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CONNECT: Ways of writing numbers

SCIENTIFIC NOTATION; SIGNIFICANT FIGURES; DECIMAL PLACES First, a short review of our Decimal (Base 10) System. This system is so efficient that we can use it to write any number we like, no matter how big or how small. But, when numbers get either TOO big or small, then the amount of digits becomes unwieldy. That’s where Scientific Notation, Significant Figures and Decimal Places all come in. You probably remember from school that the place value of a number is important because it determines how big the number is. I am sure you also remember the names of all the place values, especially of the whole numbers: millions hundred

thousands ten thousands

thousands hundreds tens units .

Reading from the right (units), each column to the left has been obtained by multiplying by 10 (each column represents a number that is 10 times bigger than the one on its right). For example, 10 is 1 x 10, 100 is 10 x 10, 1000 is 100 x 10 and so on. (Same as 20 is 2 x 10, 350 is 35 x 10.) Going in the other direction, each column to the right has been obtained by dividing the one on its left by 10. The decimal point which goes straight after the units column tells us that we no longer have a whole number, and that the unit

has been divided by 10. That’s why 0.1 is the same as 1 ÷10, or 101 . 1 has been divided by 10 in the same way as 100 ÷ 10 =

10, or 1000 ÷ 10 = 100. The number is moved across a column to make it smaller. So, we get new headings for our place values AFTER (to the right of) the decimal point.

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millions hundred

thousands ten thousands

thousands hundreds tens units . tenths hundredths thousandths

One tenth can be written as 101 or 0.1, a hundredth can be written as

1001 or 0.01, and a thousandth can be written as

10001 or

0.001. DECIMAL PLACES Decimal places are the positions of numbers that appear to the right of the decimal point. They can give us a way of stating how accurately we want to measure something. For example, when you fill up the car at the petrol pump, you know that the price of petrol is given to three places of decimals (in cents), yet we round that to 2 places when we pay – in fact we even round to the nearest 5 cents. So the calculation that the machine does is accurate right up until the very last step. For example, my car might take 32.4 litres when the price is 139.9 cents per litre. If we calculate this (and of course you can use your calculator!) we get 4532.76. Thankfully, this is cents, so we need to divide by 100 and get ($)45.3276. Now, rounding to 2 decimal places (the nearest cent), we look at the 3rd decimal place – that is the 7 – and ignore everything that comes after it. If the 3rd decimal place is 5, 6, 7, 8, or 9, then we “round up” and we would get $45.33. This is because the 7 is more than halfway up to the next number (this is similar to 327 being closer to 330 than it is to 320). I like to put a little line to the right of the number I’m going to round to: 45.32|76. Then I look at the next number (7) and work out what to do.∗ Let’s say we have an answer to some other calculation and it looks like this: 5894.874128. That is too many digits for the answer to make sense in our calculation so let’s try to round it to four decimal places. So we look at 5894.8741|28. This time the ∗ (Of course, in cash, we would actually pay $45.35.)

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2 to the right of the line tells us to keep the 4th number just as it is, so we get 5894.8741. So our number represents 5 thousand, 8 hundred and ninety-four point 8 tenths, 7 hundredths, 4 thousandths, 1 ten thousandth. We rounded off the 2 hundred thousandths and 8 millionths! Of course this all depends on how precise we want an answer to be. (Is it rocket science or nanophysics?) Try these for yourself: You can check the answers to these using the solutions at the end of the resource. Round to 2 decimal places:

1. 14.803 2. 167.1509 3. 38.598 And, some problems:

1. How much would you pay for 1.8 kg apples if they cost $4.59 per kg?

2. A sheet of glass measures 650 mm by 455 mm. Calculate the area in m2, correct to 3 decimal places. (Hint: change the mm measurements to metres first by dividing by 1000, then multiply.)

