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1
Topic 1
Rounding and Scientific Notation
Contents
1
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1
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2
5
1
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1
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8
9
Learning Objectives
By the end of this topic, you should be able to:
• round to a given number of significant figures;
• perform calculations using scientific notation
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ROUNDING AND SCIENTIFIC NOTATION
1
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Rounding to decimal places
When £28 is divided by 13 the calculator gives the result 2·153846153
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The digits in the other decimal places have no practical meaning
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To decide
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If this digit is bigger than 4 then we choose the higher answer
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If not we choose the lower
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Since 3 is less than 4 we choose £2·15 as the answer and write £28 ÷ 13 = £2·15 (to 2
d
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)
Now try some examples
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0425678
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Solution:
1
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25·043 (to 3 decimal places)
3
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25·0 (to 1 decimal places)
5
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ROUNDING AND SCIENTIFIC NOTATION
6
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Q3:
A calculator reads 74·20345
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...
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Solution:
354 = 350 (to 2 significant figures)
354 = 400 (to 1 significant figure)
3·02 = 3·0 (to 2 significant figures)
3
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Q4:
A calculator reads 3·00345
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...
It
will depend on the story-line
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this number has only 1 significant
figure
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The zero just makes sure the 5 is in the 10s column
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ROUNDING AND SCIENTIFIC NOTATION
This coin is worth 50p exactly
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The zero counts
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There are roughly 360 days in a year
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the 3 and 6
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There are 360 ◦ in a revolution
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its not 359 ◦ or 361◦ but
360◦ exactly
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The calculator will give 1·083206917 × 10 12
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If we round this to 3 significant figures we get 1·08 × 10 12
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Only the decimal part of the number is rounded
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Q5:
Evaluate 1099 2 × 87412, giving your answer correct to 3 significant figures
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© H ERIOT-WATT U NIVERSITY
TOPIC 1
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b) Round 85·88235294 to 2 decimal places
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d) Round 85·88235294 to the nearest whole number
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b) Round 7922 to 3 significant figures
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d) Round 23·98421 to 3 significant figures
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Express your answer in scientific
notation
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Express your answer in scientific
notation
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Express your answer in scientific
notation
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Express your answer in scientific
notation
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1
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You should already be able to convert and interpret
numbers in scientific notation but here is a reminder
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Carrying out calculations involving standard form requires you to be able to enter
numbers in scientific notation into your calculator
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Check out how your calculator works
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ROUNDING AND SCIENTIFIC NOTATION
Try entering
9 · 3 × 107
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Examples
1
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How long will it take to travel from
the TV to you which is a distance of 400 cm? Give your answer correct to 3 significant
figures
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© H ERIOT-WATT U NIVERSITY
TOPIC 1
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Problem:
The mass of an electron is 9·11 × 10 -31 kg
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Give your answer correct to 2 significant figures
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Venus is 0·815 of Earth
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Calculate the volume of Mars correct to 3 significant figures
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Scientific notation exercise
Q11: A grain of rice weighs 2·8 × 10 -2 grams
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Give your answer in scientific notation correct to 2 significant figures
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Q12: The radius of Saturn is 58,232 km
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a) Calculate the surface area of Saturn giving your answer in scientific notation correct
to 3 significant figures
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ROUNDING AND SCIENTIFIC NOTATION
b) Give your answer in full correct to 2 significant figures
...
Q13: If the mass of a grain of sand 0·5mm in diameter is 5·5 × 10 -5 g, calculate the
number of grains of sand would there be in a 25kg bag of sand? Give your answer in
scientific notation correct to 3 significant figures
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1
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• 1532864 = 2000000 to 1 significant figure
• 0·00002598 = 0·000026 to 2 significant figures
• 23·98421 = 24·0 to 3 significant figures
• 6·32587 × 10 7 = 6·33 × 107 to 3 significant figures
Scientific Notation
To perform calculations involving scientific notation you must know how to enter them
into your calculator and how to interpret the answer
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ROUNDING AND SCIENTIFIC NOTATION
1
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b) Round 5045 to 3 significant figures
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d) Round 4·00675 to 2 significant figures
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b) Express 0·00000000000193 in standard form
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How many seconds are there in 50 years
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2
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a) Calculate in scientific notation, to 2 significant figures the mass of 2 ×
106 protons
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Q3:
74·2
Answers from page 3
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Q5:
The calculator gives 9·228213324 × 10 13 which rounds to 9·23 × 10 13
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Note: to enter the answer into SCHOLAR online you would type 4·867*10^24
Q10: Volume of sphere = 4 /3 r3
V =
4
× π × 33903 = 1 · 63 × 1011 km3
3
Note: to enter the answer into SCHOLAR online you would type 1·63*10^11
Scientific notation exercise (page 7)
Q11: 5·4 × 104
Note: to enter the answer into SCHOLAR online you would type 5·4*10^4
Q12:
a) 4·26 × 1010
b) 43000000000
Q13: 4·55 × 10 8
End of topic 1 test (page 9)
Q14:
a) 2·5
b) 5050
c) 0·6681
d) 4·0
e) 2·70 × 1019
Q15:
a) 2·9533 × 10 11
b) 1·93 × 10-12
c) 1672621780
Q16:
i
1·5768 × 10 9
ii
a) 3·3 × 10-21
b) 60
© H ERIOT-WATT U NIVERSITY