Mathematical thinking is important for everyone. It is as important as any thinking that we do with just words. It is also something that we often do without realising we do it. In his History of the Mathematical Sciences, Ivor GrattanGuinness speculates about the intuitive mathematical thinking that would have been necessary for early humans. He thinks that early humans did not use any thinking with elements of statistics, algebra or calculus, but that their intuitive thinking did include counting and sorting (which lead to arithmetic); spacing and distancing (which lead to geometry); balancing and weighing (which lead to statics); moving and hitting (which lead to dynamics); guessing and judging (which lead to probability theory) and collecting and ordering (which led to partwhole theory). ( GrattanGuinness, I. 1997 p.19) This part of the study guide tries to explain mathematical ideas in plain English and to link these explanations to the other parts of the Study Guide, such as the science time line, the general words we use for thinking, and the advice about research.
A number is a word or symbol used for counting or to say where something comes in a series. Counting is either
Different Kinds of Numbers Natural Numbers: Natural numbers are the numbers we use for counting, and for calculating. There is an endless chain of these numbers, usually starting with one: 1, 2, 3, 4, 5, 6, etc. Sometimes zero (0) is included. Integers: are the natural numbers plus their negatives, including zero. Whilst the natural numbers form an endless chain going upwards from zero, the integers form an endless chain going up and down from zero: <<< 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, >>> Another name for integers is Whole Numbers. They are the numbers that are not broken into bits (fractions). Cardinal Numbers: Cardinal numbers are numbers we use for counting. Any number that shows us how many there are (e.g. two, forty, 627) is a Cardinal Number. Cardinal numbers do not tell us anything about the order of the objects being counted. That is the job of Ordinal Numbers. Cardinal numbers are numbers we use for counting, but not for calculating. They are the natural numbers used only for counting. If you count the buttons on your jacket, you are using cardinal numbers. If you count the buttons on your jacket and then add five, you are no longer using cardinal numbers, because you are calculating.
Ordinal Numbers: Ordinal numbers are used to show the
places in
a sequence: 1st, 2nd, 3rd, 4th, 5th, 6th, etc.
"Pi is the
ratio
of the circumference of a
circle
to its
diameter... it is
an irrational number that cannot be expressed as the ratio of two whole
numbers and has an apparently random decimal string of infinite length"
3.14159265......... for ever
(external link)
Complex Numbers: Complex Numbers are Real Numbers plus numbers called Imaginary Numbers that cannot be placed in a series. The square root of minus one is an imaginary number. It is not a "real" number, but it is real enough to be used in an equation that describes the behaviour of an alternating current of electricity. Some of the different kinds of numbers we use are related to the different kinds of scales we use to classify, quantify, and measure things.
An attribute is a property. Redness, being on and being a
baby, are all
attributes. Something either has or does not have a particular
attribute.
As a scale there are only two positions: yes, the attribute is
present; or
no, the attribute is not present. "Binary" means having two
possibilities.
The convention for binary
coding
is to use
1 where the attribute is present (something is red, for
example) and
0 when it is absent.
Arithmetical operations are meaningless on nominal scales. To add a 44 bus to a 102 bus and get a 146 bus is nonsense. But if data has been classified nominally you can make a Frequency Distribution  you can count how many 102 buses and how many 146 buses there are, for example. Interval scales do not have "absolute zeros" or real starting points. The year 1996 is numbered from the birth of Christ (which is labelled 1), but there are years before that. We can use real numbers to add to, subtract from, multiply and divide by items on the interval scale. The year 1996 divided by 2 is the year 998. This is not "half" of the year 1996, however, because an interval scale does not have an absolute zero.
But we can take two numbers
on the scale and divide by two (average) to find out the date
that is half
way between the two. 998 is half way between the year 10 and
the year 1986
because 10 plus 1986 = 1996, which divided by two equals 998.
Similarly, we
can take any other arithmetic mean: The average of 1991, 1993,
1997 and
1998 is (1991 + 1993 + 1997 + 1998) divided by four, which
equals =
1994.75 (the autumn of 1994).
With a ratio scale we can say that one point on the scale is so much more or less than another. 10 centimetres is twice 5 centimetres and half of 20 centimetres. We cannot do this with simple interval scales. The year 1998 is not twice a previous year or half of a future year. Ratio scales are real numbers. They have order, equal intervals and an absolute zero. It is because real numbers have these characteristics that we can perform the operations of mathematics with them. Real numbers and ratio scales can be added, subtracted, multiplied and divided. They can also be raised to powers or brought down to roots, and this means that we can take logarithms and use calculus with them.
Statistics: see time
series
External links on series: Wikipedia (main)  simple English Wikipedia  Wikipedia (maths)
Scale can have two meanings. Two bowls hung on a piece of string can be used to find out which of two objects is the heaviest. These are scales as in the "scales of justice". A ladder or stairs has a series of steps at equal intervals, these are like the marks on a ruler that enable us to measure the "scale" (size) of an object and which are also like "musical scales". The first idea of scale comes from germanic words for bowls and cups. The second comes from the Latin word for staircase or ladder  and so goes back to the idea of walking, where your steps have an equal length.
