Very Basic Mathematics
Topic 1: Types of numbers
http://www.easymaths.com/
Anyone who thinks of maths - thinks of numbers first of all.
In maths, numbers are called different names:
Real Numbers These are rational and irrational numbers (see below for an
explanation of these).
Rational Numbers These will always have terminating or recurring decimals. This
means they can always be turned into a fraction. e.g. 0.25 is a rational number and
it can be also represented as the fraction 1/4.
Irrational Numbers These can not be turned into a fraction. They are nonterminating (i.e. they go on forever after the decimal place with no apparent
pattern). 2 (the square root of two) is an example of an irrational number. These
also include numbers such as Pi These include numbers such as Pi ∏
Integers These are whole positive or negative numbers (-3, -2, -1, 0, 1, 2, 3).
Prime Numbers These are numbers with only TWO factors (factors are numbers
which exactly divide into another number). Prime numbers are only divisible by
themselves and 1. Note that oddly enough, 1 itself is not classed as a prime
number.
Square Numbers These numbers are formed by a number multiplied by itself (1, 4,
9, 16...).
Whole numbers (e.g. -2, -1, 0, 1, 2, 3....) can only be written in one way, however
other numbers can be written in two different ways:
1) As a fraction e.g. the number half way between 2 and 3 is shown as 2½. Note
that sometimes you will see two and a half written as 2 1/2 ('cos it's easier to do
this on the computer).
2) Or as a decimal e.g. 2.5.
Decimal Places
Many answers you work out may give a long string of decimal places which don't
seem to end. Questions in an exam may ask you to give answers to a specific
degree of accuracy by writing answers to a number of significant figures or to a
number of decimal places.
Topic 2: Operations
BODMAS is the secret code which enables us to know exactly the right
sequence of doing things mathematically. In particular electronic
calculators have to use a rule (known in computing circles as an algorithm)
to know which answer to calculate when given a string of numbers to add,
subtract, multiply, divide etc.
What do you think the answer to 2 + 3 x 5 is? (read this aloud)
Is it (2 + 3) x 5 = 5 x 5 = 25 ?
or 2 + (3 x 5) = 2 + 15 = 17 ?
BODMAS can come to the rescue and give us rules to follow so that we
always get the right answer:
(B)rackets
(O)rder
(D)ivision
(M)ultiplication
(A)ddition
(S)ubtraction
According to BODMAS, multiplication should always be done before
addition, therefore 17 is actually the correct answer according to BODMAS
and will also be the answer which your calculator will give if you type in 2 +
3 x 5 <enter>.
I am assuming that you know what everything in BODMAS means apart
from "Order". Order is actually a poor word to use here "Power" would be
much better though BPDMAS doesn't quite have the same ring to it!
Order means anything raised to the power of a number.
You may have heard of Einstein's famous equation E = mc2 here it can be
said that c is raised to the power 2, or c has order 2 or c is squared (they
all mean the same thing!).
(read this aloud too:)
Here's an example to show how to use all the features of BODMAS:
Explain the answer that a calculator would give to the calculation 4 + 70/10
x (1 + 2)2 - 1 according to the BODMAS rules. (four plus seventy divided by ten
multiplied by one plus two in brackets squared minus one) exam!!!
Brackets gives 4 + 70/10 x (3)2 - 1
Order gives 4 + 70/10 x 9 - 1
Division gives 4 + 7 x 9 - 1
Multiplication gives 4 + 63 - 1
Addition gives 67 - 1
Subtraction gives 66
Topic 3.
http://www.bbc.co.uk/education/asguru/maths/
Looking at Indices
The manner in which we count is based on the number of fingers (digits) that we
have.
Numbers are put together so that the position of any particular digit in a whole
number represents its value multiplied by 10, 100, 1000, etc.
These multiples of ten can often be written more conveniently as 10, 102, 103, etc.
(read: ten to the power of two/ten to the second/ ten squared; ten to the power of
three/ten to the third/ten cubed; …ten to the millionth)
Ten is called the base and the small number above and to the right is called the
index (when there are several, the word is indices).
This notation logically extends to include indices of zero and of negative numbers.
100 = 1 (ten to the power of zero)
Using negative indices allows the position of digits after the decimal point to represent
fractions of whole numbers.
10-1 = 1/10 and 10-2 = 1/100 (ten to the power of minus two)
Read this:
873 59/100 is the same as
8 × 102 + 7 × 10 + 3 × 100 + 5 × 10-1 + 9 × 10-2
The idea of bases and indices can be extended to algebra.
a is the base and m is the index.
This whole expression should be read as:
"a to the power of m"
Topic 4. The Co-ordinate System
An equation of the first degree in x and y represents a straight line.
Straight Line Graphs:
Descartes introduced the custom of employing the first letters of the alphabet to
denote known quantities and the last letters to denotes unknown values.
y = ax + b
y = mx + c
Ax + By + c = 0
We now have the means for expressing straight lines via an equation.
A: a line parallel to the x-axis y = 1
B: a line parallel to the y-axis x = 2
C: a line through the origin at 45° to the x-axis x = y
D: a line with both x- and y-intercepts. Y = 2x – 2
Gradient and Intercept
All straight lines will fit the general equation y = mx + c , where m is the gradient
and c is the y-intercept.
The gradient is found by measuring the degree of steepness from the x-axis. To do
this, we draw any convenient right-angled triangle using the line as hypotenuse.
Then m = vertical distance ÷ horizontal distance
The equation of the x-axis is y = 0 (m = 0, c = 0).
