Title: Fractions
Course: UGQuants (SSPS) > Section: Maths Self Test

FRACTIONS

If we think of 1, 2, 3, 4 etc. as being whole numbers then we use fractions to express parts of these whole numbers, i.e. as a way to express numbers between 0 and 1, 1 and 2 etc. There are two basic ways we can do this:

1)     We use the common ways of representing division: either the / sign, or the horizontal line sometimes used in equations and formulae. So e.g. 1/4 (sometimes written as ¼)  means one-quarter of 1. Actually, there is a third (less common) way of expreeing this using the other division sign (1 ÷ 4). The result is the same but conventionally, and for reasons of economy, we do not express one-quarter in this way. Fractions written using the / or horizontal line are known as ‘common’ or ‘vulgar’ fractions. In such fractions we describe the number to the left or above the line as the numerator and the number to the right or below the line as the denominator. We can also combine whole numbers and fractions: e.g. 2½ means two and one-half.

2)     We express fractions as decimals. We often use the decimal system so naturally that we do not think about the principles behind it. It is, as the name implies, a system based on tens. If we want to multiply any number by ten, we simply add a zero to the right of that number. So 2 multiplied by 10 is 20. A different way to think of this number is as two ‘tens’ and zero ‘ones’ (or ‘units’) or as zero in the ones ‘column’ and two in the tens ‘column’. Each time we add another number to the left we add another ‘column’ with a value ten times that of the column to its right. So, e.g., another way of thinking of a four-digit decimal number like 7482, is as 7 in the thousands column, 4 in the hundreds column, 8 in the tens column and 2 in the ones column.

We apply a similar principle to express fractions, except this time every ‘column’ to the right of the ‘units column’ has a value one-tenth of the column immediately to its left. And, to indicate the place where we move from whole units to fractions of a unit we add a decimal point. In some countries (including the UK) this is conventionally written as a full stop (or ‘period’ in North America). In some other countries a comma is used. So, e.g. a number like 24.5 means 2 tens, 4 units and 5 tenths; 1.25 means 1 unit, 2 tenths and 5 hundredths, etc. To express fractions less than 1 we simply use a 0 before the decimal point: e.g. 0.6. Incidentally, in the UK and many other countries the conventional use of a comma is to indicate multiples of one thousand. So, e.g., 7,450,000 means seven million four hundred and fifty thousand.

Rounding is a way of presenting numbers to an appropriate level of precision. E.g. it is often appropriate to round to the nearest whole number: we round 83.4 to 83, because it is nearer to 83 than 84. The only convention to be aware of here is that when a number ends in exactly .5, and thus is equally close to two whole numbers, we round to the higher number: e.g. 83.5 would be rounded to 84. Of course sometimes it is certainly not appropriate to round to the nearest whole number: e.g. doing this with the heights of adults in metres would mean that nearly everyone was described as 2 metres tall ...

In quantitative analysis we usually express fractions as decimals as opposed to common fractions. However a disadvantage of doing this is that some fractions do not convert very well into the decimal system. E.g. 1/3 is represented by a 0 and a decimal point followed by an infinite series of 3s. Conventionally, we often express this as 0.33, even though this does not represent the fraction exactly. We call this rounding to two decimal places.