What are Numbers?

We use the word ‘numbers’ to refer to numerical ‘digits’ or ‘numerals’.

Digits are unique symbols or characters (such as ‘0’, ‘1’, ‘3’ or ‘7’), that are used alone or in groups (such as ‘37’ or ‘1073’) to identify a number.

We use the term ‘numerals’ to refer to the digits in a ‘numerical system’.

For example, you might have heard the term ‘Roman numerals’. The Roman system is an ancient system that uses letters, such as I, V and X and is sometimes still used today. We will look at some examples later.

However, the numerals that many of us are familiar with are from the base 10 system, also known as the ‘decimal’ system. These are the numerals 0 (zero) through to 9 (nine). We don’t usually refer to these as ‘numerals’ because it is the system that we use most of the time. We simply call them ‘numbers’ or sometimes ‘digits’.

No matter which numerical system we use, numbers are a useful language for counting, measuring and identifying. We use numbers in an unlimited range of ways: in mathematical calculations, to make phone calls and to identify our bank accounts.

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Numbers in Mathematics

In mathematics, numbers are used to count, measure and calculate.

The introduction mentioned the decimal or base 10 system, which many of us use and recognise.

In the decimal system, we use 10 digits to represent numbers:

0 zero | 1 one | 2 two | 3 three | 4 four | 5 five | 6 six | 7 seven | 8 eight | 9 nine

Numbers that cannot be represented by a single digit are arranged in columns called place values. The place values in the following examples are shown as labelled boxes for each column. Usually we don’t have labelled columns to help us, so we have to imagine them.

When we count from zero to nine, we run out of single digits to describe the numbers from ten onwards. To display the number ten we need two columns. Ten is made up of one ten and zero units:

Similarly the number twenty seven is made up of two tens and seven units and therefore is displayed as:

We run out of columns again when our tens and units columns both reach 9 (ninety nine, 99). So when we want to express one hundred we have to use a third column:

So the number three hundred and fifty eight would be displayed in three columns as:

As we count upwards to larger and larger numbers, we need to add more and more columns. Numbers continue to infinity, so the system of columns continues infinitely as well.

One million, two hundred and fifty four thousand, eight hundred and twenty six for example, would be written as:

This system also works for negative numbers, that is, numbers less than zero. Negative numbers are usually shown with a preceding ‘ − ‘ symbol so minus 1 would be written as −1.

It is not necessary to do this, but it can be kinder to the reader. It is more comfortable to read large numbers in groups of three digits. The commas or spaces are conveniently positioned to separate thousands, millions, billions, trillions etc.

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Whole Numbers and Fractions

Integers

An integer is the term used to describe a ‘whole’ number that can be written without the need for a decimal point or fraction. Integers can be either positive or negative. 1, 7, 375, −56, 12, −8 are all integers.

1.5 or 1½ are not integers because they include a fraction of a whole number.

Fractional Numbers

See our pages Fractions and Decimals for more information.

There are two ways of displaying fractional values in mathematics.  Usually in modern mathematics, a decimal point ‘.’ is used to indicate that the digits after the ‘.’ are a fraction.  The number ‘one and a half’ for example is written as 1.5, and ‘one and three-quarters’ as 1.75.

The ‘.’ symbol is also used when dealing with money, usually to denote the fraction of the main currency unit, in the UK £1.23 is 1 pound and 23 pence. When talking about money, it is correct to say ‘one pound, twenty three’ and not ‘one point two three’.

Fractions are written as division operations*, for example, ½  is 1 divided by 2 (0.5).  ¾ is three divided by 4 (0.75).

When dealing with a decimal place we can use the same columns as we do when dealing with whole numbers (integers); we simply continue the columns to the right, as each number is smaller than the one before.  So 350.75 is:

Negative fractions work in the same way with the inclusion of a minus (‘-‘) symbol.  Minus 1.5 is therefore written as -1.5.

When writing decimal numbers, it is not necessary to include ending 0’s after the decimal place.  For example, 3.50 is the same as 3.5 and 5.00 is the same as 5.  If a 0 occurs before the end of the number then this must be kept, so 5.01 is correct. 

Sometimes, especially with money, we include ending 0’s for clarity, $3.50 for example is more commonly used than $3.5.

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Other Number Systems

Roman Numerals

Roman numerals are still used in some disciplines but most commonly to count or show numbers of years. We often also see them on clock faces.

For example, the BBC uses Roman numerals to show the copyright date of TV programmes. It is common to see at the end of a BBC programme © MMXX, for example (meaning © 2020).  Most word-processors allow users to number pages in Roman numerals, and this is commonly used in books for supplementary pages such as appendices.

Common Roman Numerals used today are:
I = 1
V = 5
X = 10
L = 50
C = 100
D = 500
M = 1,000

Other numbers are written using a combination of the above, II = 2, III = 3, IV = 4, VI = 6, VII = 7, VIII = 8 and IX = 9.  If the smaller symbol comes before the larger then it is subtracted from the larger number (IV = 5 – 1 = 4).  Usually Roman numerals are written in order (largest symbol first) but there is no universal standard.

Tally Systems

Tally systems are still used commonly today for simple counting, and can be helpful when, for example, something has to be counted quickly.  An example could be counting garden birds over a ten-minute period. There are numerous different birds that you may see during this period and it may prove difficult to remember how many of each has been spotted. It is therefore easier to make a list and use a symbol (in this case a vertical line) as a counter.

After the watching has been completed, the totals can quickly be reached by seeing how many symbols have been marked against each category.

To make totalling quicker, it is common to draw a diagonal line through four previous lines to denote 5.

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