The equation of the y-axis is x = 0.
The equations of the diagonals through the origin are y = ± x
Lines with a +ve gradient slope upwards to the right: lines with a -ve gradient
slope upwards to the left.
Topic 5: Data
(note: the use of data as uncountable noun is most common but there is a form datum for one piece
of data but I would never use it)
Scales
Data is information collected according to some principle. We can make some
statements about the 'quality' of our data in terms of our ability to carry out
arithmetical operations.
1. Nominal Scale
This is where data is simply in terms of names or descriptions. For example
colours red, blue, green would be nominal data.
2. Ordinal Scale
This is where data can be recognised as being in some order. For example, a
collection of names might be ordered in ascending alphabetical order. Or a list of
entrants in an exam might be ordered by their marks. The order does not mean
that items can be added because the gap between the items is unspecified.
3. Interval Scale
This is where the gaps between whole numbers on the scale are equal. This
permits the arithmetic operations of addition and subtraction. An example of this
kind of scale is temperature. 20° C is not twice as hot as 10° C because 0° is not
an absolute zero. It is simply the amount of heat beyond which water turns from
solid to liquid. (think about it)
4. Ratio scale
A ratio scale permits full arithmetic operation. Measuring the time something
takes is an example of using a ratio scale. If a train journey takes 2 hr and 35 min,
then this is half as long as a journey which takes 5 hr and 10 min.
We need to be very careful about the scale used, and it is not uncommon for
people to do arithmetic on scales which are simply ordinal - marks for
assignments at school for example!
Discrete vs. continuous
A full ratio scale can be modelled on a number line. A number line is continuous
which means that there are no gaps between the numbers. However, empirically
we can only measure things with a degree of inaccuracy. That means there are
gaps in our measure, and this we refer to as discrete.
Continuous measures:
Time
Height
Weight
Discrete measures:
The number of cars in a car park
The results of an examination
The cost of anything
Topic 6: Fractions
"Fractions" - there I've said it. This word is sure to create a mild sense of
panic in most people who have ever been taught maths but not really
understood what was going on.
OK..so the next thing you say is "What's the point of learning fractions
when we can do everything on our calculators".
If you have a cake to share and each person gets 0.14285714….. (as your
calculator will reliably tell you) of the cake, how many people share the
cake?
The answer isn't very easy to see, but if you say each person gets a
seventh of the cake (1/7 as it's written as a fraction) things become a lot
clearer.
Simplifying fractions can be a pain, until you get used to calculations with
numbers, however 7/8 makes much more sense than 43/48 (but both give
the same answer 0.875).
To simplify fractions all you need to do is find a number that will divide into
the numerator – the number at the top of the division line - and into the
denominator - number at the bottom of the division line leaving a whole
number. (a good knowledge of your times tables will help - remember
those?).
Now I'm not going to rant on about lowest common denominators etc. as
they are just confusing gobbledegook. The easy way to learn how to do
calculations with fractions is to see how the process works with an
explained example of each:
MULTIPLYING FRACTIONS
Multiplying fractions is the easiest thing to do:
all you do is multiply the two numbers above the division line and the two
numbers below the division line. Then you need to simplify the answer:
12/40 divided by 2 is 6/20 divide by 2 again to give 3/10 as your final
(simplified) answer.
DIVIDING FRACTIONS
Dividing fractions is almost as easy as multiplying:
Did you spot the trick? - all you do is turn the second fraction upside down
and multiply the two together! Simplifying further you can say that the final
answer is 2 2/15 (two and two fifteenths).
Topic 7: Decimals
Twenty Percent in shorthand is written as 20%. It represents a proportion
out of 100 - in other words 20% means 20 in every 100.
Percentages are used in the workplace in various ways. When you are
presented with a bill for a car service at a garage, building work or if you
buy a new computer in a shop etc. you will see a line added to the bill after
the actual price which is VAT (this is Value Added Tax) it is an extra
amount you need to add to your bill which goes to the Government.
Consider this Garage bill:
New Starter Motor £40.00
Service £60.00
Labour £50.00
Sub Total £150.00
VAT (17.5%) £26.25
Total £176.25
If you have a calculator handy, to calculate the VAT simply multiply 150 by
.175.
Topic 8 Ratios
The mystical world of horses, bookmakers and horse racing in general
revolves around the central theme of betting - and more specifically the
odds on a particular horse.
Imagine that you decide to stake £4 of your "hard earned cash" on your
favourite horse "Donkey Doo" and you find that the "odds" are 6/4 - what
on earth does that mean?
6/4 is called "six to four against" and means that if you bet £4 and your
horse wins, then you will receive £10 - £6 winnings plus your £4 stake
returned to you.
Now, if you bet £8 how much do you win if “Dinkey Doo” wins? Well, as
you are betting twice as much money, you get twice as much back i.e. £12
plus your original stake of £8 which gives you a grand total of £20.
Of course, the reality is normally that you end up with £-4 or if you were
foolish enough to bet £8, you end up with £-8!
Ratios can present themselves in all sorts of ways, for example:
Consider you are a TV researcher and have conducted a survey from a
group people and found that their preferred TV soap operas out of 3
choices (Eastenders : Coronation Street ) were in the ratio 5 : 6 (five to
six). Out of a group of 120 people, how many people watch Coronation
street? (these are incredibly popular ‘soaps’ both have been going for more years and
years and years)
Topic 8 matrixes
Topic 9 statistics