STRAND A: Computation A3 Scientific Notation

Transcription

1 UNIT A Standard Form: Text STRAND A: Computation A Scientific Notation Text Contents Section A. Standard Form A.2 Calculations with Standard Form

2 UNIT A Scientific Notation: Introduction A Scientific Notation A. Standard Form Standard form is a convenient way to write very large or very small numbers. It is particularly useful when working with a scientific calculator. In standard form a number is written as n a 0 where a < 0 and n is an integer That is, a is a number greater than or equal to and less than 0, and n is a positive or negative whole number. Note Because of the way that the powers of 0 are used in standard form, it is important to remember that: 4 0 = and 4 0 = = = = = = 0 0 = = Worked Example Write each of the following numbers in standard form (c) = = = = (c) = = Worked Example 2 Find =. 0 = (c) = (d) =

3 A. UNIT A Scientific Notation: Introduction = = = = (c) 6. 0 = = (d) = = Worked Example Write each of the following numbers in normal decimal form (c) = = = = (c) = = Most calculators can work with numbers in standard form. On a calculator display would look like Worked Example 4 Calculate and write the answer exactly, in standard form. 2

4 A. UNIT A Scientific Notation: Introduction = = 2 = =. 0 0 in standard form. Exercises. Write each of the following numbers in standard form (c) (d) (e) 420 (f) (g) 000 (h) (i) (j) (k) 0.07 (l) (m) 0.24 (n) 0.07 (o) (p) (q) 0.47 (r) Write each of these numbers in standard form. million thousand (c) 6.4 million (d) 0.4 million (e) 4 million (f) 0.4 million. Write each of the following numbers using normal decimal notation (c) (d) (g) (h) 2. 0 (j) (k) (m) (n) (e) (f) (i) (l) (o) For each of the numbers below state whether or not it is in standard form. If it is not in standard form, write it in standard form (c) (d) (e). 0 8 (f) (g) (h) (i)

5 A. UNIT A Scientific Notation: Introduction. Give the answers to the following calculations in standard form = = (c) 0 = (d) = (e) = (f) = (g) = (h) = (i) = (j) 0 2 = (k) = (l) 00 2 = 6. There are 000 m in km. Convert the following distances to metres, giving your answers in standard form. 0 km 620 km (c) 46 km 7. Find the number of: hours in a year minutes in a week (c) seconds in a day, giving your answers in standard form. 8. The radius of the earth is km. (c) Write this in a normal decimal form. Find the radius of the earth in metres and express it in both decimal form and standard form. Find the circumference of the earth in metres, giving the answer in standard form. 9. The mass of the earth is kg. Write this as a decimal number. 0. The width of a thin strip of metal is mm. Write this in standard form. 00. Scientists estimate the mass of a newly discovered planet as kg. Write this in standard form. 2. The distance of the earth from the sun varies between. 0 8 km and km. Write these numbers in a decimal format. Convert both distances to metres and write them in standard form. 4

6 A. UNIT A Scientific Notation: Introduction. Using a calculator, or otherwise, determine the exact value of 2 2 (i) (ii) (iii) 2 2 (i) Write your answer in part (i) correct to one significant figure. (ii) Write your answer in part (ii) in standard form. A.2 Calculations with Standard Form When using standard form it is possible to multiply and divide numbers, taking advantage of the form in which they are written. Note that we can use the rules where a and b are numbers. a b a + b 0 0 = 0 a b a b 0 0 = 0 Worked Example 8 4 Find To do this calculation, you multiply together the 4 and the and then multiply together the 0 8 and the = = ( ) This result is not in standard form so the final stage is to convert the result to standard form = = =. 2 0

7 A.2 UNIT A Scientific Notation: Introduction Worked Example 2 Find ( ) ( ) (c) ( ) ( ) Multiply together the.2 and the and then multiply together the 0 4 and the = = This number is not in standard form so converting gives = = Division follows a similar approach to multiplication. First divide 6 by and then divide 0 8 by 0 4. ( ) ( ) = ( ) ( ) = This result is in standard form so no further work is required. (c) First divide 7.2 by 6 and then divide 0 by 0 4. ( ) ( ) = ( ) ( ) This result is in standard form. = 2. 0 Problems can be done directly on a calculator, or by entering numbers using the EE or EXP keys. Challenge! In astronomy, the distance between stars is measured in light years, which is the distance travelled by light in a year. One light year = km. This is approximately km. How long would it take for light to travel from the Sun to the Earth if their distance apart is. 0 8 km? 6

