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Pure Mathematics 01 – Arithmetic

01 Arithmetic Operations

Pure Mathematics 01 – Arithmetic
01 Arithmetic Operations
Mathematics has many different branches
...
An understanding of the rules of arithmetic is essential for tackling
everyday calculations, but also for the study of other branches of mathematics
...
Real numbers can be divided into groups
...
The set of counting numbers is called the set of Natural
numbers
...
The sets of
real numbers are:
Natural numbers:

1, 2, 3, 4, 5 …

Whole numbers:

0, 1, 2, 3, 4, 5 …

Integers:

… -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 …

Rational numbers:

any number that can be expressed as a ratio (fraction) of two
Integers, i
...
in the form

𝒑
𝒒

(𝒒 ≠ 𝟎)
...


All finite decimal numbers (those that have a last decimal place):
𝟑
𝟏𝟎

because we can write 𝟎
...


𝟐𝟓𝟔
𝟏𝟎𝟎

etc
...


Mixed numbers and fractions in which both numerator and
denominator are integers: 𝟐

iv
...
𝟓𝟔 =

, −𝟑 =

−𝟑
,
𝟏

i
...
𝟑̇ = n:
𝟑

Irrational numbers:

any real number that does not belong to the set of rational numbers
...


Real numbers:

All real numbers belong to either the set of rational numbers or the set of
irrational numbers
...
This line may be a horizontal line
or a vertical line
...
From any given point on the number line we can add a positive number by moving the
many places to the right (or up), or subtract a positive number by moving that many places to the
left (or down)
...

To find the difference of two numbers, the second number is subtracted from the first
...


1

Pure Mathematics 01 – Arithmetic

01 Arithmetic Operations

To find the quotient of two numbers, the first number is divided by the second
...
These rules for combining positive
and negative signs are part of the algebraic logic of any scientific calculator but it is useful to be
familiar with them
...
As a simple example, consider evaluating 𝟐 + 𝟑 × 𝟒
...
The BODMAS rule tells the order
in which we must carry out operations of addition, subtraction, multiplication and division
...
If an expression only contains multiplication
and division, we evaluate by working from left to right
...


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Pure Mathematics 01 – Arithmetic

03 Factors and Highest Common Factor

03 Factors and Highest Common Factor

A Factor of a number is an integer that divides into that number exactly, i
...
there is no remainder
...

A Common Factor of two or more numbers is an integer that divides into each number exactly, i
...
is
a factor of each of the numbers
...

The Highest Common Factor of two or more numbers is the largest number that is a facto of each of
the numbers
...
For
example the factors of 7 are 1 and 7, therefore 7 is a prime number
...

An integer is a Prime Factor of a number if it is a factor of the number and also a prime number
...
Of these only 2 and 5 are prime factors of 10
...

The multiples of the number 6 are 6, 12, 18, 24, 30, and so on
...

Any number that is a multiple of 2 or more different numbers is a common multiple of these
numbers
...






The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48…
The multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64…
The common multiple of 6 and 8 are: 24, 48, 72…
The Lowest Common Multiple of 6 and 8 is 24
...

A number that is expressed in scientific notation is composed of two parts that are multiplied
together
...

Example:
𝟐𝟑𝟔 = 𝟐
...


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Pure Mathematics 01 – Arithmetic

06 Fractions

06 Fractions

Fractions of a Whole
A fraction describes a part of a whole
...
It shows how many parts
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It shows how many parts there are altogether
...
The table below shows commonly used names
...
In the right hand diagram

𝟔
𝟏𝟎

is shaded
...
They show the same fraction of a whole
...


Fractions Greater than 1
𝟑

𝟒

You can have fractions that show amounts that are greater than a whole
...
An improper fraction has a numerator
larger than the denominator, also called top heavy fraction
...
In general
the reciprocal of the number 𝒙 is

𝟏
𝒙

When solving equations involving a reciprocal it is useful to remember that if:
𝟏
𝒙

= 𝒚

then

𝟏
𝒚

= 𝒙

Surds
A surd is the square root of a number that is not a perfect square
...
√𝟑𝟔 is not a surd because 36 is a perfect square
...
A very limited range of problems can be solved, or
even described, using linear equations only
...
The most common application of indices are as a shorthand means of indicating
repeated multiplication or division
...
The power indicates the number of times the base
is multiplied by itself
...
E
...
𝟐 𝟏 = 𝟐, 𝟕𝟐 𝟏 = 𝟕𝟐, 𝒂 𝟏 = 𝒂
Any number raised to the power ‘0’ is unity
...
g
...

𝟏
𝟐𝟑

𝟐−𝟑 =
𝟐𝟑 =

𝟏
𝟐−𝟑

Fractional Indices
A power of

𝟏
𝟐

is the same as taking the square root:
√𝟗 = 𝟗 𝟏⁄ 𝟐

In general:
𝒚

√ 𝒙 = 𝒙 𝟏⁄ 𝒚 (𝒙 ≥ 𝟎)

Laws of Indices
In the following, 𝒙, 𝒚, 𝒎 and 𝒏 are real numbers
...
If 𝟐 𝒙 = 𝟐 𝟑 what is the value of 𝒙 ?
The same base is raised to a power on each side
...


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Pure Mathematics 01 – Arithmetic

07 Reciprocals, Surds and Indices

A similar example is, find 𝒙 when 𝟓 𝟑𝒙+𝟏 = 𝟓 𝒙−𝟕
...
e
...
The solution is then:

𝟑𝒙 + 𝟏 = 𝒙 − 𝟕
𝟐𝒙 = −𝟖
𝒙 = −𝟒
A slightly more complicated example is 𝟐 𝒙 = 𝟒 𝟐
...
To solve this equation we must write 4 as a power of 2 and then
simplify:

𝟐𝒙 = 𝟒𝟐
𝟐 𝒙 = (𝟐 𝟐 ) 𝟐 = 𝟐 𝟒
𝒙= 𝟒

Rules for indices
The Product Rule: When multiplying powers of the same quantity add the indices
...


𝒂𝒎
𝒂 ÷ 𝒂 = 𝒏 = 𝒂
𝒂
𝒎

𝒏

𝒎−𝒏

The Power Rule: When raising on power of a quantity to another power, multiply the indices
...


𝒂𝟎 = 𝟏

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Pure Mathematics 01 – Arithmetic

07 Reciprocals, Surds and Indices

A Negative Index: This indicates the reciprocal of a quantity
...


𝒂 𝟏⁄ 𝒏 = √ 𝒂
𝒏

𝒂

𝒎⁄ 𝒏

𝒏

=√𝒂

𝒎 or

𝒂

𝒎⁄ 𝒏

𝒏

= ( √ 𝒂)

𝒎

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