Download Mathematics for engineering technicians Unit 4

Mathematics for engineering
technicians
Unit 4
Handout No. 1
I Ford
1
Numbers
• Engineers use a lot of numbers as they
are precise.
• For example, instead of saying a ‘large
storage tank’ we would be more specific
such as ‘a tank with a capacity of 4,500
litres’.
• When engineers draw up a specification
they do so using numbers and drawings in
preference to written descriptions.
2
Units and symbols
• Engineering as in science uses a
large number of units and symbols.
• It is important that you get to know
the common units and get to
recognise their abbreviations and
symbols.
• The units used are part of the
International System (known as SI) of
units.
3
Fundamental SI units
Name
Energy (work)
Force
Frequency
Mass
Length
Power
Pressure
Time
Electric current
Temperature
Amount of
substance
Luminous intensity
Symbol
Unit
Abbreviation
W
F
f
M
L
P
p
T
I
θ
N
Joule
Newton
Herz
kilogram
metre
Watt
Pascal
second
Ampre
Kelvin
mole
J
N
Hz
Kg
M
W
Pa
s
A
K
mol
j
candela
cd
4
Multiples and sub-multiples
• Unfortunately, the numbers that we deal with in engineering can
sometimes be very large or very small.
Example 1: the voltage of a VHF radio could be as little as
0.0000015 V.
Example 2: the resistance present in an amplifier stage could
be as high as 10,000,000 Ώ
• Having to take into account all the zero’s can be a bit of a problem.
We can make life a lot easier by using a standard range of multiples
and sub-multiples.
• These use a prefix letter that adds a multiplier to the quoted value.
5
Some common prefixes & multipliers
Prefix
Tera
Giga
Mega
Kilo
(none)
centi
Milli
Micro
Nano
Pico
Femto
Atto
Symbol
T
G
M
k
(none)
c
m
μ
n
p
f
a
Multiplying Factor
1012
109
106
103
100
10-2
10-3
10-6
10-9
10-12
10-15
10-18
Multiplier
1,000,000,000,000
1,000,000,000
1,000,000
1,000
1
0.01
0.001
0.000,001
0.000,000,001
0.000,000,000,001
0.000,000,000,000,001
0.000,000,000,000,000,001
6
Examples
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
An amplifier requires an input voltage of 50 mV. Express this in V.
An aircraft strut has a length of 1.25m. Express this in mm.
A marine radar operates at a frequency of 9.74 GHz. Express this
in MHz.
A generator produces a voltage of 440 V. Express this in kV.
A manufacturing process uses a coating with a thickness of 0.075
mm. Express this in μm.
A radar transmitter has a frequency of 15.62 MHz. Express this in
kHz.
A current of 570 μA flows in a transistor. Express this current in
mA.
A capacitor has a value of 0.22μF. Express this in nF.
A resistor has a value of 470 kΏ. Express this in MΏ.
A plastic film has a thickness of 0.0254 cm. Express this in mm.
(Answers: 1=0.05V 2=1250mm 3=9740mm 4=0.44kV 5=75m
6=15,620kHz 7=0.57mA 8=220nF 9=0.47M Ώ 10=0.254mm) 7
INTEGERS
Positive
Negative
(+)
(-)
1, 2, 3, 4, …..,
-1, -2, -3, -4, …..,
You do not need to show the + sign as we assume that it is there!
Fig.1 The number line (showing positive & negative integers)
•
•
•
The number of units that a number is from zero (regardless of direction or
sign) is known as the absolute value)
Positive values are to the right whilst negative are to the left
The number zero (0) is neither a positive or negative integer.
8
Numbers between two integer
numbers
• Engineers frequently have to deal with numbers
that lie between two integer numbers
• Therefore, integers are not precise enough for
engineering applications
• We can get over this problem in two ways:
– Use fractions
– Use a decimal point
• For example, the number that sits half way
between 3 and 4 can be expressed as 3½ or 3.5
9
Laws of signs
(Directed numbers)
There are four basic laws for using signs:
First Law
Second Law
To add two numbers with like signs, add
their absolute values & attach their
common sign.
