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Mathematics 1 11 May, 2015M L1 MH

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your maths teacher for Maths 1 Dr Michael Hughes (Mike) m.s.hughes@exeter.ac.uk The required textbook A2 Pure Mathematics C3/C4 11 May, 2015M L1 MH

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Lesson 1- Basics 11 May, 2015M L1 MH Objectives : - Scientific Notation - Error estimation - Surds recap - Algebraic expression recap

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Standard Form 11 May, 2015M L1 MH 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 11 May, 2015M L1 MH Step 2 Write number without zeros

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Step 3 Count how many places you moved decimal Make that your power of ten 11 May, 2015M L1 MH

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Standard Form 11 May, 2015M L1 MH Example: Partial pressure of CO 2 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 11 May, 2015M L1 MH This is the same as scientific notation except the POWER is replaced by the letter E Examples NumberScientific Notation/ Standard Form Engineering Notation 1001.x10 2 1.E2 1000 (1 sig fig)1. x 10 3 1.E3 1000 (2 dec pl)1.00x 10 3 1.00E3 -0.00123-1.23x 10 -3 -1.23E-3 10071.007x10 3 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 π, √2 They have a non repeating infinite pattern of decimal places. 11 May, 2015M L1 MH

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Rules for Surds Try not to be lazy and therefore express them in their lowest form Example 11 May, 2015M L1 MH Surd Rules

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Rationalise the denominator If you have the following Rationalise it by multiplying by 1 11 May, 2015M L1 MH Example 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 11 May, 2015M L1 MH

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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 11 May, 2015M L1 MH

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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 272.55m 2 and the householder measured the area as 264m 2 (b) What is the absolute error (c) What is the relative (Percentage) error 11 May, 2015M L1 MH

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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 11 May, 2015M L1 MH

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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. 11 May, 2015M L1 MH

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Solution 11 May, 2015M L1 MH

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Algebraic expressions Adding 11 May, 2015M L1 MH

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Subtracting 11 May, 2015M L1 MH

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Multiplying and Dividing Remember our index rules here 11 May, 2015M L1 MH

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Summary We have recapped on the following topics - Scientific Notation - Rational Numbers and Surds - Absolute and Relative error - Algebraic Expressions 11 May, 2015M L1 MH

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Scientific Notation A short-hand way of writing large numbers without writing all of the zeros.

Scientific Notation A short-hand way of writing large numbers without writing all of the zeros.

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