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Home , Hundredth, Order of magnitude

Q Skills Review Dr. C. Stewart Measurement 4:

The System

We are so very familiar with our decimal notation for writing that we usually take it for granted and do not think about what it actually means. For example, we know that 6 means six “ones” or “units”, but when we write 60, then the 6 indicates six tens, or sixty, of the units. The zero must be written to ensure we know that the 6 is in the tens’ place. When we write the digits in a row, we add the values, so that 64 indicates “six tens plus four” of the units. If the quantity we want to indicate is not a whole then we sometimes use fractions, for example 6⅓ indicates “six units plus one third of a unit”, but this takes us away from the idea of simply writing digits in a row.

Let us now consider only those fractions in which the denominator is a power of ten, such as 1 4 9 10 ,1000 , 1000000 . Fractions of this type are called decimal fractions. They are a continuation of the base ten place-value notation, and with this understanding we can simply continue to write down the digits of the numerators in the correct place. We keep track of the value associated with each digit by using a ‘decimal point’ to separate the digits representing whole numbers from those representing parts of a whole. For example, the number 2453.176 represents 2 thousands, 4 hundreds, 5 tens, 3 ones (or units), 1 tenth, 7 and 6 thousandths. We can expand 2453.176 to show the meaning of this notation as follows:

1 7 6 2453.176= 2000 + 400 +++++ 50 3 10 100 1000

If the numerator of a decimal fraction has more than one digit, then the fraction can be expressed as the sum of decimal fractions with single digit numerators by considering equivalent fractions, 57 50 7 5 7 0 5 7 e.g. 1000 = 1000 + 1000 = 100 + 1000 . This can then be written as 10 + 100 + 1000 to allow us to write it as 0.057 in decimal notation. The ‘0’ after the decimal point is necessary to ensure the 5 and 7 are in the correct places. By convention we very rarely write ‘0’ at the beginning of a number larger than one, and so seeing ‘0’ at the beginning of a number indicates that it has a value less than one. It is important to remember to write this ‘0’, is the convention in Canada: according to the Government of Canada’s terminology and linguistic data bank (www.btb.termiumplus.gc.ca, writing tools, The Canadian Style, section 5.09), “Normally, no number should begin or end with a decimal point. A zero is written before the decimal point of numbers smaller than 1, while in whole numbers the decimal point should either be dropped or be followed by a zero: $0 .64 not $.64; 11 or 11 .0 not 11 .”. This is also an internationally agreed convention (www.bipm.org/en/si/si_brochure, section 5.3.4).

One feature of the decimal system is that each digit represents the value of the digit multiplied by a power of ten. We can use this feature to multiply or divide numbers by powers of ten very easily. For example, what is the value of 56.437× 10 2 ? If we look at the meaning of the notation 4 3 7 2 we see that 56.437 means 50++ 6 10 + 100 + 1000 , and so when we multiply by 10= 100 , we get (50++++ 64 3 7 ) × 100 =× 50 100 +× 6 100 + 4 ×100 + 3 × 100 +7 × 100 10 100 1000 10 100 1000 7 =5000 + 600 +×+ 4 10 3 + 10 The end result is that 56.437× 102 = 5643.7 Isn’t this amazing? 2

[You’re probably wondering what the big deal is, but think about the notations used before the decimal system. How would you do this in Roman Numerals, for example? Apart from difficulties in expressing 0.437, what would 56× 100 = 560 look like? It would be written as LVI× C = DCLX . Because the decimal system is positional (the place where the digits are written indicates their value), then when we multiply (or divide) by a power of ten, we move the digits to a different place, but we use the same digits in the same order!] We often say we are moving the decimal point by the number of places indicated in the exponent of the ten. Although this is technically incorrect, since the decimal point always remains between the whole numbers and the fractional part, it relates easily to what we see written down. In 56.437× 102 = 5643.7 we feel that the decimal point has been moved 2 places to the right.

472.3 400 70 2 3 1 Consider the following example: 472.3÷= 10 2 = + ++× 100 100 100 100 10 100 7 2 3 =4 +++ = 4.732 10 100 1000

472.3 1 But 472.3÷= 102 = 472.3 ×= 472.310 × −2 , and so 472.3× 10−2 = 4.723 . Here it looks 102 10 2 like the decimal point has been moved 2 places to the left. In summary: When we multiply by a positive power of ten, 10 n , we see the decimal point is now n places to the right. When we divide by a positive power of ten, 10 n , or multiply by a negative power of ten, 10 −n , we see the decimal point is now n places to the left.

