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Lesson 1- Basics Objectives : To know how to approx numbers to a required accuracy by 2-3 1.Basic Number Types 2.Decimal Places 3.Significant Figures 4.Writing numbers in Standard Form 5.Writing numbers in Engineering notation Oct 2011Foundation - L1

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Number Types There are two types of numbers (Scientist) - Exact -> Amount of money in your pocket - Approximate -> Measurements like weight height Mathematicians have more definitions of numbers........... Oct 2011Foundation - L1

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Number Types Counting Numbers Positive Whole Numbers1, 2, 3, 4, 5…… Natural Numbers N Counting numbers and zero 0, 1, 2, 3, 4…… Integers Z All positive and negative whole numbers …-2, -1, 0, 1, 2… Rational Numbers Q Numbers which can be written as a fraction where m and n are integers -1, 0, ½, 2¾, Real Numbers R All rational and irrational numbers -1, 2¾, π, Oct 2011Foundation - L1

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Rational Numbers Most real numbers can be written as a fraction in its lowest form Example: Express 0.123123123123......... 123 as a fraction Trick x 1000 to get rid of decimals Subtract to get rid of decimals Oct 2011Foundation - L1

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Irrational Numbers ─ But some numbers can not be expressed as fractions ─ Examples include These are numbers where the patterns in the decimals do not repeat We can not express numbers like this in faction form. The irrational number set is much smaller than the set of rational numbers Oct 2011Foundation - L1

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Proof that is irrational Method- We will assume that it is rational and then we will contradict this assumption m and n are integers and the fraction can not be simplified further (i.e lowest form) So m 2 is an even number m 2 – even this implies that m is even m m 2 1 2 2 4 3 9 4 16 5 25 so “m” can be written as “2 × a” (as m even) so (so n is even too!!) So is Both numerator and denominator are divisible by 2 and therefore is not in lowest form and can be simplified Contradiction!! Oct 2011Foundation - L1

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Starter − You need to buy some carpet for your bedroom − You measure the width and length of your room as 7.22m x 6.58m − You do not have a calculator or a pen and you have to estimate the area quickly in your head! − How do you estimate the area? − What values did you use for the length and width? − The carpet cost £5.80 per square meter, consider how much money you should take to the shop? Oct 2011Foundation - L1

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Area is 7.22m x 6.58m The area must be smaller then 8m x 7m => 56m The area must be larger then 7m x 6m => 42m 42 m < Area < 56m But a better guess might be 7m x 7m => 49m These workings are all to 1 significant figure (sf) Obviously taking more (sf) will result in a more accurate answer How much money should you take? It is easy to how much exactly if you are good with mental aritmetic or have a calculator, but in principal if you take more than you need you cant go wrong!! If bad with numbers take 60 x 6 = £360

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Significant Figures Consider the Real number 37.500 All the digits to the left of the decimal point are important Only the 5 to the right of the decimal point is important as 37.5 is the same as 37.500 Consider 37.5001, then all of the digits are important SIGNIFICANT FIGUREs (SF) means IMPORTANT DIGITS Oct 2011Foundation - L1

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Sig Figs 37.5 3-> This is the 1 st Sig Fig 7-> This is the 2 nd Sig Fig 5 -> This is the 3 rd Sig Fig The significance of numbers decreases from left to right Oct 2011Foundation - L1

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Rounding to Sig Figs Example Approximate 37.5 to 1 significant figures Look at the next most significant number (2 nd number) 304037 Round up if ≥ 35 Round down < 35 37.5 is 40 to 1 significant figures we write 40 (1 sf) Oct 2011Foundation - L1

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Rounding to Sig Figs Example Approximate 37.5 to 2 significant figures Look at the next most significant number (this is now 3 rd No.) 373837.5 Round up if ≥ 37.5 Round down < 37.5 37.5 is 38 to 2 significant figures we write 38 (2 sf) Oct 2011Foundation - L1