See Maths is Fun's most common mathematical symbols See Wikipedia's list of mathematical symbols Includes Σ
Special Numbers Zero and the arabic number system Zero is the number 0 (nothing). It is one of the great discoveries of mathematics made in India. It made possible a system of numbers that we call "arabic", because it came to Western Europe through arab mathematicians. To understand this revolution in mathematics, you will first need to look at some of the problem of the Roman system of numbers that Western Europe used before we adopted arabic numbers. The Romans used letters for numbers. To make two you put two I s together: II To make three you put three I s together: III . To make four they put I in front of V to indicate that the one should be taken away from five: IV . To make six they put I after V to indicate that the one should be added to five: VI .
The Romans used
The same principles of adding and taking away letters are used throughout, so
MM
is 2,000,
MCMXCVII , for example, is 1997.
The
M
is 1,000,
But how would you work the answer out? If I find out, I will tell you. I suspect that you need a special instrument, like an abacus (a counting machine made of beads on wires). [See article by Steve Stephenson about The Roman HandAbacus and timeline on the Roman Empire] You can use an electronic calculator on the world wide web to convert roman to arabic numbers and the other way round: See Nova Romana: Roman Numbers The Arabic system can also be called the Hindu or Indian system. The Arabic system has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 1 is one 10 is ten 100 is one hundredThis system makes arithmetic much easier. For example, when work out how many years before 2000 the year 1997 was in arabic numerals, we write it down like this:
The calculation begins on the right of the numbers and moves to the left (the opposite direction to the one we read in, because the smallest number is on the right). The first 0 in 2000 is the ten position. A 7 is immediately beneath it, so take 7 from 10 = 3. We write that down. When we treat 0 as ten (10  7 =3) instead of as nothing (0  7 = 3) we carry the 1 from the ten along to the next digit on the second line and add it. 9 + 1 = 10, so taking the second line from the first in the second position is 0  0 = 0 The same happens with the third position. For the fourth position we are carrying one forward, so 1 becomes 2 and 2  2 = 0. Most people will be so used to doing this that thinking about what they actually do will be quite difficult. A little reflection, however, should convince you that the arabic system is much easier than the Roman. It is also much more powerful.
Absolute: (Collins Plain English Dictionary): "Absolute is used to talk about amounts when they are considered independently of other amounts. For example, if defence spending this year has risen in absolute terms, the actual amount has gone up, although as a proportion of government expenditure it may have gone down"
Formulas A formula is a rule that you can follow to get a result. "Scratch to stop itching" is a simple formula, but it does not always work. Mathematics and science (especially chemistry) contain many formulas written in symbols. Most of these work if you use them properly. A = r ² is the formula to find out the area of a circle. To use it you have to understand the symbols, remember the rule and know when to apply it.
A is area
If we write a formula as simple steps, it becomes an algorithm This is an algorithm for the area of a circle: Measure the radius Multiply the radius by itself Multiply the result by an approximation to pi Write down the answer as the area of the circle
Mathematics The study of ideas like number, quantity, arrangement, shape and space. The parts of mathematics include
Model See the Wikipedia article See also pattern and isomorph
Mathematical Models Mathematics provides us with models of external reality. One can think of the models as being
A model has a clearer shape than the real thing. It does not have all the confusing detail of the external reality and it allows us to manipulate an idea of the reality in our minds. Example:
In reality the speed of the train varies considerably on the journey. Sometimes it stands still at stations and sometimes it goes much faster than fifty miles an hour. It is only by using a simplified model of complex reality that you are able to make calculations about the speed of the train and when it will arrive at different stations. (Based on Open University DDB) Catchup Mathematics Catchup mathematics is an opportunity for people who may have missed out on the chance to appreciate and enjoy mathematics, the first time around. It allows people to reclaim mathematics, in areas that interest them, at a speed they find congenial, and in an environment where they can voice any difficulties and feelings they may have experienced with mathematics previously. ( Jeff Evans)
Numeracy The word numeracy is used, by different people, to cover a different ranges of skill related to mathematics. Here are some definitions:
Study Link
Andrew Roberts likes to hear from users: © Andrew Roberts 7.1999  3.2001

*****************Mathematics index*****************
3,4,5
binary = using two
cent = hundredth
Chart
deca = ten times
deci = tenth
Decimals
giga (giant)
hecto = 100 times
Histogram
kilo = 1,000 times
Litre
mega (great)
metric = using ten
milli = thousandth
Mode
*****************

Quapropter bono christiano, sive mathematici, sive quilibet impie divinantium, maxime dicentes vera, cavendi sunt, ne consortio daemoniorum irretiant [Augustine] has recently been translated as "Hence, a devout Christian must avoid astrologers and all impious soothsayers, especially when they tell the truth, for fear of leading his soul into error by consorting with demons and entangling himself with the bonds of such association" (see external link) Mathematicus is mathematical, but mathematici can be mathematician or astrologer. 