Scientific Notation

Very large or very small numbers can be difficult to read and so we sometimes use words like million or trillion to help us. Another technique is to give prefixes to the units, for example milligram (mg) or gigabyte (GB). A useful notation for writing very large or very small numbers is “scientific notation”. A number in scientific notation has the form: a ×10 n where 1≤a < 10 , and n is an integer This means that in the number a, there is exactly one digit before the decimal point (but this cannot be ‘0’), and the power of ten (10 n) is used to ensure the correct value of each digit. Here are some examples: 472.3= 4.723 × 10 2 , 0.00345= 3.45 × 10 −3 , 0.16= 1.6 × 10 −1 .

How would we write the value 2.7 in scientific notation? We need to find a value for n such that 2.7= 2.7 × 10 n . We know that 2.7= 2.7 × 1 , and that 100 = 1 , and so we can write 2.7= 2.7 × 10 0 .

There different ways to enter numbers in scientific notation on a calculator, for example, some have a button marked EXP or EE which you press before entering the power of ten. Also, calculators differ in the way they present numbers in scientific notation: some miss out the ‘×10’, for example, showing 3.6× 10 7 as 3.6 07 or 3.6 E07; others may use a different notation. It is very important to get to know your own calculator. 3

In to knowing how to enter numbers in scientific notation for use with a calculator or computer, it will help your understanding of this notation if you learn how to calculate without using a machine.

Multiplying and Dividing When multiplying or dividing numbers given in scientific notation it is easiest to deal with the number part and the powers of ten separately. Example 1: (610×23) ××( 710 5) =×× 6710 235 × 10 [re-order the multiplications] (23+ 5 ) 28 =×42 10 =× 4210 [using laws of exponents] 1 28 29 =××4.2 10 10 =× 4.2 10 [write in correct scientific notation]

Example 2: 17 17 2 8 ×10 2× 10 810×17 ÷× 410 − 8 = = [write the division as a fraction and cancel] ()() −8 − 8 1 4 ×10 10 (17−( − 8 )) =2 × 10 [using laws of exponents] =2 × 10 25

Adding and Subtracting When adding or subtracting numbers given in scientific notation we must first write the numbers with the same power of ten - we cannot use laws of exponents to combine the powers of ten when adding or subtracting numbers. Once the power of ten is the same we simply add or subtract the number part (and re-write in correct scientific notation if necessary). Example 3: 13 12 13 13 (a) (2.34× 10) +×( 3.110) =( 2.34 × 10) +×( 0.3110 ) [get the same power of ten – the largest] 13 =()2.34 + 0.31 × 10 [the power of ten is a common factor] 13 =2.65 × 10 OR – 13 12 12 12 (b) (2.34× 10) +×( 3.110) =( 23.4 × 10) +×( 3.110 ) [the same power of ten – the smallest] 12 =()23.4 + 3.1 × 10 [the power of ten is a common factor] 12 13 =26.5 × 10 = 2.65 × 10 [write in correct scientific notation]

Method (a), writing both numbers with the largest power of ten, is slightly shorter, but some people find it easier to use the smaller power of ten, method (b). The important thing is to write both numbers in a form with the same power of ten – you can choose whichever method you find easier. Look at the following examples and decide which you find easier when there are negative powers of ten involved. 4

Example 4: −5 − 8 − 5 −− 35 (a) (7.594× 10) −×( 3.8 10) =( 7.594 × 10) −××( 3.8 10 10 ) [‘split up’ one power of ten ...] −5 − 5 =(7.594 × 10) −( 0.0038 × 10 ) [... to get the same power of ten] −5 =()7.5940 − 0.0038 × 10 [write with the same # of d.p.] −5 =7.5902 × 10 OR – −5 − 8 −− 35 − 8 (b) (7.594× 10) −×( 3.810) =( 7594 ×× 10 10) −×( 3.810 ) [change one number part ...] −8 − 8 =(7594 × 10) −×( 3.8 10 ) [... to get the same power of ten] −8 =()7594.0 − 3.8 × 10 [write with the same # of d.p.] −8 =7590.2 × 10 3− 8 − 5 =7.5902 ××= 10 10 7.5902 × 10

