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We don't all need to be great mathematicians, and we're not all rocket scientists, but an understanding of the basic principles of day-to-day numeracy, arithmetic and maths will open many doors.
You may want to start
(1) Introduction to Numbers, which explains some of the fundamentals needed to understand numeracy, how numbers work.
Progress to further series on basic arithmetic, learning how to use the common arithmetic functions (addition, subtraction, multiplication and division) learning tips and tricks to help solve common problems along the way like working out the best deals on products and services, planning a household budget and working out your share of a restaurant bill.
you need to learn Percentages and Percentage Change as well as Calculating Area and Averages which covers mean, median and mode.
lets start with Introduction to Numbers:
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:
| Tens | Units |
| 1 | 0 |
Similarly the number twenty seven is made up of two tens and seven units and therefore is displayed as:
| Tens | Units |
| 2 | 7 |
We run out of columns again when we want to express one hundred and have to use a third column:
| Hundreds | Tens | Units |
| 1 | 0 | 0 |
So the number three hundred and fifty eight would be displayed in three columns as:
| Hundreds | Tens | Units |
| 3 | 5 | 8 |
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:
| Millions | Hundred Thousands | Ten Thousands | Thousands | Hundreds | Tens | Units |
| 1 | 2 | 5 | 4 | 8 | 2 | 6 |
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.
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
Integers
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.
Fractional Numbers
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.
Note:
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 Rupee 1.23 is 1 rupee and 23 paisa, in the case of money it is correct to say ‘one rupee, 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).
| Hundreds | Tens | Units | Point | tenths | hundredths |
| 3 | 5 | 0 | . | 7 | 5 |
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, Rs 3.50 for example is more commonly used than Rs3.5.
Other Number Systems
Roman Numerals
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
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
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.
| Blackbird | |||| |
| Magpie | ||| |
| Chaffinch | | |
| Sparrow | ||||| ||| |
| Wren | |
| Robin | ||| |
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.
Prime Numbers
A prime number can only be divided by itself and 1 (one) to leave a whole number (integer) answer.
A mathematician may say: A prime number is a number that has only two integer divisors: itself and one.
Prime Number Example
Examples of prime numbers include 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29, but there are an infinite amount of larger prime numbers too.
7 is a prime number since it can only be divided by itself or 1 to leave a whole number.
7 ÷ 7 = 1 and 7 ÷ 1 = 7
If you divide 7 by any other number the answer is not a whole number.
7 ÷ 2 = 3.5 or 7 ÷ 5 = 1.4
9 is not a prime number. 9 can be divided by itself, 1 and 3 to leave a whole number.
9 ÷ 9 = 1 and 9 ÷ 1 = 9 and 9 ÷ 3 = 3
Some quick facts about prime numbers:
- 2 is the only even prime number. All other even numbers, of course, divide by 2.
- The 1000th prime number is 7,919.
- Euclid, the Greek mathematician demonstrated in around 300BC that there are an infinite number of prime numbers.
Prime numbers are important in mathematics and computing, but for most of us, their use is probably limited to interest, and to knowing when you’ve reached the limit of dividing down a fraction.
Squares and Square Roots
The square of a number is the number that you get if you multiply that number by itself. It is written as x2 (where x is any number).
For example:
52 = 5 x 5 = 25.
52 = 5 x 5 = 25.
Square numbers are used in area calculations as well as other areas of mathematics. Suppose you want to paint a wall which is 5 meters high by 5 meters wide. Multiply 5m × 5m to give you 25m2. You would need to buy enough paint for 25m2.
The square root of a number is the number that is squared to obtain that number. The square root symbol is √
Square roots are easier to understand with examples:
√25 = 5, i.e. 5 is the square root of 25 since 5 x 5 =25
√4 = 2, i.e. 2 is the square root of 4 since 2 x 2 =4
√4 = 2, i.e. 2 is the square root of 4 since 2 x 2 =4
Not all numbers have a whole square root. For example, √13 is 3.60555.
Exponents and Powers
Squares are particular types of exponents, also known as powers. For exponents, the superscript number, instead of always being 2 as it is for squares, can be any number, and that tells you how many times to multiply the number itself by.
For example:
23 = 2 x 2 x 2 = 8
510 = 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 9,765,625
23 = 2 x 2 x 2 = 8
510 = 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 = 9,765,625
The other way that exponents are often used is to express very large and very small numbers in terms of the number of times that they are multiplied by 10.
For example:
2 x 106 = 2 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 2,000,000.
5 x 10-5 = 0.00005
2 x 106 = 2 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 2,000,000.
5 x 10-5 = 0.00005
The use of exponents here reduces the number of digits that very large and very small numbers are expressed in.
For example:
1.23 x 1012 = 1,230,000,000,000
4 x 10-15 = 0.000000000000004
1.23 x 1012 = 1,230,000,000,000
4 x 10-15 = 0.000000000000004
Warning!
When the power is positive, it tells you how many zeros to add to the number that is being multiplied by 10.
For 2 x 106, add 6 zeros to 2, and get 2,000,000.
However, when the power is negative, the number of zeros after the decimal point is one less than the superscript number.
1 x 10-3 is 0.001
This is because you have to divide by 10 once to move the number itself to the other side of the decimal point.
Also note: When expressing a numbers in terms of the number of times it is multiplied by 10, you always need the number itself to be between 1 and 9 inclusive.
Factors and Multiples
Factors are numbers that divide or ‘go’ a whole number of times into another.
For example, 2, 3, 5 and 6 are all factors of 30.
Each of them goes into 30 a whole number of times.
Multiples are the numbers that you get when you multiply one number by another.
4, for example, is a multiple of 2.
30 is a multiple of 15, 6, 5, 3 and 2.
Infinite Numbers (Irrational Numbers)
The phrase ‘infinite numbers’ does not refer to the fact that there are an infinite number of numbers. Instead, it refers to numbers that do not themselves ever end.
The best-known infinite number is probably pi, π, which starts 3.142 and goes on from there. Not even the most powerful computer programme in the world could ever map all of its numbers, because it is infinite.
These numbers are also called irrational numbers.
Finite numbers are numbers that have a finite number of digits. After a certain point, the only number that can be added is zero. 1, 3, 1.5, and 0.625 are all examples of finite numbers.
Recurring numbers are one particular form of infinite numbers. Here, the same one or few digits repeat infinitely in the decimal form of the number.
Some numbers which can be expressed easily as fractions turn out to be recurring numbers in the decimal form.
Examples include 1/3, which is 0.33333 recurring in decimals, and 1/11 which is 0.090909090909 recurring.
Real, Unreal and Complex Numbers
Real numbers are numbers that actually exist and can have a physical value placed on them.
Real numbers can be positive or negative, and may be integers (whole numbers) or decimals. They may even be infinite numbers, but they can be written as numbers and expressed in numerals.
Unreal numbers or imaginary numbers do not actually exist, but are a mathematical construct to solve certain problems.
The simplest example is the square root of a minus number. This makes sense because you can only obtain a minus number by multiplying a negative number by a positive number. If you multiply two negative numbers or two positive numbers, you get a positive number. It therefore follows that the square root of a negative number cannot exist.
However, it can in mathematics! The square root of minus one is given the notation i.
Complex numbers follow from real and unreal numbers. They are numbers composed of a real number multiplied by an unreal or imaginary number, usually denoted by some multiple of i.
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