Topic&1& &Number&Skills& &

Transcription

1 Year%10%Mathematics%2016! Name: Topic1 NumberSkills Workbook#2

2 Scientific(Notation:! What%is%it?% Physists!deal!with!quantities!ranging!from!the!truly!microcosmic!to!the!macrocosmic.!For!example,!for! Astronomers! it! is! very! inconvenient! to! always! have! to! write! out! the! age! of! the! universe! as! 14,000,000,000!years!or!the!distance!to!the!Sun!as!149,600,000,000!metres.!! To!save!effort,!powers'of'ten+notation!OR!scientific+notation!is!used.!! For+example,!! 10!=!10 1 ;!the!exponent!tells!you!how!many!times!to!multiply!by!10.!! So,!3!x!10 2!tells!you!to!times!3!by!10 2!and!since!10 2!=!100.!So,!3!x!10 2!=!3!x!100!=!300.! Another+example,!! 10 K2!=!1/100;!in!this!case!the!exponent!is!negative,!so!it!tells!you!how!many!times!to!divide!by!10.! So,!10 K2!=!1/10 2!=!1/100.!! The!only!trick!is!to!remember!that!10 0!=!1.!! Using!powersKofKten!notation,!the!age!of!the!universe!is!1.4!x!10 10!years!and!the!distance!to!the!Sun!is! 1.496!x!10 11!meters.!! General%Form% The!general!form!of!a!number!in!scientific+notation!is!a!x!10 n,!where!a!must!be!between!1!and!10,!and!n! must!be!an!integer.!! (Thus,!for!example,!these!are!not!in!scientific!notation:!34!x!10 5 ;!4.8!x! ).!! If+the+number+is+between+1+and+10.+ For!example,!the!number!4.56!already!is!in!scientific!notation.!It!is!not!necessary!to!write!it!as!4.56!x!10 0,! which!would!be!4.56!x!1!which!is!of!course!4.56!(but!you!may!write!it!this!way!if!you!wish).!! If!the!number!is!a!power!of!10,!then!it!is!not!necessary!to!write!that!it!is!multiplied!by!1.!For!example,!the! number!100!can!be!written!in!scientific!notation!either!as!10 2!or!as!1!x!10 2.!(Note,+however,+that+the+ latter+form+should+be+used+when+entering+numbers+on+a+calculator.)++ The! use! of! scientific! notation! has! several! advantages,! even! for! use! outside! of! the! sciences:! Scientific! notation! makes! the! expression! of! very! large! or! very! small! numbers! much! simpler.! For! example,! it! is! easier!to!express!the!u.s.!federal!debt!as!$3!x!10 12!rather!than!as!$3,000,000,000,000.!! Because!it!is!so!easy!to!multiply!powers!of!ten!in!your!head!(by!adding!the!exponents),!scientific!notation! makes!it!easy!to!do!"in!your!head"!estimates!of!answers.!! Use!of!scientific!notation!makes!it!easier!to!keep!track!of!significant!figures;!that!is,!does!your!answer! really!need!all!of!those!digits!that!pop!up!on!your!calculator?!! Converting%from%"Normal"%to%Scientific%Notation% Place!the!decimal!point!after!the!first!nonKzero!digit,!and!count!the!number!of!places!the!decimal!point! has!moved.!if!the!decimal!place!has!moved!to!the!left!then!multiply!by!a!positive!power!of!10;!to!the!right! will!result!in!a!negative!power!of!10.!! Example:,To!write!3040!in!scientific!notation!we!must!move!the!decimal!point!3!places!to!the!left,! so!it!becomes!3.04!x!10 3.!! Example:!To!write! !in!scientific!notation!we!must!move!the!decimal!point!4!places!to!the! right:!1.2!x!10 K4.!!!!! 1

3 Questions! Normal!to!scientific!notation.! Write!the!following!numbers!in!scientific!notation.! 1.!1001! 7.! ! 2.!53! 8.!K0.0038! 3.!6,926,300,000! 9.! ! 4.!K392! 10.!K0.567! 5.! ! , % Converting%from%Scientific%Notation%to%"Normal"% If! the! power! of! 10! is! positive,! then! move! the! decimal! point! to! the! right;! if! it! is! negative,! then! move!it!to!the!left.!! Example:!! Example:+ Convert!4.01!x!10 2.!We!move!the!decimal!point!two!places!to!the!right!making!401.!! Convert!5.7!x!10 K3.!We!move!the!decimal!point!three!places!to!the!left!making! !!! Questions! Scientific!notation!to! normal! 1.!1.92!x!10 3! 7.!1.03!x!10 K2! 2.!3.051x10 1! 8.!8.862!x!10 K1! 3.!K4.29!x!10 2! 9.!9.512!x!10 K8! 4.!6.251!x!10 9! 10.!K6.5!x!10 K3! 5.!8.317!x!10 6! 11.!3.159!x!10 2! !x!10 K2! !x! ! %! 2

