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Mathematics 1

Mathematics 1. your maths teacher for Maths 1. The required textbook A2 Pure Mathematics C3/C4. Dr Michael Hughes (Mike) m.s.hughes@exeter.ac.uk. Lesson 1- Basics. Objectives : - Scientific Notation - Error estimation - Surds recap

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Mathematics 1

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  1. Mathematics 1 M L1 MH

  2. your maths teacherfor Maths 1 The required textbook A2 Pure Mathematics C3/C4 Dr Michael Hughes (Mike) m.s.hughes@exeter.ac.uk M L1 MH

  3. Lesson 1- Basics Objectives : - Scientific Notation - Error estimation - Surds recap - Algebraic expression recap M L1 MH

  4. 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 M L1 MH

  5. Step 1 • Move decimal left • Leave only one number in front of decimal 93,000,000 -> 9.3000000 Step 2 • Write number without zeros 93,000,000 -> 9.3 M L1 MH

  6. 7 93,000,000 = 9.3 x 10 Step 3 • Count how many places you moved decimal • Make that your power of ten M L1 MH

  7. 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. M L1 MH

  8. Engineering Notation This is the same as scientific notation except the POWER is replaced by the letter E Examples M L1 MH

  9. 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. M L1 MH

  10. Rules for Surds • Try not to be lazy and therefore express them in their lowest form Example Surd Rules M L1 MH

  11. Rationalise the denominator Example • If you have the following • Rationalise it by multiplying by 1 Exercise 5a page 130 M L1 MH

  12. 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 M L1 MH

  13. 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 M L1 MH

  14. 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 M L1 MH

  15. 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 M L1 MH

  16. 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. M L1 MH

  17. Solution M L1 MH

  18. Algebraic expressions • Adding M L1 MH

  19. Subtracting M L1 MH

  20. Multiplying and Dividing • Remember our index rules here M L1 MH

  21. Summary We have recapped on the following topics - Scientific Notation - Rational Numbers and Surds - Absolute and Relative error - Algebraic Expressions M L1 MH

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