Review of past work  (Page 6/8)

Fractions and decimal numbers

A fraction is one number divided by another number. There are several ways to write a number divided by another one, such as $a÷b$ , $a/b$ and $\frac{a}{b}$ . The first way of writing a fraction is very hard to work with, so we will useonly the other two. We call the number on the top (left) the numerator and the number on the bottom (right) the denominator . For example, in the fraction $1/5$ or $\frac{1}{5}$ , the numerator is 1 and the denominator is 5.

Definition - fraction

The word fraction means part of a whole .

The reciprocal of a fraction is the fraction turned upside down, in other words the numerator becomes the denominator and the denominator becomesthe numerator. So, the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$ .

A fraction multiplied by its reciprocal is always equal to 1 and can be written

This is because dividing by a number is the same as multiplying by its reciprocal.

Definition - multiplicative inverse

The reciprocal of a number is also known as the multiplicative inverse.

A decimal number is a number which has an integer part and a fractional part. The integer and the fractional parts are separated by a decimal point , which is written as a comma in South African schools. For example the number $3\frac{14}{100}$ can be written much more neatly as $3,14$ .

All real numbers can be written as a decimal number. However, some numbers would take a huge amount of paper (and ink) to write out in full! Some decimal numberswill have a number which will repeat itself, such as $0,33333...$ where there are an infinite number of 3's. We can write this decimal value by using a dotabove the repeating number, so $0,\dot{3}=0,33333...$ . If there are two repeating numbers such as $0,121212...$ then you can place dots or a bar, like $0,\overline{12}$ on each of the repeated numbers $0,\dot{1}\dot{2}=0,121212...$ . These kinds of repeating decimals are called recurring decimals .

[link] lists some common fractions and their decimal forms.

Scientific notation

In science one often needs to work with very large or very small numbers. These can be written more easily in scientific notation, which has the general form

where $a$ is a decimal number between 0 and 10 that is rounded off to a few decimal places. The $m$ is an integer and if it is positive it represents how many zeros should appear to the right of $a$ . If $m$ is negative, then it represents how many times the decimal place in $a$ should be moved to the left. For example $3,2\times {10}^3$ represents 32 000 and $3,2\times {10}^{-3}$ represents $0,0032$ .

If a number must be converted into scientific notation, we need to work out how many times the number must be multiplied or divided by 10 to make it into anumber between 1 and 10 (i.e. we need to work out the value of the exponent $m$ ) and what this number is (the value of $a$ ). We do this by counting the number of decimal places the decimal point must move.

For example, write the speed of light which is $\mathrm{299\; 792\; 458}m·s{}^{-1}$ in scientific notation, to two decimal places. First, determine where the decimalpoint must go for two decimal places (to find $a$ ) and then count how many places there are after the decimal point to determine $m$ .