Mathematics

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• Mathematics
• 1

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your maths teacherfor Maths 1
The required textbook A2 Pure Mathematics C3/C4
Dr Michael Hughes (Mike) m.s.hughes_at_exeter.ac.uk
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Lesson 1- Basics
Objectives -
Scientific Notation - Error estimation
- Surds recap - Algebraic
expression recap
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Standard Form
A short-hand way of writing large or small
numbers without writing all of the zeros

Example The Distance From the Sun to the Earth
93,000,000
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Step 1

• Move decimal left
• Leave only one number in front of decimal

93,000,000 -gt 9.3000000
Step 2

• Write number without zeros

93,000,000 -gt 9.3
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Step 3

• Count how many places you moved decimal
• Make that your power of ten

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Standard Form
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.
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Engineering Notation
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
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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 p, v2
• They have a non repeating infinite pattern of
  decimal places.

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Rules for Surds

• Try not to be lazy and therefore express them in
  their lowest form
• Example

Surd Rules
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Rationalise the denominator
Example

• If you have the following
• Rationalise it by multiplying by 1

Exercise 5a page 130
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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

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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

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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

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Solution

• Max Area is 12.5 x 22.5 281.25
• Min 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

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Question

• Exercise
• Find the percentage error when p is given the
  following approximate values
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• (i) 3 (ii) (iii) 3.14 (iv) v10
• Take the true value of p to be the number stored
  on your calculator.

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Solution
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Algebraic expressions

• Adding

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Subtracting
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Multiplying and Dividing

• Remember our index rules here

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Summary

• We have recapped on the following topics
• - Scientific Notation
• - Rational Numbers and Surds
• - Absolute and Relative error
• - Algebraic Expressions