Standard form
🎯 In this topic you will
- Write large and small numbers in standard form
🧠 Key Words
- scientific notation
- standard form
Show Definitions
- scientific notation: A way of writing very large or very small numbers using powers of ten (e.g., 3 × 10⁸).
- standard form: Another name for scientific notation, especially in UK math, written as a number between 1 and 10 multiplied by a power of 10.
🔢 Multiplying by powers of ten
Look at these numbers
4.67 × 10 = 46.7
4.67 × 102 = 467
4.67 × 103 = 4670
4.67 × 106 = 4 670 000
🔎 Reasoning Tip
Expanding powers of 10: \( 4.67 \times 10^2 \) is the same as \( 4.67 \times 100 \) or \( 4.67 \times 10 \times 10 \).
📏 Standard Form and Powers of 10
You can use powers of 10 in this way to write large numbers. For example, the average distance to the Sun is 149 600 000 km. You can write this as 1.496 × 108 km. This is called standard form.
You write a number in standard form as a × 10n where 1 ≤ a < 10 and n is an integer.
You can write small numbers in a similar way, using negative integer powers of 10. For example:
4.67 × 10–1 = 0.467
4.67 × 10–2 = 0.0467
4.67 × 10–3 = 0.00467
4.67 × 10–7 = 0.00000467
🔎 Reasoning Tip
Negative powers of 10: Think of \( 4.67 \times 10^{-1} \) as \( 4.67 \div 10 \).
🔬 Scientific Use of Standard Form
Small numbers occur often in science. For example, the time for light to travel 5 metres is 0.000 000 017 seconds. In standard form, you can write this as 1.7 × 10–8 seconds.
🔎 Reasoning Tip
Standard form: Standard form is also sometimes called scientific notation.
❓ EXERCISES
1. Write these numbers in standard form:
a) 300 000
b) 320 000
c) 328 000
d) 328 710
👀 Show answer
b) $3.2 \times 10^5$
c) $3.28 \times 10^5$
d) $3.2871 \times 10^5$
2. Write these numbers in standard form:
a) 63 000 000
b) 488 000 000
c) 3 040 000
d) 520 000 000 000
👀 Show answer
b) $4.88 \times 10^8$
c) $3.04 \times 10^6$
d) $5.2 \times 10^{11}$
3. These numbers are in standard form. Write each number in full:
a) $5.4 \times 10^3$
b) $1.41 \times 10^6$
c) $2.337 \times 10^{10}$
d) $8.725 \times 10^7$
👀 Show answer
b) 1 410 000
c) 23 370 000 000
d) 87 250 000
4. Here are the distances of some planets from the Sun.
Write each distance in standard form.
| Planet | Distance (km) |
|---|---|
| Mercury | 57 900 000 |
| Mars | 227 900 000 |
| Uranus | 2 870 000 000 |
👀 Show answer
Mars: $2.279 \times 10^8$ km
Uranus: $2.87 \times 10^9$ km
5. Here are the areas of four countries.
| Country | Area (km²) |
|---|---|
| China | $9.6 \times 10^6$ |
| Indonesia | $1.9 \times 10^6$ |
| Russia | $1.7 \times 10^7$ |
| Kazakhstan | $2.7 \times 10^6$ |
a) Which country has the largest area?
b) Which country has the smallest area?
c) Copy and complete this sentence with a whole number:
The largest country is approximately … times larger than the smallest country.
👀 Show answer
b) Indonesia has the smallest area: $1.9 \times 10^6$
c) $1.7 \times 10^7 \div 1.9 \times 10^6 \approx 8.95 \approx \mathbf{9}$
So the largest country is approximately 9 times larger than the smallest.
6. Write these numbers in standard form:
a) 0.000007
b) 0.000812
c) 0.00006691
d) 0.000000205
👀 Show answer
b) $8.12 \times 10^{-4}$
c) $6.691 \times 10^{-5}$
d) $2.05 \times 10^{-7}$
7. These numbers are in standard form. Write each number in full:
a) $1.5 \times 10^{-3}$
b) $1.234 \times 10^{-5}$
c) $7.9 \times 10^{-8}$
d) $9.003 \times 10^{-4}$
👀 Show answer
b) 0.00001234
c) 0.000000079
d) 0.0009003
8. The mass of an electron is $9.11 \times 10^{-31}$ kg.
This is 0.000…911 kg.
a) How many zeros are there between the decimal point and the 9?
b) Work out the mass of 1 million electrons.
Give the answer in kilograms in standard form.
👀 Show answer
b) $9.11 \times 10^{-31} \times 1\,000\,000 = 9.11 \times 10^{-25}$ kg
9. Here are four numbers:
$w = 9.81 \times 10^{-5}$ $x = 2.8 \times 10^{-4}$ $y = 9.091 \times 10^{-5}$ $z = 4 \times 10^{-4}$
a) Which number is the largest?
b) Which number is the smallest?
👀 Show answer
b) Smallest number is $y = 9.091 \times 10^{-5}$
10. a) Explain why the number $65 \times 10^4$ is not in standard form.
b) Write $65 \times 10^4$ in standard form.
c) Write $48.3 \times 10^6$ in standard form.
👀 Show answer
b) $65 \times 10^4 = 6.5 \times 10^5$
c) $48.3 \times 10^6 = 4.83 \times 10^7$
11. Write these numbers in standard form:
a) $15 \times 10^{-3}$
b) $27.3 \times 10^{-4}$
c) $50 \times 10^{-9}$
👀 Show answer
b) $2.73 \times 10^{-3}$
c) $5.0 \times 10^{-8}$
12. Do these additions. Write the answers in standard form:
a) $2.5 \times 10^6 + 3.6 \times 10^6$
b) $4.6 \times 10^5 + 1.57 \times 10^5$
c) $9.2 \times 10^4 + 8.3 \times 10^4$
👀 Show answer
b) $6.17 \times 10^5$
c) $1.75 \times 10^5$
13. Do these additions. Write the answers in standard form:
a) $4.5 \times 10^{-6} + 3.1 \times 10^{-6}$
b) $5.12 \times 10^{-5} + 2.9 \times 10^{-5}$
c) $9 \times 10^{-8} + 7 \times 10^{-8}$
👀 Show answer
b) $8.02 \times 10^{-5}$
c) $1.6 \times 10^{-7}$
14. a) Multiply these numbers by 10. Give each answer in standard form:
i) $7 \times 10^5$
ii) $3.4 \times 10^6$
iii) $4.1 \times 10^{-5}$
iv) $1.37 \times 10^{-4}$
b) Generalise your results from part a.
c) Describe how to multiply or divide a number in standard form by 1000.
👀 Show answer
i) $7 \times 10^6$
ii) $3.4 \times 10^7$
iii) $4.1 \times 10^{-4}$
iv) $1.37 \times 10^{-3}$
b) When you multiply by 10, the power of 10 increases by 1.
c) When you multiply by 1000, increase the power of 10 by 3.
When dividing by 1000, decrease the power of 10 by 3.
⚠️ Be careful!
A number is only in standard form if it is written as $a \times 10^n$, where $1 \leq a < 10$. For example, $65 \times 10^4$ is not in standard form — you must rewrite it as $6.5 \times 10^5$.