Scientific notation and decimal conversion | Year 9

01 / LEARN

Understand the idea.

One standard form

For a positive number, scientific notation has the form $a \times 10^{n}$ with $1 \leq a < 10$ and integer $n$ . The coefficient records the leading digits; the exponent records the place-value scale.

The coefficient and exponent work together. For example, $42 \times 10^{3} = 4.2 \times 10^{4}$ : dividing the coefficient by $10$ requires multiplying the power-of-ten factor by $10$ . Zero is written as $0$ and has no unique standard exponent.

Place value and scientific notation
Ordinary form	Scientific notation
420000	4.2 × 10⁵
42	4.2 × 10¹
0.042	4.2 × 10⁻²
0.00042	4.2 × 10⁻⁴

Decide the sign before counting places

A positive exponent makes the coefficient larger, so it represents a number at least $10$ when the exponent is positive. A negative exponent divides by a power of $10$ and represents a positive number below $1$ .

Choose the coefficient first, then count the scaling needed to recover the original value. Check by converting back. Counting zeros alone fails when there are digits within the number, such as the zero in $8.04$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A large distance

Write $7300000$ in scientific notation.

Worked solution

1
Choose a coefficient between 1 and 10.
$1 \leq 7.3 < 10$
Using 7.3 preserves the leading digits 7 and 3.
2
Find the factor needed to recover the original number.
$7 300 000 = 7.3 \times 1 000 000$
The coefficient must be scaled up by one million.
3
Write that factor as a power of ten.
$7 300 000 = 7.3 \times 10^{6}$
One million is 10 multiplied by itself six times.

Answer

$7.3 \times 10^{6}$

EXAMPLE 02

A small measurement

Write $0.000056$ in scientific notation.

Worked solution

1
Choose the standard coefficient.
$1 \leq 5.6 < 10$
The nonzero digits become the coefficient 5.6.
2
Identify the scaling from the coefficient.
$0.000056 = \frac{5.6}{100000}$
The original value is five decimal places smaller than 5.6.
3
Use a negative exponent and check.
$0.000056 = 5.6 \times 10^{- 5}$
Multiplication by 10 to the power -5 divides by 100000.

Answer

$5.6 \times 10^{- 5}$

03 / YOUR TURN

Try it yourself.

Check your understanding

Write $8.04 \times 10^{- 3}$ as an ordinary decimal.

Show answer & reasoning

$0.00804$

$8.04 \times 10^{- 3} = \frac{8.04}{1000} = 0.00804$ . Divide every place value by 1000, preserving the zero between 8 and 4.

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Write $56000$ in scientific notation.
2 marks
Write $0.0031$ in scientific notation.
2 marks
Write $2.07 \times 10^{5}$ as an ordinary number.
2 marks
Write $9 \times 10^{- 6}$ as an ordinary decimal.
2 marks
Rewrite $47 \times 10^{3}$ in standard scientific notation.
2 marks
A student writes $0.00072 = 7.2 \times 10^{4}$ . Explain and correct the error.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Write $56000$ in scientific notation.
2 marks
$56000 = 5.6 \times 10000 = 5.6 \times 10^{4}$ .
$5.6 \times 10^{4}$
Write $0.0031$ in scientific notation.
2 marks
$3.1 \div 1000 = 0.0031$ .
$3.1 \times 10^{- 3}$
Write $2.07 \times 10^{5}$ as an ordinary number.
2 marks
$2.07 \times 100000 = 207000$ .
$207000$
Write $9 \times 10^{- 6}$ as an ordinary decimal.
2 marks
$9 \div 1000000 = 0.000009$ .
$0.000009$
Rewrite $47 \times 10^{3}$ in standard scientific notation.
2 marks
Change 47 to 4.7, dividing the coefficient by 10.
Increase the power by 1 to preserve the value.
$4.7 \times 10^{4}$
A student writes $0.00072 = 7.2 \times 10^{4}$ . Explain and correct the error.
3 marks
The original number is below 1, so the power must be negative.
$7.2 \div 10000 = 0.00072$ .
$7.2 \times 10^{- 4}$

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Year 9 numerical integer exponent laws and their extension to variables; scientific notation and practical measurement applications. Excludes fractional-index and full surd algebra taught in Year 10.

Official curriculum source ↗

In this lesson

Understand the idea.

One standard form

Decide the sign before counting places

See the method in action.

A large distance

Choose a coefficient between 1 and 10.

Find the factor needed to recover the original number.

Write that factor as a power of ten.

A small measurement

Choose the standard coefficient.

Identify the scaling from the coefficient.

Use a negative exponent and check.

Try it yourself.

Check your understanding

Keep practising