The basic mathematical processes are addition, subtraction, multiplication and division. As long as only 'whole numbers' are involved, such sums are simple.
However, very often we must consider quantities which are which are less than one (unity), for instance ¼, ½, ⅛, etc. Here ⅛ means one eighth part of the whole.
⅛ is called a vulgar fraction and has two parts: the '8' (the bottom part) is called the 'denominator' and the '1' (the top part) is called the 'numerator'. The magnitude of a fraction is not changed if we multiply top and bottom by the same number, ie
As the '2' is on the top and bottom we can 'cancel' it thus
A fraction should always be cancelled down to its simplest form
Here top and bottom have been divided by 4.
Fractions can be
(a) Multiplied
(b) Divided
Dividing by 1/2 is the same as multiplying by 2/1, ie
In other words, dividing by a fraction is the same as multiplying by that fraction 'upside down'. Another example is:
Here we divide top and bottom by 4.
(c) Added
If the denominators are different, we must make them the same, ie 'bring them to a common denominator' and normally the lowest common denominator is used. For example
Here we have multiplied top and bottom of 2/3 by 2, making it 4/6. Hence we can add it to 5/6, making 9/6, which is then simplified to 1½ . Another example is
It is generally preferable to divide out fractions greater than one, as we have done above.
(d) Subtracted
Exactly the same rules apply to the subtraction of fractions.
We can also express parts of the whole as 'decimals' or 1/10 parts, written as 0.1, 0.2, 0.3 etc (these are equivalent to 1/10, 2/10, 3/10 etc). The 'full stop' is known as the 'decimal point'. In a decimal, the 'nought' before the decimal point should never be omitted.
The denominator of any fraction can be divided into the numerator to give a decimal, eg
1/8 = 0. 125
3/8 = 0.375
The more common fractions and decimal equivalents should be memorised, eg
| 1/10 = 0.1 | 1/8 = 0. 125 |
| 2/10 = 1/5 = 0.2 | 2/8 = 1/4 = 0.25 |
| 3/10 = 0.3 | 3/8 = 0.375 |
| 4/10 = 2/5 = 0.4 | 4/8 = 1/2 = 0.5 |
etc.
Numbers can be expressed to 'so many significant figures' or 'so many decimal places'. Significant figures is basically the degree of accuracy required.
12345 is a number to five significant figures
12340 is the same number to four significant figures
12300 is the same number to three significant figures
Note also that 5.71 is a number to three significant figures (the decimal point is ignored).
12.345 is a number to three decimal places
12.34 is the same number to two decimal places
12.3 is the same number to one decimal place
Before shortening, decimals are usually 'rounded off'; that means
3.3267 to three decimal places is 3.327 (the 7 is greater than 5, so 6 becomes 7)
3.327 to two decimal places is 3.33 (the 7 is greater than 5, so 2 becomes 3)
3.33 to one decimal place is 3.3 (the 3 is less than 5, so is ignored)
So, what if the number in question is 5? Well in that case you can do what you like really, round it up or leave alone. There is a convention though, and it is good practice to follow such convention where possible.. Usually, if the preceding number is odd, it is rounded up, if even it is left alone. Let's look at a couple of examples,
3.64577 expressed to two decimal places. The third decimal, the one we are going to "chop off" is "5". The second decimal, which will be the last one when we're done, is even (4) so we leave it alone to give 3.64.
3.63577 expressed to two decimal places. The third decimal, the one we are going to "chop off" is "5". The second decimal, which will be the last one when we're done, is odd (3) so we round it up to give 3.64.
