engineering mathematic
CHAPTER 1: NUMBERS AND ARITHMETIC
1.1 TYPE OF NUMBERS
NEGATIVE INTEGER

POSITIVE
AND
REAL NUMBERS (R)
•
•
Numbers that can be expressed as
decimals
Real Number System:
•
Consist of positive and
negative natural numbers
including 0
Example:
…, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, …
•
All numbers including natural
numbers, whole numbers,
integers, rational numbers
and irrational numbers are
real numbers
Example:
4 = 4.0000...
−
5
= −0.8333...
6
1
= 0.5000...
2
• Classification of Real Numbers
Numbers
Example
Natural Numbers (N)
1, 2, 3, 4, 5, …
– counting numbers
Whole Numbers (W)
0, 1, 2, 3, 4, 5, …
– a set of zero together with
the natural numbers
Rational Numbers (Q)
– any number that can be
written in the form of
a
b
8 0 5
, , ,7
4 9 3
where a and b are integers
with b ≠ 0
a) Terminates: end in an
infinite string ‘0’
3
= −0.75
4
65
= 65
1
−
b) Repeats: end with a block
of digits that repeat over
and over
Irrational Numbers (I)
 the decimal represented of
irrational numbers do not
repeat in cycles (pattern)
10
= 3.3333...
3
5
= 0.8333...
6
0.1010010000100001...
3 = 1.7320508075...
log10 5 = 0.698970004336...
3 = 1.37050...
•
Real Numbers can be
represented geometrically as
points on a number line called
Real Line
Example
Prime Numbers
 any natural number,
greater than 1, only divisible
by itself and 1
Integers (Z)
– any positive and negative
natural number including ‘0’
Zero
–
the
number
represents
‘none’
‘nothing’
Even Numbers
– any number
divisible by 2
SES
2,3, 5, 7,11,13,...
..., −3, −2, −1, 0,1, 2, 3,...
zero 0
or
that
is
2, 4, 6,8,10,...
2 Engineering Mathematics 1 (AQB10102)
Odd Numbers
– any number that is not
divisible by 2
Composite Numbers
– natural numbers but not a
prime number
LOWEST COMMON MULTIPLY (LCM)
1, 3, 5, 7, 9,...
The LCM of a set of integers is the smallest
integer that they will all divide into.
1, 4, 6,8, 9,10,12,...
To find LCM:
•
1.2 FACTORS,
MULTIPLES,
COMMON FACTOR AND
COMMON MULTIPLY
HIGHEST
LOWEST
Some natural numbers can be produced by
multiplying smaller natural numbers together.
These smaller numbers are called factors.
•
•
•
Do a prime factorization of each
integer
List them in column
For each column, write down the
value of the factor
Multiply the values together to get
the LCM
Example:
Example:
Find the LCM of 10, 15, 45 and 75
12 = 3 x 4 = 6 x 2 = 12 x 1
LCM
The number 12 above has been factorized in 3
different ways. The numbers 12 is known as a
multiple of 3 or 4 or 2 or 6, etc
10
15
45
75
2
HIGHEST COMMON FACTOR (HCF)
2
3
3
3
3
3
3
5
5
5
5
5
5
5
450
The HCF of a set of integers is the largest
integer which is a factor of each of them.
1.3 ARITHMETIC OPERATION (BEDMAS)
To find HCF:
FUNDAMENTAL PROCESSES
•
•
•
Do a prime factorization of each
integer
List them in column
Find the full columns (these are the
common factors)
Multiply all the common factors to
get the HCF
The 4 fundamental processes of arithmetic
are:
Addition

Subtraction
x
Multiplication
÷
•
+
Division
Example:
Find the HCF of 24, 28 and 40
ORDER OF OPERATIONS
HCF
24
28
40
SES
2
2
2
2
2
2
2
2
2
3
7
2
5
4
When a calculation involves more than one
operation, the order in which the numbers
are combined is important. The sequence in
which operations are preformed is called
BEDMAS
3 Engineering Mathematics 1 (AQB10102)
B
 the contents of any brackets
must be evaluated before
performing
any
other
operation
 the power of the number
Bracket
E Exponent
D Division or
M Multiplication  whichever appears first
from left to right
A Addition or
S Subtraction
 whichever appears first
from...
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