Numbers | An Introduction to Numeracy
What are Numbers?
We use the word ‘numbers’ to refer to either mathematical or notational digits or numerals.
When numerals are used for things like telephone numbers and code numbers, they are not intended to be used for mathematics and therefore these numbers are notational. A customer or other code number may combine other characters, such as a surname, to create a unique code referring to a particular customer (also referred to as a ‘key’ in computer databases) for example a customer number may be SMITH8761. UK postcodes also contain a combination of letters and numerals - SW1A 2AA is the postcode for 10 Downing Street.
In mathematics numbers are used to count and measure.
A digit is a single character that we use to represent a number. We usually use 10 digits to represent numbers, namely:
0 zero | 1 one | 2 two | 3 three | 4 four | 5 five | 6 six | 7 seven | 8 eight | 9 nine
This numbering system is called the decimal system, or base 10. Numbers that cannot be represented by a single digit are arranged in columns (although usually these columns are not displayed). These columns are called place values.
To display the number ten we need two columns as there is not a single digit for ten. Ten is made up of one ten and no 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 we want to express one hundred and have to use a third column:
So the number three hundred and fifty eight would be displayed in three columns as:
This system continues infinitely adding a new column when it is no longer possible to write the number using the ten available digits. One million, two hundred and fifty four thousand, eight hundred and twenty six for example, would be written as:
A new column is needed when the number to be displayed is at least ten times the number of the previous column.
1 x 10 = 10 (2 columns) 10 x 10 = 100 (3 columns) 10 x 100 = 1000 (4 columns) etc.
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.
Note: When writing numbers of a thousand or more it is common to use a comma or space to make the number easier to read, commas separate three digits. So the number 1000 can be written as 1,000 and one million as 1,000,000. Commas have no mathematical significance; they are simply used to make longer numbers easier to read.
Whole and Fractional Numbers
An Integer is 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 as they include a fraction of a whole number.
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.
In speech it is common to use words like half and quarter, in mathematics it is more usual to say ‘one point five’ for one and a half and ‘one point seven five’ for one and three quarters.
To say ‘one point seventy five’ is incorrect, except in the case of currency.
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, in the case of money it is correct to say ‘one pound, twenty three’ and not ‘one point two, three’.
Fractions are written as division sums, ½ for example 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 (integer) numbers, we simply continue the columns to the right, as each number is smaller than the one before. So 350.75 is:
Largest (most significant numbers) ... Smallest (least significant numbers).
Negative fractions work in the same way with the inclusion of a ‘-‘ symbol. So minus 1.5 would be 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, 5.00 is the same as 5. If a 0 occurs before the end of the number then this must be kept – 5.01 for example is correct. Sometimes, especially with money, we include ending 0’s for clarity, £3.50 for example is more commonly used than £3.5.
Other Number Systems
Roman numerals are still used in some disciplines but most commonly to count or show numbers of years.
The BBC for example uses Roman numerals to show the copyright date of TV programmes, it is common to see at the end of a BBC programme © MMXII, for example (meaning © 2012). Most word-processors allow users to number pages in Roman numerals, this is common in books for supplementary pages such as those in an appendix.
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.
Still used commonly today for simple counting tally systems 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 a numerous different birds that you may see during this period and it may prove difficult to remember how many of each has been spotted, therefore a list can be created and then a symbol used as a counter.
After the watching has been completed the totals can quickly be reached by seeing how many symbols fall against each category.
To make totalling quicker it is common to draw a diagonal line through four previous lines to denote 5.