To add two numbers with unlike signs,
ignore the signs and subtract the smaller
number from the larger one. Attach the
sign of the larger number.
Third Law
Fourth Law
To subtract one number from the other,
change the sign of the number to be
subtracted and follow the rules of
addition.
To multiply (or divide) one signed number
by another:
•Multiply (or divide) their absolute values
•If signs are both the same, attach a (+)
sign
•If signs are unlike, attach a (–) sign
10
Examples on directed numbers
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
8+7
-13 – 12
-8 – 9
5+8
7 – 11
8 – 16
-5 – 12
-4 + 8
11 – 5
-8 – 10
-7 -5 -4
-6 + 8 -3
17 – 8 – 5
20 – 19 – 8 + 3
8 – (+5)
-4 – (-7)
8 – (-3)
-6 – (-2)
(15)
(-25)
(-17)
(13)
(-4)
(-8)
(-17)
(4)
(6)
(-18)
(-16)
(-1)
(4)
(-4)
(3)
(3)
(11)
(-4)
19) -5 – (+6)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
36)
(-11)
-3 – (-6) – (-4)
(7)
5x4
(20)
(-5) x 4
(-20)
5 x (-4)
(-20)
(-5) x (-4)
(20)
(-3)²
(9)
(-8)²
(64)
3 x (-4) + 2 x (-3)
(-18)
(-3) x (2) – (-2) x 4 (2)
6/3
(2)
6 / (-3)
(-2)
(-6) / (3)
(-2)
(-6) / (-3)
(2)
(-10) / (-5)
(2)
1 / (-1)
(-1)
(-1) / 1
(-1)
(-3) x (-4) / (-2)
(-6)
11
Formulae
• Formulae play a very important part in
engineering as by using them, it is possible to
give a clear and accurate statement of physical
laws
• E.G. OHMS LAW :
V=IxR
• The value that we are attempting to find is
known as the SUBJECT.
12
BODMAS
• This gives the order (sequence) in which
operations should be done when solving an
equation.
Example :
4 + 3 x 2 = ??
• Which operation do we do first, Add or multiply?
• We get a different answer depending on what
we do first.
• In order to get the right answer, we multiply first:
4 + 6 = 10
13
BODMAS RULE
B : BRACKETS
O : ORDER (Powers and square roots)
D : DIVISION
M : MULTIPLICATION
A : ADDITION
S : SUBTRACTION
14
BODMAS EXAMPLE
30 – (2 x 32 + 5)
BRACKETS 1st then use ODMAS inside the brackets
30 - (2 x 9 + 5)
(ORDER 32)
30 – (18 + 5)
(MULTIPLY 2 x 9)
30 – 23
(ADD 18 + 5)
Answer = 7
15
Examples
1.
2.
3.
4.
5.
6.
7.
8.
42 + (6 x 3 – 32)
24 – (72 + 42 – 82)
12 x (22 x 5 – 15)
6 ( 9 – 3)
4 (32 x 5 – 15)
32 (15 – 4 x 3)
7 – 4 (60 – 2 x 7 + 3 - 29)
(6 x 3 – 32) + (22 x 5 – 15)
25
23
60
36
120
27
60
14
16
Significant figures
• When doing calculation using formulas, the answers are
sometimes very long.
• Complete the following table:
Number
2.3333
3.6666
1.4923
6.8744
7.6296
To four
significant
figures
2.333
To three
significant
figures
2.33
To two
significant
figures
2.3
To one
significant
figures
2
17
Decimal Places
• Sometimes we are more interested in the number of digits (to the
right) after the decimal point.
• Complete the following table:
Number
2.3333
3.6666
1.4923
6.8744
7.6296
To four decimal To three
places
decimal places
2.3333
2.333
To two decimal
places
2.33
To one decimal
places
2.3
18