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your maths teacherfor Maths 1
Dr Michael Hughes (Mike)
m.s.hughes@exeter.ac.uk
The required textbook
A2 Pure Mathematics C3/C4
18 April 2023 M L1 MH
18 April 2023 M L1 MH
Objectives : - Scientific Notation - Error estimation - Surds recap - Algebraic expression recap
18 April 2023 M L1 MH
A short-hand way of writing large or small numbers without writing all of the zeros
xExample :The Distance From the Sun to the Earth
93,000,000
Step 1Move decimal leftLeave only one number in front of decimal
18 April 2023 M L1 MH
Step 2• Write number without zeros
18 April 2023 M L1 MH
Example: Partial pressure of CO2 in atmosphere 0.000356 atm. This number has 3 sig. figs, but leading zeros are only place-keepers and can cause some confusion.
So expressed in standard form this is
3.56 x 10-4 atm
This is much less ambiguous, as the 3 s.f. are clearly shown.
18 April 2023 M L1 MH
This is the same as scientific notation except the POWER is replaced by the letter E
Examples
Number Scientific Notation/
Standard Form Engineering Notation
100 1.x102 1.E2
1000 (1 sig fig) 1. x 103 1.E3
1000 (2 dec pl) 1.00x 103 1.00E3
-0.00123 -1.23x 10-3 -1.23E-3
1007 1.007x103 1.007E3
Rational/Irrational Numbers
Rational numbers can be expressed as a fraction with no common factors
Irrational numbers can not be expressed as a fraction in its lowest terms
Surds are irrational numbers like π, √2
They have a non repeating infinite pattern of decimal places.
18 April 2023 M L1 MH
Rules for Surds
Try not to be lazy and therefore express them in their lowest form
Example
18 April 2023 M L1 MH
38
364364192
baba
b
a
b
a
baba
Surd Rules
Rationalise the denominator
If you have the following
Rationalise it by multiplying by 1
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3
1
a
ba
ba
15
10
Example
Exercise 5a page 130
Errors
Suppose a cars petrol tank holds 50 litres of petrol and you think the car does 12km for each litre of petrol.
Is it safe to travel 600 km on a full tank of Petrol?
Solution
In practice the car may travel as little as 10km / ltr
or as much as 12.5 km/ltr
Therefore one might be able to drive anywhere between
500 ≤ distance ≤ 650
18 April 2023 M L1 MH
Example
If we say a piece of wood is 5.0 m long
We are implying that it is 4.95 ≤ length ≤ 5.05
if we say a piece of wood is 5.23 m long
We are implying that it is 5.225 ≤ length ≤ 5.235
18 April 2023 M L1 MH
Relative and absolute error
A lawn is said to be 12m x 22m
(a) Between what bounds does the area lie
The true Area is 272.55m2 and the householder measured the area as 264m2
(b) What is the absolute error
(c) What is the relative (Percentage) error
18 April 2023 M L1 MH
Solution
Max Area is 12.5 x 22.5 = 281.25Min area is 11.5 x 21.5 = 247.25
247.25 ≤ Area ≤ 247.25
Absolute error is |272.55-264| = 8.5m2
Relative error is |272.55-264| = absolute error = 3.1%
272.55 true value
18 April 2023 M L1 MH
Question
Exercise
Find the percentage error when π is given the following approximate values:
(i) 3 (ii) (iii) 3.14 (iv) √10
Take the true value of π to be the number stored on your calculator.
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7
22
Multiplying and Dividing
Remember our index rules here
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10
20
49
102
4 xyz
yzx
x
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