Engineers will often decide whether to use exact values or approximated values. When calculating a length of metal, an engineer may decide \sqrt{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 cm. This is an approximation of the exact value that has been rounded to the nearest whole number.

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 to two decimal places.

Units	.	Tenths	Hundredths	Thousandths	Ten-thousandths
0	.	0	5	3	7

The place value in the third decimal place is 3, so round down to 0.05. This number is now 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 then round down.
If it's greater than or equal to 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.

Examples

Example 1

Indicate if the following is true or false:

\sqrt{530} is an exact value.

Worked Solution

Create a strategy

Observe if 530 is one more than a perfect square.

Apply the idea

530 is one more than a perfect square.

So, it is true that \sqrt{530} is an exact value.

My calculator states that \sqrt{530} is 23.021\,728\,866. Is this still exact?

Worked Solution

Create a strategy

Check if we multiply 23.021\,728\,866 by itself will give us 530.

Apply the idea

Since in calculator we only get a rounded value, this is not an exact value.

Example 2

Round 85.144\,6 to the nearest hundredth.

Worked Solution

Create a strategy

Use a place value table to round the value.

Apply the idea

Tens	Units	.	Tenths	Hundredths	Thousandths	Ten-thousandths
8	5	.	1	4	4	6

To round to the nearest hundredth we look at the thousandths column.

4 is less than 5, so we round down and leave the digit in the hundredths place as 4.

So the rounded answer is 85.14.

Example 3

Round 7.034\,500 to four decimal places.

Worked Solution

Create a strategy

Use a place value table to round the value.

Apply the idea

Units	.	Tenths	Hundredths	Thousandths	Ten thousandths	Hundred thousandths	Millionths
7	.	0	3	4	5	0	0

To round to the nearest ten-thousandth we look at the hundred-thousandths column.

0 is less than 5, so we round down and leave the digit in the ten-thousandths place as 5.

So the rounded answer is 7.034\,5.

Idea summary

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 then round down.
If it's greater than or equal to 5 then round up.

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 95\,446 people at the football game. A newspaper may report 95\,000 attended the game, in this case they have rounded to 2 significant figures (or to the nearest thousand). Rounding to 3 significant figures would be 95\,400.

The image shows a significant figures that discussed different parts. Ask your teacher for more information.

Significant figures are:

all non-zero digits
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 is slightly different than rounding.

Consider the width of a human hair is approximately 0.023 cm. Rounding this to 1 significant figure it would be 0.02 (two decimal places). Rounding to 2 significant figures it would be 0.023 (three decimal places).

Examples:

10\,432 to 3 significant figures =10\,400 The zero between the 1 and the 4 is counted as significant.
1.040\,052 to 3 significant figures =1.04 The zero between the 1 and 4 is counted as significant.

Examples

Example 4

Round off 461\,585 to three significant figures.

Worked Solution

Create a strategy

Count the number of significant figures you want from the first nonzero digit. If the next digit is greater or equal to 5 round up, otherwise round down.

Apply the idea

Starting from 4 we count the first 4 significant figures as 461.The next digit is 5 so we need to round up and replace the remaining digits by zeros:

\displaystyle 461\,585

\displaystyle \approx

\displaystyle 462\,000

Example 5

Round off 0.004\,039\,531 to two significant figures.

Worked Solution

Create a strategy

Count the number of significant figures you want from the first nonzero digit. If the next digit is greater or equal to 5 round up, otherwise round down.

Apply the idea

Starting from 0 we count the first 3 significant figures as 0.004.The next digit is 0 so we need to round down and replace the remaining digits by zeros:

\displaystyle 0.004\,039\,531

\displaystyle \approx

\displaystyle 0.004\,0

Idea summary

Significant figures are:

all non-zero digits
all non-zero digits zeros appearing between two non-zero digits
trailing zeros to the right in a number containing a decimal point.

Scientific notation (standard form)

Scientific notation, or standard form, is a way of writing very big or very small numbers in a compact way. For example, the mass of the sun is 1\,988\,000\,000\,000\,000\,000\,000\,000\,000\,000 kg. This is a very big number and it would be easy to accidentally leave off a zero. Using scientific notation the number can be written as 1.988 \times 10^{30} kg.

Scientific notation uses the form: a \times 10^{b}, where a is a number between 1 and 10 and b is an integer (positive or negative) that is expressed as an index of 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.

