*} \Tred{R} &= 10.08\Tgreen{33\dot{3}},\quad \Tblue{10} \Tred{R} = 108.\Tgreen{\dot{3}}, \quad \Tblue{10} \Tred{R} = 1083.\Tgreen{\dot{3}},\\ & \Tblue{90} \Tred{R} = 1083.\Tgreen{\dot{3}} - 108.\Tgreen{\dot{3}} = 1083 - 108 = \Torange{9075},\\ &\qquad\qquad \Tred{R} = \frac{\Torange{9075}}{\Tblue{90}} =\frac{121\times 75}{12\times 75} = \frac{121}{12} \end{align*} - If the decimal is infinite and
**not recurring**, the number is not a fraction. It cannot be simplified.