Practice Questions Set 1 [Answers to these questions can be found at the end.] 1) Expand the following decimal numbers to show the meaning of the notation: (a) 0.49647 (b) 52 038 (c) 72.01 (d) 0.333 2) Write the following in ordinary decimal notation: (a) 4.65× 10 1 (b) 67.394× 10 6 (c) 91.563× 10 −3 (d) 7.2× 10 −8 3) Write the following in scientific notation: (a) 3 896.21 (b) 0.000 000 34 (c) 2 710 000 000 (d) 0.189 004 4) Write each of the following in correct scientific notation: (a) 23× 10 4 (b) 600× 10 −2 (c) 0.673× 10 8 (d) 915.63× 10 −4 5) Calculate the following, giving your answer in scientific notation, without using a calculator . (a) (2.110×16) ×( 4 × 10 13 ) (b) (810×−4) ×( 510 × 7 ) (c) (6.9× 108) ÷( 3.0 × 10 9 ) (d) (210×−5) ÷( 410 × − 11 )

6) Calculate the following, giving your answer in scientific notation, without using a calculator . (a) (9.3× 1053) +( 1.04 × 10 56 ) (b) (8.2× 10−12) +( 5.1 × 10 − 13 ) (c) (6.132× 106) −( 3.0 × 10 4 ) (d) (510×−32) −( 610 × − 34 )

[ More practice questions like these can be found online at http://janus.astro.umd.edu/astro/scinote/ ] 5

Significant Digits and

Very few measurements are exact. Continuous quantities, such as length, time, mass, etc., are measured and quoted to a certain precision, for example, to the nearest second or tenth of a kilogram. Even discrete data may not be exact: perhaps we can say exactly how many students are registered in a particular class, but the exact number of students attending a university may change too frequently (as some join and others leave) for an up-to-date figure to be available. It is important to be aware of the precision of the numbers we read and to indicate the precision of the numbers we write. Sometimes we talk about the number of decimal places (d.p.) to which a value is given: 3.4 has been given to the nearest tenth of a unit (1 d.p.), whereas 3.41 is more precise since it has been given to the nearest (2 d.p.). Although this is a simple concept, there are many situations in which it is not useful. Consider the length of a room given as 320 cm. This number has no decimal places. All we can deduce is that the length is closer to 320 cm than it is to 310 cm or 330 cm. The first two digits (3 and 2) are called the significant digits (s.d., or sometimes s.f for significant figures), and the ‘0’ is there as a ‘place holder’ (to indicate that the 2 is a number of tens, not ones). If the room was actually measured to the nearest centimetre (closer to 320 cm than to 319 cm or 321 cm), then it should be described as 320 cm to 3 s.d., indicating that the ‘0’ is now more than just a place holder. Unless the precision of a value is given, or the context makes it clear, we assume that zeros at the end of a whole number are simply place holders. However, zeros after the decimal point indicate that the measurement has been made to that precision, e.g. 15.00 has 4 s.d. and is closer to 15.00 than to 14.99 or 15.01 (it has been measured ‘to the nearest hundredth’). [For more about significant digits and precision, see the review “Approximation and Precision”.] When writing numbers in scientific notation, the digits given should be those which are significant. For example, we would assume that 2,960,000,000 is a number that has been rounded to the nearest ten million and only the first three digits are ‘significant’ (the zeros are simply place holders), and so we would write this as 2.96× 10 9 . Writing 2.960000000× 10 9 would indicate that the original number was correct to the nearest one – highly unlikely! However, if we know the number is given to the nearest million (closer to 2960 million than to 2959 million or 2961 million), then we would write 2.960× 10 9 to indicate that there are four significant digits. Frequently we do not need to know precise values of numbers, but are more concerned with their general size; perhaps we are estimating rather than measuring accurately. We may not care what the amount of oil spilt was to the nearest litre, but we do want to know if it was hundreds or thousands or tens of thousands of litres! This broad description of size is called the ‘order of magnitude’ of a quantity. An easy way to determine the order of magnitude of a number is to look at the power of ten when the number is written in scientific notation. For example, 86,150,000= 8.615 × 10 7 is a number with order of magnitude 10 7 , or in words: tens of millions. We may describe one quantity as “an order of magnitude larger” than another quantity. This indicates that it is about ten times larger. An amount such as $3.2billion ( 3.2× 10 9 ) is described as three orders of magnitude larger than an amount such as $6.1million ( 6.1× 10 6 ). When talking about the order of magnitude we are looking at the ‘big picture’, rather than the detail of the ‘3.2’ and ‘6.1’.