Significant figures in scientific notation:

6.531\,26 \times 10^{7} to 4 significant figures = 6.531 \times 10^{7}
6.000\,02 \times 10^{8} to 3 significant figures = 6.00 \times 10^{8}

Examples

Example 6

Express 0.07 in scientific notation.

Worked Solution

Create a strategy

We convert first the given decimal into fraction and express in the form a \times 10^{b} where a is a number between 1 and 10, and b is an integer.

Apply the idea

\displaystyle 0.07	\displaystyle =	\displaystyle \dfrac{7}{100}	Convert the decimal into fraction
	\displaystyle =	\displaystyle 7 \times 10^{-2}	Rewrite as scientific notation

Example 7

Express the following number as a basic numeral: 6\times 10^{7}

Worked Solution

Create a strategy

Use the fact that 10^7=10\,000\,000.

Apply the idea

\displaystyle 6\times 10^7	\displaystyle =	\displaystyle 6\times 10\,000\,000	Evaluate 10^7
	\displaystyle =	\displaystyle 60\,000\,000	Evaluate the multiplication

Example 8

If we round to one significant figure, sound travels at a speed of approximately 0.3 kilometres per second, while light travels at a speed of approximately 300\,000 kilometres per second.

Express the speed of sound in kilometres per second in scientific notation.

Worked Solution

Create a strategy

Scientific notation uses the form: a \times 10^{b}.

Apply the idea

0.3 = 3 \times 10^{-1}\text{ km/s}

Express the speed of light in kilometres per second in scientific notation.

Worked Solution

Create a strategy

Scientific notation uses the form: a \times 10^{b}.

Apply the idea

300\,000 = 3 \times 10^{5}\text{ km/s}

How many times faster does light travel than sound?

Worked Solution

Create a strategy

Divide the speed of light from part (b) by the speed of sound from part (a).

Apply the idea

\displaystyle \text{Number of times faster}	\displaystyle =	\displaystyle \dfrac{3 \times 10^{5}}{3 \times 10^{-1}}	Divide the speed of light by the speed of sound
	\displaystyle =	\displaystyle \dfrac{ 10^{5}}{10^{-1}}	Cancel out like terms
	\displaystyle =	\displaystyle 10^{5-(-1)}	Simplify the expression using index rules
	\displaystyle =	\displaystyle 10^{6}	Evaluate
	\displaystyle =	\displaystyle 1\,000\,000	Simplify

Idea summary

Scientific notation uses the form:

\displaystyle a \times 10^{b}

\bm{a}

is a number between 1 and 10

\bm{b}

is an integer (positive or negative) that is expressed as an index of 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.

Calculation with scientific notation

Multiplying either 1\,000\,000 by 1\,000\,000\,000 or 0.000\,001 by 0.000\,000\,001 is made easier when these numbers are written using scientific notation.

For example, the first product above can be rewritten as 10^{6}\times 10^{9}=10^{15}.

The first law of indices states if multiplying when bases are the same, add the powers. a^{m} + a^{n} = a^{m + n}

The second product above can be rewritten as 10^{-6}\times 10^{-9}=10^{-15}

Examples

Example 9

Use your calculator to find the value of 208 \times 10^{6}\div (6.5 \times 10^{6})

Worked Solution

Create a strategy

Enter into the calculator the expression after the \div sign into your calculator should be placed inside brackets.

Apply the idea

\displaystyle \text{Value}	\displaystyle =	\displaystyle 208 \times 10^{6}\div (6.5 \times 10^{6})	Write the expressions
	\displaystyle =	\displaystyle 32	Evaluate

Idea summary

The first law of indices states if multiplying when bases are the same, add the powers. a^{m} + a^{n} = a^{m + n}

Outcomes

U2.AoS4.8

scientific notation, exact and approximate answers, significant figures and rounding

U2.AoS4.15

distinguish between exact and approximate answers and write approximate answers correct to a given number of decimal places or significant figures

10.01 Significant figures and scientific notation

Ideas

Approximation and round numbers

Examples

Example 1

Example 2

Example 3

Significant figures

Examples

Example 4

Example 5

Scientific notation (standard form)

Examples

Example 6

Example 7

Example 8

Calculation with scientific notation

Examples

Example 9

Outcomes

U2.AoS4.8

U2.AoS4.15

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