8 A.2 UNIT A Scientific Notation: Introduction Exercises. Do the following calculations, making sure that your answer is in standard form. Do not use a calculator = = (c) = (d) = 2 8 (e) = (f) = (g) = (h) = 8 2 (i) = 6 (k) = 7 (j) = 4 9 (l) = 2. Give the answers to the following calculations in standard form. Do not use a calculator. ( 6) ( 2) = 2 ( 9 0 ) ( 0 ) = ( 4) ( 2) = (d) 2 (. 6 0 ) ( 2 0 ) = (c) ( 8) ( 2) = (f) ( ) ( 4 0 ) = (. 4) ( ) = (h) 4 0 (. ) ( 9 0 ) = (e) (g) ( ) ( ) = (i) ( ) ( ) = (k) ( ) ( ) = (j) ( ) ( ) = (l) Do the following using a calculator, giving your answers in standard form. ( ) = ( ) 2 = ( ) = (c) (d) = 4 2 (e) = (f) = (g) ( ) ( ) = (h) = 2. 0 (i) = (j) = (k) = (l) = 4. There are seconds in one day. How many seconds are there in: 0 days week (c) year? 7

9 A.2 UNIT A Scientific Notation: Introduction. The mass of an electron is 9. 0 kg. Find the mass of: 0 8 electrons electrons (c) electrons. 6. A rectangle has sides of length 0 mm and mm. Find the area of the rectangle. 7. The speed of sound is ms. How far would the noise from a 'bang' travel in: (i) 0 seconds (ii) 0 seconds (iii) How long would it take the noise from a 'bang' to travel: (i) 0 metres (ii) 2 0 metres (iii) seconds? metres? 8. The speed of light is 0 8 ms. How far would light travel in 00 seconds? The mean distance of the earth from the sun is. 0 m. How long does it take for light to travel from the sun to the earth? 9. The distance from the earth to the moon is km. Find this distance in metres. How long would it take a spaceship to travel to the moon from earth if its average speed was 400 ms? 0. The density of air is. 0 6 kg/ cm. How many cubic centimetres are in one cubic metre? Find the mass of (c) m of air. Find the volume of air that will have a mass of grams. (d) The density of hydrogen is kg / cm. Repeat and (c) for hydrogen.. The population of the world was estimated to be in If the population increases by % each year, estimate the population at the beginning of the year The approximate population of Jamaica in 2008 is given in standard form as Write this as an ordinary number. The thickness of grade A paper is cm. Grade B paper is twice as thick as grade A. Calculate, in centimetres, the thickness of grade B paper. Write your answer in standard form. 8

10 A.2 UNIT A Scientific Notation: Introduction. Between 970 and 200 the number of people living in towns and cities in a group of countries increased from to Calculate the increase in the number of people, giving your answer in standard form. 4. Centuries ago, a man promised to give his wife some grains of rice. He took a chess board and placed one grain on the first square, two grains on the second square, four grains on the third square, eight grains on the fourth square, and so on. If he had completed all 64 squares on the chess board he would have used approximately grains of rice. One grain of rice weighs about 0.0 grams. Calculate an estimate of the weight of rice used. Give your answer in tonnes, correct to one significant figure. [ tonne = 000 kg]. Evaluate Evaluate (. ). Give your answer in standard form. 6. The mean distance of the earth from the sun is 49.6 million kilometres. Write the number 49.6 million in standard index form. The earth travels a distance, D km, in one day. The value of D is given by the formula D = 2π distance of earth from sun 6 Calculate the value of D, giving your answer in standard index form. 7. The number 0 00 is called a googol. Write the number, 0 googols, in standard index form. A nanometre is 0 9 metres. Write 0 nanometres in metres. Give your answer in standard index form. Investigation Han Sin, a Chinese general, devised a method to count the number of soldiers that he had. First, he ordered his soldiers to form groups of, followed by groups of and then groups of 7. In each case he noted down the remainder. Using the three remainders, he was able to calculate the exact number of soldiers he had without doing the actual counting. D Do you know how he did it? 9