Scientific notation and decimal conversion | Year 9

01 / LEARN

The idea, step by step

One standard form

Scientific notation writes a positive number as $a \times 10^{n}$ , where $1 \leq a < 10$ and $n$ is an integer. This format makes the order of magnitude visible and avoids counting long strings of zeros.

A coefficient such as 42 or 0.42 is not in standard scientific notation. Adjust the coefficient and exponent together so that the value remains unchanged. Zero itself is written as 0 rather than having a unique scientific-notation 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 large number needs a positive power of ten because the coefficient is multiplied to become larger. A small positive number below 1 needs a negative power because a reciprocal power makes the coefficient smaller.

When converting back, think of multiplication by a power of ten rather than a rule about moving a decimal point without context. Check your answer roughly: $6 \times 10^{- 3}$ is close to six thousandths, not six thousand.

A common trap

Counting only the zeros when choosing an exponent.

Instead: Count the place-value scaling from the coefficient; the number of zeros alone is not reliable.

02 / WORKED EXAMPLES

See the thinking

Example 1

A large distance

Write $7300000$ in scientific notation.

Choose coefficient 7.3, between 1 and 10.
To recover 7300000 from 7.3, multiply by 1000000.
$1000000 = 10^{6}$ , so the number is $7.3 \times 10^{6}$ .

7.3 \times 10^{6}

Example 2

A small measurement

Write $0.000056$ in scientific notation.

Choose coefficient 5.6.
The value is below 1, so the exponent must be negative.
$5.6 \times 10^{- 5} = 5.6 \div 100000 = 0.000056$ .

5.6 \times 10^{- 5}

Check your understanding

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

Show answer

$0.00804$

Divide 8.04 by 1000. The zero between 8 and 4 remains in its correct place.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF 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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Comparing/calculating very large and small quantities

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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 ↗

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