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Practice Questions Set 2 [Answers to these questions can be found at the end.]

1) State the precision (number of significant digits) of each of the following measurements: (a) 5× 10 4 kg (b) 6.2× 10 −2 mg (c) 6.736× 10 8 km (d) 9.12× 10 −3 s (e) 45.61 cm (f) 435,000 people (g) $43.6 million (h) 0.00026 g (i) 1.00026 g

2) State the order of magnitude, both in symbols and in words, of each of the quantities in Q1.

3) State the order of magnitude of each of the following (note that the values are not given in correct scientific notation): (a) 470× 10 12 (b) 320× 10 −5 (c) 0.02× 10 7 (d) 0.00019× 10 −15

4) By considering the order of magnitude, and without converting to ordinary notation, write the following from smallest to largest: 1.03× 1016 , 2.67 ×× 10 32 ,1.3 10− 2 , 9.67 × 10 15 , 2.57418 ×× 10 33 ,4.8 7 10 − 8

5) For each of the following, state the order of magnitude without carrying out the actual calculation: (a) (1.8× 1023) ×( 2 × 10 15 ) (b) (6.3× 103) ×( 1.1 × 10 − 7 ) (c) (7.2× 1018) ÷( 3.56 × 10 9 ) (d) (5.7× 1014) +( 6.02 × 10 27 ) (e) (8.2× 10−5) −( 7.6 × 10 − 9 )

6) In each of the following sentences, the quantity is given to an unreasonable precision considering the quantity or the method of measurement. Give an appropriate number of significant digits. (a) The distance from Squamish to Vancouver is 73.2 km. (b) My bathroom scales gave my weight as 64.317 kg. (c) The population of Canada was 35,141,542 in April 2013. (d) I used a ruler to measure a ream (500 sheets) of unopened paper to have a height of about 45 mm. With the paper wrapping top and bottom, we have 502 sheets of paper. Each piece of paper has a thickness of 45÷ 502 = 0.08964 mm.

7) Give an appropriate order of magnitude for each of the following: (a) The number of students at Quest. (b) The distance across Canada (in km). (c) The price of an apartment. (d) The capacity of a teaspoon (in litres).

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Answers Practice Questions Set 1

1) (a) 0.49647 =++49 6 + 4 + 7 (b) 52038= 50000 + 2000 +×++ 0 100 30 8 10 100 1000 10000 100000 (c) 72.01= 70 ++ 2 0 + 1 (d) 0.333 =3 + 3 + 3 10 100 10 100 1000

2) (a) 4.65× 101 = 46.5 (b) 67.394× 106 = 67394000 (c) 91.563× 10−3 = 0.091563 (d) 7.2× 10−8 = 0.000000072

3 −7 3) (a) 3896.21= 3.89621 × 10 (b) 0.00000034= 3.4 × 10 9 −1 (c) 2710000000= 2.71 × 10 (d) 0.189004= 1.89004 × 10 2+( − 2 ) 4) (a) 23× 104 = 2.3 ×× 10 10 4 = 2.3 × 10 5 (b) 60010×−222 =×× 610 10 − =× 610( ) =× 610 0 (−1) + 8 (c) 0.673×=××=× 10818 6.73 10− 10 6.73 10( ) =× 6.73 10 7 2+( − 4 ) −2 (d) 915.63×= 10−4 9.1563 ××= 10 2 10 − 4 9.1563 × 10( ) = 9.1563 × 10 ()

16 13(16+ 13 ) 29 5) (a) (2.110×) ××( 410) =×× 2.1410 =× 8.410 −47((−4) + 7 ) 34 (b) (810×) ××( 510) =× 4010 =××=× 41010 410 8 8 9 6.9× 10 6.9  ()()8− 9 − 1 (c) ()()6.910×÷×= 3.010 =  × 10 =× 2.310 3.0× 10 9 3.0  −5 −5 − 11 2× 10 ()()()−5 − − 11 − 1 6 5 (d) ()()210× ÷× 410 = =× 0.510 =××=× 510 10 510 4× 10 −11

