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The ABC Study Guide, University education in plain English alphabetically indexed. Click here to go to the main index |
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ABC Arithmetic |
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Arithmetic and practical geometry are the oldest parts of mathematics
The four basic parts of arithmetic are:
Addition
Subtraction
Multiplication
Division
These are called the four basic operations of arithmetic. They are also the four basic operations of algebra.
"When we have to add one given number to another, this is indicated by the
sign +, which is placed before the second number, and is read plus. Thus
5+3 signifies that we must add 3 to the number 5".. "instead of the
expression... is equal to; this sign .. =, .. is read is equal to: thus,
when I write a=b, this means that a is equal to b:"
(Leonhard Euler 1765.. See also 1557:
arithmetical signs and
algebra)
Multiplication is a short way to
add
several of the same numbers. It can be done mentally or on an electronic
calculator.
To multiply mentally, you need to know
multiplication tables
For example:
To add 3 + 3 + 3 + 3 + 3 + 3 + 3 (which equals 21)
Division: Finding out how many time one number contains another.
For example:
Aproximation is simplifying an answer by rounding it off, when
exact accuracy is not required, but quick comprehension is.
For example:
Decimals
A decimal is something based on the number ten. It particularly refers to
showing parts
(fractions) of numbers as
tenths by writing numbers on both sides of a point, called a decimal
point
For example:
Digits
In arithmetic, digits are the individual numbers for counting in tens:
Your digits are your fingers, thumbs and toes. If we take our socks off, we
can count to twenty on them. It is more convenient, however, to just use
the digits on our hands. This gives us ten as a basic unit for counting. By
turning down a digit every time we get to ten, we can count to one hundred.
The word fraction means a broken bit. If you fractured a saucer in two
you would have two fractions of the saucer. If the two pieces were the same
size, the fractions would be halves.
One divided by two equals a half, or
1/2 = 0.5
(decimal fraction)
1 / 2 = ½
The bottom number in a fraction is called the denominator
A fraction is part of a
whole
Factors and
Multiples
A factor, or divisor, of
a
(whole)
number is one which divides exactly into it. For example, 3 is a
factor of 12 because four 3s are 12.
All numbers have 1 as a factor. Numbers apart from one have other factors
as well.
The largest number that is a factor of two numbers is called the highest
common factor or greatest common divisorof the
two numbers.
Example:
The lowest common multiple or least common multiple
is the smallest number that is a multiple of a group of numbers. This is
the same as saying, the smallest number into which every one of a group of
numbers will divide exactly.
For example:
For example:
Their denominators (bottom line) are 4, 2 and 4
4 is the lowest common denominator
The money that someone lends is called the capital.
In simple interest the capital remains the same each year and so
the interest also remains the same.
In compound interest the interest at the end of each year is
added to the capital, and so the interest increases from year to year.
Addition: To calculate the total of two or more numbers.
For example: 2 + 3 = 5.
The total is called the sum of the numbers added.
Subtraction: To find the difference between two numbers by
taking one away from the other. For example:
Subtracting 3 from 7 leaves 4.
Or: 7 - 3 = 4
Multiplication:
You remember a multiplication table that tells you that:
7 threes are 21.
Or, in symbols, 7 x 3 = 21.
15 contains 5 three times.
The 15 is the dividend (The number divided)
The 5 is the divisor (The number dividing)
The result (3) is the quotient
Approximation:
2517 is approximately 2520 or 2500.
5.5 means five and five tenths (five and a half)
5.9 means five and nine tenths
5.09 means five and nine hundredths
5.59 means five, five tenths and nine hundredths
See the
Decimal
System
0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
Fraction
In arithmetical language this is written:
The top number in a fraction is called the numerator
The common factors of two numbers are the numbers that will
divide exactly into both.
12 has factors
1, 2, 3, 4, 6 and 12.
42 has factors 1, 2, 3, 6, 7, 14, 21 and 42.
The common factors of 12 and 42 are 1, 2, 3 and 6
6 is the highest common factor of 12 and 42.
A multiple is a number that can be divided exactly by another. For
example, 12 is a multiple of 3 because 12 can be divided by 3 four times,
leaving no remainder.
30 is the lowest common multiple of 3, 5 and 6.
Lowest Common Denominator or Least Common Denominator
The
lowest common multiple
of the
denominators
of several
fractions
If the fractions are: ¼ ½ and
¾
Lowest Common Denominator is sometimes used as a
metaphor.
When a report on study skills
spoke of students working towards the "lowest common denominator", it
appears to have meant that if students can get away with something in one
module, they will try to get away with it in another.
Interest
Money that you pay in exchange for borrowing money.
Or (the other way round)
money that someone pays you in exchange for borrowing your money.
The interest is usually expressed as a
percentage
of the capital.
The interest added to the capital makes a sum of money called the
amount.
Percentage
|
Per cent means [parts] per hundred.
So five per cent means five in every hundred Five per cent can be written as 5% |
5% | five in every hundred |
If A and B are any two numbers, A is A/B x 100 percent of B
If a number is part of a total and you want to calculate what percentage it is, you divide the number by the total and multiply by 100.
| This can be expressed as the formula: (n/t) x 100 |
number
----- total |
all times 100 |
|---|
For example, if there are 100 students taking an exam and 17 chose a question from the organic chemistry section, it is clear that 17% chose an organic chemistry question.
17/100 = .17
.17 x 100 = 17%
If 113 students take an exam and 17 chose a question from the organic chemistry section, it is not self evident what the percentage choosing an organic chemistry question is. But the formula calculates it:
17/113 = .15
.15 x 100 = 15%
Percentage change or percentage difference
may be a
percentage increase or a percentage decrease
The change (increase or decrease) in a value is expressed as a percentage of the original value.
If the price of something increases from 100 pence to 110 pence, there is a 10% increase.
To calculate the change you need the difference between the original and final figure.
110 - 100 = 10
10/100 = .1
.1 x 100 = 10
Or (when it is not so obvious)
£79.86 is increased to £91.22 the increase is what percent?
91.22 - 79.86 =
11.36
11.36/79.86 =
14.22 [I used a calculator for that]
So the percentage increase is 14.22%
Progression, like progress, is about moving from where you are by
steps.
In mathematics, a progression is a series of numbers that increases
or decreases by steps. The steps are
proportional
to one another. Different kinds of progression have different proportions.
Arithmetic Progression
1, 4, 7, 10, 13, 16, 19, 22, 25.....
Geometric Progression
1, 4, 16, 64, 256, 1,024.....
For example,
Any number divided into itself is one. So the ratio of a quantity to the
same quantity of something else is one. Look at
the graph of the
British balance of trade. This has the ratio
of export to import volumes as its vertical axis. When exports and imports
are equal, the ratio is one.
A rate is often a ratio: the quantity of something expressed
as a proportion. For example, Durkheim defines the
rates of births, marriages, and suicides as
Parsons distinguishes between rates and incidence
A frequent use of rates in
statistics is to construct
time series showing the rate of change over time
Andrew Roberts likes to hear from users:
Progression
An example of an arithmetic progression is:
An example of geometric progression is:
Proportion
The proportion or ratio that one number bears to another is the
result
of dividing one by the other.
The proportion of 3 to 7 is three divided by seven = 0.42
The proportion of 6 to 14 is six divided by fourteen
This is also 0.42
"
the number obtained by dividing the average annual total of marriages,
births, suicides, by the number of persons whose ages lie within the range
in which marriages, births, and suicides occur."
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