Standard form is a way of writing very large or very small numbers using powers of 10.
For GCSE Grade 5, you need to be able to write numbers in standard form, convert them back to ordinary numbers, and calculate with them.
What is standard form?
A number in standard form looks like this:
\[ \boxed{a\times10^n} \]
Where:
- $a$ is a number between 1 and 10, but not including 10.
- $n$ is an integer (a whole number, zero, or a negative whole number).
For example:
\[ \begin{aligned} 4500 &= 4.5\times10^3\\ 0.0062 &= 6.2\times10^{-3} \end{aligned} \]
Writing large numbers in standard form
Example 1: Write 56,000 in standard form.
- Move the decimal point until the number is between 1 and 10.
- $56,000\rightarrow5.6$
- Count how many places the decimal point moved: 4 places to the left.
$56,000=5.6\times10^4$
Remember: Large numbers have a positive power of 10.
More examples:
| Ordinary number | Standard form |
|---|---|
| 300 | $3\times10^2$ |
| 7,200 | $7.2\times10^3$ |
| 8,500,000 | $8.5\times10^6$ |
Writing small numbers in standard form
Example 2: Write 0.0048 in standard form.
- Move the decimal point until the number is between 1 and 10.
- $0.0048\rightarrow4.8$
- Count how many places it moved: 3 places to the right.
$0.0048=4.8\times10^{-3}$
Remember: Small numbers between 0 and 1 have a negative power of 10.
More examples:
| Ordinary number | Standard form |
|---|---|
| 0.7 | $7\times10^{-1}$ |
| 0.025 | $2.5\times10^{-2}$ |
| 0.000091 | $9.1\times10^{-5}$ |
Converting standard form into ordinary numbers
This is the reverse process.
- A positive power means move the decimal point to the right.
- A negative power means move the decimal point to the left.
Example 3: Write $3.42\times10^4$ as an ordinary number.
The power is $+4$, so move the decimal point 4 places to the right.
\[ 3.42\rightarrow34,200 \]
Answer: 34,200
Example 4: Write $6.7\times10^{-3}$ as an ordinary number.
The power is $-3$, so move the decimal point 3 places to the left.
\[ 6.7\rightarrow0.0067 \]
Answer: 0.0067
Calculating with standard form
At Grade 5, you may also need to multiply and divide numbers in standard form.
Multiplication
Multiply the numbers at the front, then add the powers.
Example:
\[ (2\times10^3)\times(4\times10^2) \]
Step 1: Multiply the front numbers.
\[ 2\times4=8 \]
Step 2: Add the powers.
\[ 10^{3+2}=10^5 \]
Answer:
\[ \boxed{8\times10^5} \]
Division
Divide the numbers at the front, then subtract the powers.
Example:
\[ \frac{8\times10^6}{2\times10^2} \]
Step 1: Divide the front numbers: $8\div2=4$.
Step 2: Subtract the powers: $6-2=4$.
Answer:
\[ \boxed{4\times10^4} \]
If your answer is not in standard form, adjust it so the front number is at least 1 but less than 10.
The three rules to remember
Large numbers
Positive powers: $5,000=5\times10^3$.
Small numbers
Negative powers: $0.005=5\times10^{-3}$.
Calculations
Multiply: add the powers. Divide: subtract the powers.
GCSE exam tip: Always check that the number at the front of your final answer is between 1 and 10. For example, $45\times10^3$ is not standard form; write it as $4.5\times10^4$.