6) (a) (9.3×+×= 1053) ( 1.04 10 56) ( 0.0093 ×+×= 10 56) ( 1.04 10 56) 1.0493 × 10 56 OR (9.3×+×=×+×= 1053) ( 1.04 10 56) ( 9.3 10 53) ( 1040 10 53) 1049.3 ×= 10 53 1.0493 × 10 56 (b) (8.210×−12) +×( 5.110 − 13) =×( 8.210 − 12) +×( 0.5110 − 12) =× 8.7110 − 12 OR (8.210×−12) +×( 5.110 − 13) =×( 8210 − 13) +×( 5.110 − 13) =× 87.110 − 13 =× 8.7110 − 12 (c) 6.132×−×= 1064 3.0 10 6.132 ×−×= 10 66 0.03 10() 6.132 −×= 0.03 10 66 6.10 2 × 10 ( ) ( ) ( ) ( ) OR 6.132×−×= 1064 3.0 10 613.2 ×−×= 10 44 3.0 10 610.2 ×= 10 4 6.102 × 10 6 ( ) ( ) ( ) ( ) (d) 510×−−−32 −× 610 34 =× 510 32 −× 0.0610 − 32 =−×() 50.06 10 −− 32 =× 4.9410 32 ( ) ( ) ( ) ( ) OR (510×−−32) −×( 610 34) =×( 50010 −− 34) −×( 610 34) =× 49410 − 34 =× 4.9410 − 32

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Practice Questions Set 2 [OofM means “order of magnitude” in the following solutions.]

1) & 2) (a) 5× 104 = 50,000 1 s.d. OofM 4, tens of thousands (b) 6.2× 10−2 = 0.062 2 s.d. OofM –2, hundredths (c) 6.736× 108 = 673,600,000 4 s.d. OofM 8, hundreds of millions (d) 9.12× 10−3 = 0.00912 3 s.d. OofM –3, thousandths (e) 45.61= 4.561 × 10 1 4 s.d. OofM 1, tens (f) 435,000= 4.35 × 10 5 3 s.d. OofM 5, hundreds of thousands (g) 43.6 million= 4.36 × 10 7 3 s.d. OofM 7, tens of millions (h) 0.00026= 2.6 × 10 −4 2 s.d. OofM –4, ten thousandths (i) 1.00026= 1.00026 × 10 0 6 s.d. OofM 0, ones [Think carefully about the similarities and differences between (h) and (i)!]

3) (a) 470× 1012 = 4.7 × 10 14 OofM 14 (b) 320× 10−5 = 3.2 × 10 − 3 OofM –3 (c) 0.02× 107 = 2 × 10 5 OofM 5 (d) 0.00019× 10−15 = 1.9 × 10 − 19 OofM –19

4) Put the exponents in increasing order (–8, –2, 15, 16, 32, 33) to order the numbers: 4.87×× 10−8 ,1.3 10 − 2 , 9.67 ××× 10 15 ,1.03 10 16 , 2.67 10 32 , 2.5741 8 × 10 33

5) (a) OofM = 23 + 15 = 38 (b) OofM = 3 + (–7) = –4 (c) OofM = 18 – 9 = 9 (d) 10 27 is much larger than 10 14 , so OofM = 27 (e) 10 -5 is much larger than 10 -9, so OofM = –5

6) (a) 1 s.d. (70 km). Exactly where in each town? (b) 3 s.d. at most (64.3 kg). Bathroom scales might possibly measure to the nearest 100 g, but certainly not to the nearest gram – try standing on the scales several times: many will vary by up to 500 g. (c) 2 or 3 s.d. (35 million or 35,100,000). The note associated with this Statistics Canada figure says, “Estimates are based on 2006 Census counts adjusted for census net undercoverage and incompletely enumerated Indian reserves to which is added the estimated demographic growth for the period from May 16, 2006, to March 31, 2013”. With so many estimates involved, and the population changing by about 400,000 per year, giving the population to the nearest person does not seem meaningful! (d) 2 s.d. at most (0.090 mm). The measurement with the ruler would probably be to the nearest millimetre (2 s.d.).

7) (a) Hundreds of students at Quest: OofM 10 2. (b) Thousands of km across Canada: OofM 10 3. (c) Hundreds of thousands of dollars for an apartment: OofM 10 5. (d) 5 ml = 0.005 litres, so thousandths of a litre for a teaspoon: OofM 10 -3.