3. Calculate the tax on $15 825.14 at 35 cents in the dollar. (Hint: change the 35 cents to dollars first and multiply.)

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SIGNIFICANT FIGURES Significant Figures are different from Decimal places, although we may “round” using either of them. But significant figures can apply to whole numbers as well as decimals. Significant figures are the digits that are regarded as “important” when stating a number or measurement. All non-zero numbers are “significant”. 0 (“zero”), when at the end of a whole number or at the beginning of a decimal, becomes a place holder and is not counted as significant. An example of this might be: the crowd at a game of football is accurately counted as 38 521.∗ Do you think it is more likely that a commentator would say: “the crowd today is thirty-eight thousand, five hundred and twenty-one”, or would they say: “the crowd is nearly forty thousand”? The forty thousand is a lot easier to say, and for the listener to understand. 40 000 has only 1 significant figure (the 4) and the 0s tell us its place value – that it is in the ten thousands column. The accurate number (38 521) has 5 significant figures (ALL OF THEM!) and when we read or hear it we know that everyone in the crowd has been counted. However, zeros are significant when they are between two non-zero digits, such as 504, which has 3 significant figures. Another example: Round 0.005 218 to 2 significant figures. As with Decimal Places, we can put a line to the right of the number we need to round at, so 0.005 2|18 (the 0s are not significant, and as the 52 are the first 2 significant figures we are only interested in them). The 1 tells us to leave the 2 alone, so 0.005 218 ≈ (is approximately, or roughly equal to) 0.005 2, correct to 2 significant figures. Here are some for you to try. You can check your answers from the back of the handout. Round each of the following to 3 significant figures: ∗ (Note the space between the 38 thousand, and the 521. We don’t use commas any more and use a space instead.)

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1. 58 942 2. 632.1 3. 0.047 93 4. 409 100 And some more questions:

1. According to http://en.wikipedia.org/wiki/Lunar_distance_(astronomy), the distance from the Earth to the Moon is 384 400 km. Express this distance correct to 2 significant figures.

2. The diameter of a golf ball is given by http://en.wikipedia.org/wiki/Orders_of_magnitude_%28length%29 as 4.267 cm. Express this correct to 3 significant figures.

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SCIENTIFIC NOTATION We need to return to our Decimal (Base 10) System. For Scientific Notation, we talk about powers of 10, and we can actually express every number in terms of a power of 10. This is because of the following:

I’m sure you recognise 102 as meaning 10 x 10, which is 100. (See how it lines up with the 100 column!). 103 means 10 x 10 x 10 which is 1 000. And so on to the left. To the right, 101 is just 10. If you continue the pattern of powers from the left to the right, you will see that the next power (for units) should be 0. This is right! (If you want more information about this, and negative powers, please refer to CONNECT: Powers: POWERS, INDICES, EXPONENTS, LOGARITHMS – THEY ARE ALL THE SAME!)

The good thing about having powers of 10 to act as our place value column headings is that it gives us an efficient method of expressing ANY number, no matter how large or how small. This is called Scientific Notation, or Standard Form.

