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TPC MATHS (PART A) - NSWTMTH307A
ELEMENT 1: REAL NUMBERS
1.1 What Are Real Numbers?
This section is just a matter of learning a few definitions about real numbers.
Integers -
Integers are all the whole numbers (both positive and negative) and
zero. Another way of saying this is that integers are numbers that do
not have any decimal places.
....-3, -2, -1, 0, 1, 2, 3,....
Rational Numbers - A rational number is any number that can be made by dividing
one integer by another. The word comes from "ratio". A rational
number can also be expressed as a finite decimal number or a repeating
decimal number.
Examples:
1/2 is a rational number (1 divided by 2, or the ratio of 1 to 2)
0.75 is a rational number (3/4)
1 is a rational number (1/1)
2 is a rational number (2/1)
2.12 is a rational number (212/100)
-6.6 is a rational number (-66/10)
Irrational Numbers - An irrational number is a number that cannot be expressed as
a ratio between two integers. If written in decimal form, an irrational
number would have an infinite number of digits to the right of the
decimal point with no repetition.
Examples: √2, √5,  (pi)
Squares and Square Roots - To square a number, just multiply it by itself.
e.g. 32 (3 squared) = 9, so we say 9 is a square number - it can be
drawn as a square with 3 on each side.
Hint: It is handy to be
able to recognise square
numbers. The first ten
perfect square numbers
are 1, 4, 9, 16, 25, 36,
49, 64, 81, 100
The square root of a number is a value that, when multiplied by itself,
gives the number. The symbol for square root is √
Example: 4 × 4 = 16, so the square root of 16 is 4.
√36 = 6 (because 6 x 6 = 36)
Cubes and Cubed Roots - To cube a number, just use it in a multiplication 3 times.
e.g. 33 (3 cubed) = 3 x 3 x 3 = 27, so we say 27 is a cube - it can be
drawn as a cube with 3 on each side.
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Hint: It is handy to be
ablethat,
to recognise
cube
The square root of a number is a value
when multiplied
by itself, gives the
3
numbers. The first five
perfect cube numbers
are 1, 8, 27, 64, 125
27
3
3
The cube root of a number is a special value that when cubed gives the
original number. The cube root of 27 is 3, because when 3 is cubed
you get 27.
Example: 3 125 = 5 (because 5 x 5 x 5 = 125)
Odd and Even Numbers - An even number is any integer that can be divided exactly
by 2. The last digit of an even number must be 0, 2, 4, 6 or 8. An odd
number is any integer that is not an even number.
Multiples and Factors - The multiples of a number are what you get when you
multiply it by other numbers (such as if you multiply it by 1,2,3,4,5,
etc). Just like the multiplication tables.
Example: The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, etc ...
The multiples of 12 are 12, 24, 36, 48, 60, 72, etc ...
The factors of a number are the numbers that you multiply together to
give you the original number.
Example: The Factors of 12 are 1, 2, 3, 4, 6, 12 because all these
numbers multiply evenly into 12 (1 x 12, 2 x 6, 3 x 4)
Primes and Composites - A Prime number is a whole number greater than 1 which
can not be divided evenly by any number (except 1 or itself).
Example: 7 can only be divided evenly by 1 or 7, so it is a prime
number.
A Composite number is any whole number greater than 1 which is not
a prime number. Or it is a number which can be divided evenly by
numbers other than 1 or itself.
Example: 6 is a composite number because it can be divided evenly by
2 and 3.
Hint: All even numbers
are composite numbers
because the number
two will go evenly into
them!
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Hint: The first ten prime
numbers are 3, 5, 7,
11, 13, 17, 19, 23, 29,
31. It is good to be able
to recognise them.
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1.2 Number Lines
You need to be able to plot real numbers on a number line. (Later on, you will learn
how to plot points on a number plane.)
The purpose of plotting numbers on a number line is that it makes it easy to tell
which numbers are bigger or smaller and it can also be used for carrying out
operations (+, -, x,  ) in a visual way.
The number line can be drawn as follows:
Numbers to the left are smaller than numbers to the right.
A couple of points about the number line:

All rational numbers can be plotted on the number line (decimals & fractions)

Each whole number division can be divided up into smaller sections (e.g.
halves, quarters, etc)
1.3 Multiplying and Dividing by Powers of 10
Here are some rules for multiplying and dividing by powers of 10 i.e. 10, 100, 1000,
etc
1. How do we multiply a whole number by a power of 10?
36 × 10
Add on as many 0's as appear in the power.
Examples.
36 × 10 = 360
36 × 100 = 3600
36 × 1000 = 36,000
Add on one 0.
Add on two 0's.
Add on three 0's.
2. How do we multiply a decimal by a power of 10?
7.32 × 10
Move the decimal point right as many places
as there are 0's in the power. If there are not enough digits, add on 0's.
Examples.
7.32 × 10 = 73.2
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Move the decimal point one place right.
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7.32 × 100 = 732
Move the point two digits right: 732. However, since
all the digits fall to the left of the decimal point, the
answer is a whole number, 732, which we write
without a decimal point.
7.32 × 1000 = 7,320 Move the point three digits right. To do this, we must
add on a 0.
Again, the answer is a whole number.
3. How do we divide a decimal by a power of 10?
63.4 ÷ 10
"63.4 divided by 10."
Move the decimal point left as many places as there are 0's in the power. If
there are not enough digits, add on 0's.
Examples.
63.4 ÷ 10 = 6.34
Move the point one place left.
63.4 ÷ 100 = .634
Move the point two digits left.
63.4 ÷ 1000 = .0634
Move the point three digits left. To
do this, add on a 0.
1.4 Scientific Notation
In maths, we use scientific notation as a simpler way of writing very large or very
small numbers.
Large Numbers = Positive powers
A number such as 300 000 can be written as 3 x 105 because 300 000 = 3 x 100 000
and 100 000 = 105
The number is written in two parts:
1. The digit is written as a number between 1 and 10. i.e. put the decimal point
after the first non-zero digit
2. The power of ten is written with the power showing how many places to move
the decimal point to get it in the right spot.
Example: 56 980 000 = 5.698 x 107
Small Numbers (less than 1) = Negative powers
A number such as 0.0006 can be written as 6 x 10-4 because 0.0006 = 6 x 0.0001 and
0.0001 = 10-4
Again the number is written in two parts, but this time the power is negative.
Example: 0.0000004809 = 4.809 x 10-7
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1.5 Rounding
Decimal Places
0.5 has one decimal place
0.96 has two decimal places
15.345 has three decimal places
What is Rounding?
Rounding means reducing the digits in a number while trying to keep its value similar.
The result is less accurate, but easier to use.
Example: 73 rounded to the nearest ten is 70, because 73 is closer to 70 than to 80.
How to Round Numbers



