Approximation and rounding
Approximating a number is appropriate when it is not possible or not reasonable to write the number exactly.
Engineers will often decide whether to use exact values or approximated values. When calculating a length of metal, an engineer may decide $\sqrt{2130}$√2130 cm is needed. It may be difficult to know the length to cut, so a drawing may specify that the metal must be cut to $46$46 cm. This is an approximation of the exact value that has been rounded to the nearest whole number.
Practice question
Question 1
Indicate if the following is true or false:
$\sqrt{530}$√530 is an exact value.
True
AFalse
BMy calculator states that $\sqrt{530}$√530 is $23.021728866$23.021728866. Is this still exact?
Yes
ANo
B
When you round a number it becomes an approximate answer. A decimal number can be rounded to a specified number of decimal places depending on the required level of accuracy. Consider rounding $0.0537$0.0537 to two decimal places.
| Units | . | Tenths | Hundredths | Thousandths | Ten-thousandths |
|---|---|---|---|---|---|
| $0$0 | $.$. | $0$0 | $5$5 | $3$3 | $7$7 |
The place value in the third decimal place is $3$3, so round down to $0.05$0.05. This number is now $0.0037$0.0037 less than the exact number, so it's an approximation.
To round a decimal number to a certain number of decimal places, look at the next decimal place value to the right.
- If it's less than $5$5 then round down.
- If it's greater than or equal to $5$5 then round up.
If you are calculating to a given amount of decimal places, ensure you are using exact values throughout your calculation and only round at the end to avoid rounding errors.
Practice questions
Question 2
Round $85.1446$85.1446 to the nearest hundredth.
Question 3
Round $7.034500$7.034500 to four decimal places.
Significant figures
Using significant figures is similar to rounding decimals except it is often used for very large numbers and provides an approximation of a number with a certain level of accuracy.
Consider a crowd of $95446$95446 people at the football game. A newspaper may report $95000$95000 attended the game, in this case they have rounded to $2$2 significant figures (or to the nearest thousand). Rounding to $3$3 significant figures would be $95400$95400.
Significant figures are:
- all non-zero digits
- zeros appearing between two non-zero digits
- trailing zeros to the right in a number containing a decimal point.
Significant figures for numbers less than $0$0 is slightly different than rounding.
Consider the width of a human hair is approximately $0.023$0.023 cm. Rounding this to $1$1 significant figure it would be $0.02$0.02 (two decimal places). Rounding to $2$2 significant figures it would be $0.023$0.023 (three decimal places).
Examples
- $10432$10432 to $3$3 significant figures $=$=$10400$10400 The zero between the $1$1 and the $4$4 is counted as significant.
- $1.040052$1.040052 to $3$3 significant figures $=$=$1.04$1.04 The zero between the $1$1 and $4$4 is counted as significant.
Practice questions
Question 4
Round off $461585$461585 to three significant figures.
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Question 5
Round off $0.006037736$0.006037736 to two significant figures.
Scientific notation (standard form)
Scientific notation uses the form:
$a\times10^b$a×10b,
where $a$a is a number between $1$1 and $10$10 and $b$b is an integer (positive or negative) that is expressed as an index of $10$10.
- A negative power indicates a small number.
- A positive power indicates a large number.
- A zero power indicates that the number will not change because $10^0=1$100=1.
Significant figures in scientific notation
- $6.53126\times10^7$6.53126×107 to $4$4 significant figures $=$=$6.531\times10^7$6.531×107
- $6.00002\times10^8$6.00002×108 to $3$3 significant figures $=$=$6.00\times10^8$6.00×108
Practice questions
Question 6
Express $0.07$0.07 in scientific notation.
Question 7
Express the following number as a basic numeral:
$6\times10^7$6×107
Question 8
If we round to $1$1 significant figure, sound travels at a speed of approximately $0.3$0.3 kilometres per second, while light travels at a speed of approximately $300000$300000 kilometres per second.
Express the speed of sound in kilometres per second in scientific notation.
Express the speed of light in kilometres per second in scientific notation.
How many times faster does light travel than sound?
Calculation with scientific notation
Multiplying either $1000000$1000000 by $1000000000$1000000000 or $0.000001$0.000001 by $0.000000001$0.000000001 is made easier when these numbers are written using scientific notation.
For example, the first product above can be rewritten as $10^6\times10^9=10^{15}$106×109=1015.
The first law of indices states if multiplying when bases are the same, add the powers.
$a^m+a^n=a^{m+n}$am+an=am+n
The second product above can be rewritten as:
$10^{-6}\times10^{-9}=10^{-15}$10−6×10−9=10−15
Practice questions
Question 9
Question 10
Use your calculator to find the value of
$208\times10^6\div\left(6.5\times10^6\right)$208×106÷(6.5×106)