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Writing a number in Scientific Notation, or Standard Form For example, we may wish to write the distance of the Earth to the Moon, 384 400 km, in a more efficient form. We see that the 3 is really 3 hundred thousand, and is in the column of 105. We put a decimal point after the 3 and call our number 3.844 x 105. To check this result, you can multiply 3.844 by 10 x 10 x 10 x 10 x 10 (so you would move the decimal point to the right 5 times). Most people use this method to get the number in Scientific Notation in the first place! So, you place a decimal point after the first non-zero digit, then work out how many times you would multiply by 10 (by moving the decimal point) until you get back to the original number. Second Example: Write the number 1 390 000 km in Scientific Notation. (This is the diameter of the Sun, according to http://en.wikipedia.org/wiki/Orders_of_magnitude_%28length%29.) So, we put a decimal point after the first non-zero digit, which in this case is 1, to get 1.39 (we don’t need the zeros on the end as they are just place holders and are not “significant”). Now, to work out the power of 10, either look at the position of the first digit in the number – the 1 – it is in the 106 column – or else, count how many places you need to move the decimal point to the right to get back to the original number (6). So, 1 390 000 km is the same as 1.39 x 106 km. Decimal numbers (numbers that are between 0 and 1) will always have negative powers of 10 as their multiplier. Example: The thickness of a human hair can be as small as 0.000 017 metres. To write this in Scientific Notation, again we can put the decimal point after the first non-zero digit, which is a 1. We can look at where the 1 lines up with our powers of 10 in the table above – uh oh, our number is too small, but we can see it would be 10-5 if we continue the pattern of powers. The other method is to count the number of places we move the point to the left this time to make the original number, again it would be 5, but in the negative direction. So, 0.000 017 m is 1.7 x 10-5 metres. Remember: Decimal numbers are expressed with negative powers of 10, but numbers larger than 1 are expressed with positive powers of 10. Some for you to try. You can check your answers at the end of this resource.

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Write these numbers in Scientific Notation (Standard Form):

1. 38 950 000 2. 0.004 21 3. 128.53 4. 0.093 And some problems:

1. The distance from the Earth to the Sun has been calculated as 149 597 870.7 km. (source: http://wiki.answers.com/Q/How_far_is_Earth_from_the_Sun). Write this distance in Scientific Notation.

2. A large flea is approximately 0.003 3 m long. Write this measurement in Scientific Notation.

3. Very large or very small? (a) 6.73 x 1024 (b) 1.89 x 10-16 If you need help with any of the Maths covered in this resource (or any other Maths topics), you can make an appointment with Learning Development through Reception: phone (02) 4221 3977, or Level 3 (top floor), Building 11, or through your campus.

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ANSWERS to PROBLEMS Decimal places 1. 14.803 ≈ 14.80 (correct to 2 decimal places). 2. 167.1509 ≈ 167.15 (correct to 2 decimal places). 3. 38.598 ≈ 38.60 (correct to 2 decimal places). In this situation, the 9 goes up to “10” but this means that we need to put the 5 up to the 6. It’s like 59 going up to 60. Problems. 1. 1.8 x 4.59 = 8.262. Rounding this to 2 decimal places we would get 8.26. But in Australia now, we would pay $8.25. 2. Change the length measurements to metres:

650 mm = 650 ÷ 1000 m = 0.65 m 455 mm = 455 ÷ 1000 m = 0.455 m Area = 0.65 x 0.455 m2 = 0.29575 m2 ≈ 0.296 m2 (correct to 3 decimal places)

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3. Tax = 0.35 x 15 825.14 = 5 538.799 ≈ 5 538.80 So the tax to pay would be approximately $5 538.80 (or, probably, for the TaxPerson, $5 539). Significant Figures 1. 58 942 ≈ 58 900 (correct to 3 significant figures). Don’t forget the 0s on the end to act as place value holders. 2. 632.1 ≈ 632 (correct to 3 significant figures). 3. 0.047 93 ≈ 0.047 9 (correct to 3 significant figures). 4. 409 100 ≈ 409 000 (correct to 3 significant figures). Some more questions: 1. 384 400 km ≈ 380 000 km (correct to 2 significant figures). 2. 4.267 cm ≈ 4.27 cm (correct to 2 significant figures).

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Scientific Notation 1. 38 950 000 = 3.895 x 107 2. 0.004 21 = 4.21 x 10-3 3. 128.53 = 1.2853 x 102 4. 0.093 = 9.3 x 10-2 And, some problems: 1. 149 597 870.7 km = 1.495 978 707 x 108 km. I would probably answer this by rounding, eg 149 597 870.7 km ≈ 1.50 x 108km. 2. 0.003 3 m = 3.3 x 10-3 m. 3. (a) VERY LARGE!!! (b) VERY SMALL!