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Rounding to Sig Figs Example : Approximate 37.5 to 3 significant figures This is just 37.5 (because there are only 3 digits) Significant figures (sf) are counted from the left of a number. Always begin counting from the first number that is not zero. 9 4 6 0 3. 5 8 1 st 2 nd 3 rd 4 th 5 th 6 th 7 th significant figure 0. 0 0 0 0 0 1 4 9 0 2 0 7 Notice that a zero can be significant if it is in the middle of a number. Oct 2011Foundation - L1

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Example Round up if ≥ Round down < Round up if ≥ Round down < Write the following to 3 sf a)12.455 b)0.013026 c)0.1005 d)13445.698 e)0.1999 Oct 2011Foundation - L1

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Find the following 801296 to 1 sf 801296 to 3 sf -52.9000 to 3 sf -52.9001 to 4 sf Oct 2011Foundation - L1

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Decimal Places -This is another way numbers are approximated or rounded -The principal is the same as for sig figs but we are only interested in the numbers to the right of the decimal place 3.14159 Interested in these numbers Example : Express π (Pi) to 1 decimal place π = 3.1415926535897932384 Round up if ≥ Round down < 3.1 3.2 3.5 π is 3.1 (1 dp) Oct 2011Foundation - L1

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Decimal Places Example : Express π (Pi) to 2 decimal places π = 3.1415926535897932384 This is < 5 so do not round up π = 3.14 (2 dp) Example : Express π (Pi) to 6 decimal places π = 3.1415926535897932384 This is ≥ 5 so round up π = 3.141593 (6 dp) Oct 2011Foundation - L1

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Scientific Notation 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 Oct 2011Foundation - L1

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Step 1 Move decimal left Leave only one number in front of decimal Step 2 Write number without zeros Oct 2011Foundation - L1

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Step 3 Count how many places you moved decimal Make that your power of ten Oct 2011Foundation - L1

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Scientific Notation 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 scientific notation this is 3.56 x 10 -4 atm This is much less ambiguous, as the 3 sig. figs. are clearly shown. Oct 2011Foundation - L1

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Engineering Notation This is the same as scientific notation except the POWER is replaced by the letter E Examples NumberScientific Notation 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 Oct 2011Foundation - L1

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Summary 1- Significant figures are of more general use as they don’t depend on units used e.g. 2,301.2 m (1d.p.) = 2.3012 km (4 d.p.) 2- Answers which are money should usually be given to 2 decimal places, so, the nearest penny 3 ×£23.57895= £70.73685 = £70.74 to the nearest penny Oct 2011Foundation - L1

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3- You must use at least one more s.f. in working than in your answer -To give an answer to 3 s.f. you generally need to use at least 4 s.f. in working. -To give an answer to 4 s.f. you generally need to use at least 5 s.f. in working. Example Calculate 3.7545 x 8.91235 to 3 sig fig You should at least use 3.754 x 8.912 but I would use all the digits on the calculator unless otherwise stated. Oct 2011Foundation - L1

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4- When calculating with numbers that have been measured to different levels of accuracy, it makes sense to work the calculation to the lowest level of measurement “Treat Like with Like” Example If a cars speed has been measured as 40 to (1 sig fig) The distance travelled is measured as 10.91325 km (7 sig fig) It makes some sense to estimate the time (=dist x speed) as : 40 (1 sf) x 10 (1 sf) = 400 sec Oct 2011Foundation - L1

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If a cars velocity has been measured as 40.012 (5 sig fig) The distance travelled is measured as 10.91325 km (7 sig fig) It makes sense to estimate the time (=dist x speed) as 40.012 (5 sf) x 10.913 (5 sf) = 436.6501 sec = 436.65 (5 sig fig) or = 436.65 (2 dec pl) Try to work to at least one digit higher accuracy. Try to measure numbers to a sensible order of accuracy Oct 2011Foundation - L1

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