4 Name Date Rearranging and Understanding Formulas - Step-by-Step Lesson Lesson 1 Addition and Subtraction Problem: Rearrange all of the following formula to solve for p. f = apd Explanation: The goal of all of these is to rearrange the formulas and get p by itself. We take all the necessary steps to do so. f = apd p is being multiplied by a and d. Multiplication and division are opposite operations. = p We can divide both sides a and d to get p by itself. Tons of Free Math Worksheets at:

5 Name Date Rearranging and Understanding Formulas - Independent Practice Worksheet Rearrange all the equations to get p be itself. 1. f = wpy 2. f = kpl 3. f = mpn 4. = 7 5. = = u = (5y 8q) 8. 4u = (6y 5q) 9. 3u = (4y 9q) 10. 2u = (7y 3q) Tons of Free Math Worksheets at:

6 Significant Figures Worksheet #1 Name Block There are two rules for determining the number of significant figures: 1) If there is no decimal point--start at the RIGHT and count, beginning with the first non-zero digit. Examples s.f s.f s.f. 2) If there is a decimal point--start at the LEFT and count, beginning with the first non-zero digit. Examples s.f s.f s.f s.f. Determine the number of significant figures (s.f.) in each of the following: a) 921 b) c) d) e) f) 93,000,000 g) 93,000,003 h) 93,000,000. There are also rules for reporting numbers when you multiply and/or divide: 1) Count the sig. figs. in the numbers you are multiplying and/or dividing. Your answer should be rounded off to the smallest number of sig. figs. in your problem. Example: a) x 3.12 = calculator answer 4 s.f. 3 s.f. 6 s.f. so round to 3 s.f. Your answer will be reported as 88.4 b) = calculator answer 4 s.f. 3 s.f. 6 s.f. so round to 3 s.f. Your answer will be reported as 9.08 Reminder: Rounding-off rules: Go to next number. If it is 0-4, round down. If it is 5-9, round up. Report the answer to the following problems, paying particular attention to the correct number of sig. figs. a) / 5.12 = b) / 3.0 = c) / 4.32 = d) x = e) 98 x = f) x = g) x = h) 9.0 / 3.0 = i) 10. x 300. = j) 10. / 3 =

7 There are also different rules for reporting the answer when you add or subtract: 1) The answer should have the same number of decimal places as that of the number with the least decimal. Example: g g g g_ g = g g = 66 g is 0-4, so round down. is 5-9, so round up. NOTE: IN ADDITION AND SUBTRACTION, DECIMAL POINTS MUST BE LINED UP!! Solve the following: a) b) c) d) e) 336,000 33, = f) = Additional practice problems: How many sig. figs in the following number? a) 87 b) 190. c) d) e) Round off the following to 2 S.F. a) b) c) d) e) How many S.F. should be in the following answers: (Don t work out the problems!) a) 0.2 x = b) 43,000,000 x = c) = d) = e) = f) 3.40 x 0.04 = g) x.802 = h) x = i) x = Solve the following problems: a) b) 43.2 x 30.3 x 17.0 = c) x x d) 73 e) 8.0 f) = How many S.F. are in the following numbers? a) 3.0 x 10 9 b) c) 4.20 x 10-4 d) 900,000 e) 900,000. f) x 10 7

8 Practice Worksheet for Significant Figures 1. State the number of significant digits in each measurement. 1) 2804 m 2) 2.84 km 3) m 4) m 5) 4.6 x 10 5 m 6) 4.06 x 10-5 m 7) 750 m 8) 75 m 9) 75,000 m 10) m 11) 75,000.0 m 12) 10 cm 2. Round the following numbers as indicated: To four figures: To one decimal place: To two decimal places: Solve the following problems and report answers with appropriate number of significant digits. 1) cm cm cm cm = 2) 1.6 km m cm = 3) g g = 4) m m = 5) m kg= 6) 1.31 cm x 2.3 cm = 7) m x m = 8) 20.2 cm : 7.41 s = 9) g : g =

9 4. Express the following numbers in their equivalent standard notational form: 1) 123, ) 1,236,840 3) ) ) ) Identify the sums or differences of the following: 1) (8.41 X 10 4 ) + (9.71 X 10 4 ) = 2) (5.11 X 10 2 ) - (4.2 X 10 2 ) = 3) (8.2 X 10 3 ) + (4.0 X 10 3 ) = 4) (6.3 X 10-2 ) - (2.1 X 10-2 ) = 6. Express the product and the quotients of the following: 1) (3.56 X 10 5 ) (4.21 X 10 6 ) = 2) (2 X 10 7 ) (8 X 10-9 ) = 3) (4.11 X 10-6 ) (7.51 X 10-4 ) = 4) 8.45 X 10 7 / 6.74 X 10 3 = 5) 9.7 X 10 8 / 8.6 X 10-2 = 6) 4.7 X 10-2 / 5.7 X 10-6 =