Decide which is the last digit to keep
Leave it the same if the next digit is less than 5 (this is called rounding down)
But increase it by 1 if the next digit is 5 or more (this is called rounding up)
Rounding Decimals
First you need to know if you are rounding to tenths, or hundredths, etc. Or maybe to
"so many decimal places". That tells you how much of the number will be left when
you finish.
Examples
3.1416 rounded to hundredths is 3.14
1.2635 rounded to tenths is 1.3
1.2635 rounded to 3 decimal places is
1.264
Because ...
... the next digit (1) is less than 5
... the next digit (6) is 5 or more
... the next digit (5) is 5 or more
Rounding Whole Numbers
You may want to round to tens, hundreds, etc, In this case you replace the removed
digits with zero.
Examples
134.9 rounded to tens is 130
12,690 rounded to thousands is 13,000
1.239 rounded to units is 1
Because ...
... the next digit (4) is less than 5
... the next digit (6) is 5 or more
... the next digit (2) is less than 5
Reminder: If the
following digit is a 5,
remember to ROUND
UP!
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Rounding to Significant Digits
To round "so many" significant digits, just count digits from left to right, and then
round off from there.
Note: if there are leading zeros (such as 0.006), don't count them because they are
only there to show how small the number is.
Examples
1.239 rounded to 3 significant digits is
1.24
134.9 rounded to 1 significant digit is 100
0.0165 rounded to 2 significant digits is
0.017
Because ...
... the next digit (9) is 5 or more
... the next digit (3) is less than 5
... the next digit (5) is 5 or more
1.6 Simple Calculations
Operations
The four basic operations are , , , 
There are also terms that are used in word problems to mean each of the four
operations:
Operation
Addition (+)
Subtraction (-)
Multiplication (x)
Division ()
Hints:



Alternate Terms
Sum, Plus, Increase, Total
Minus, Less, Difference, Decrease, Take Away, Deduct
Times, Groups of, Rows of
Share, Split into equal parts
Order does not matter for Addition and Multiplication,
but it does for Subtraction and Division
o e.g. 3 x 5 = 5 x 3, but 5 - 3  3 - 5
Addition and Subtraction are opposites.
Multiplication and Division are opposites.
Order of Operations
In order to ensure that complicated sums are always done the same way, some
conventions had to be agreed on. These conventions, commonly known as Order of
Operations, set out which operations have precedence over others.
We remember this by using the abbreviation BOMDAS
B
O
MD
AS
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Brackets first
Orders (ie Powers and Square Roots, etc.)
Multiplication and Division (left-to-right)
Addition and Subtraction (left-to-right)
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After you have done "B" and "O", just go from left to right doing any "M" or "D" as you
find them.
Then go from left to right doing any "A" or "S" as you find them.
Note:
Divide and Multiply rank equally (and go left to right).
Add and Subtract rank equally (and go left to right)
Examples
Example: How do you work out 3 + 6 × 2 ?
Multiplication before Addition:
First 6 × 2 = 12, then 3 + 12 = 15
Example: How do you work out (3 + 6) × 2 ?
Brackets first:
First (3 + 6) = 9, then 9 × 2 = 18
Example: How do you work out 12  6 × 3  2 ?
Multiplication and Division rank equally, so just go left to right:
First 12  6 = 2, then 2 × 3 = 6, then 6  2 = 3
Order of Operations and Calculators
Modern calculators have inbuilt order of operations and fraction inputting which makes
things much simpler. The following is one area that can trip you up if you're not aware
of it.
When you have a complex problem involving multiply operations and a fraction bar,
you treat what is on the top and on the bottom of the fraction as if it had brackets
around it.
Example:
3  4  2 (3  4  2)

 2.75
10  6
(10  6)
If you put this into your calculator without any brackets, what do you get?
Example:
75
75

5
5  10 15
NOT 75/5+10 = 25
Do this with and without brackets after the division and see if you get
different answers.
Estimating and Checking
In this course, you are able to use calculators whenever you want, but:

most calculations are done faster by hand than on a calculator

if you learn a few short cuts and your times tables, this will make your life
easier

you need to carry out an estimate to check if your calculator answer makes
sense
To estimate what your answer should be:

round off all numbers to one significant figure
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


do the calculation by hand using the rounded numbers
compare your calculator answer to the estimated one
if they are not in the same ballpark, check your calculations, particularly your
order of operations where division is involved
Example: 78 x 32  80(rounded up) x 30 (rounded down)  2400
Calculator Answer = 2496
Check - OK
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