Standard form writes a very large or very small number as a number between 1 and 10 multiplied by a power of 10. For a small number, count how many places the decimal point moves to the right, and that count becomes a negative power. It shows up in measurements of cells, atoms, wavelengths and small masses.
This lesson follows applying index laws with negative powers, because the negative power of 10 is the same idea: 10−3 = 1/1000.
How do I convert a small number?
- Find the first non-zero digit. That digit starts your new number a.
- Place the decimal point just after it, so a is between 1 and 10.
- Count the places the decimal point moved from its old position to the new one.
- Write a × 10−n, where n is that count. The power is negative because the original number is smaller than 1.
- Check the size. 10−3 is one thousandth, so a number with 10−3 should be about a thousandth of a.
What if the measurement is in a small unit?
Convert the unit first, then fix the number, doing one thing at a time.
1 nanometre (nm) = 10−9 metres. So 85 nm = 85 × 10−9 m. This is not yet standard form, because 85 is bigger than 10.
Worked example
A virus is 85 nm long. Write its length in metres in standard form.
Step 1, convert the unit. 85 nm = 85 × 10−9 m.
Step 2, fix the first number. 85 = 8.5 × 101.
Step 3, combine the powers. 8.5 × 101 × 10−9 = 8.5 × 101 + (−9) = 8.5 × 10−8.
Step 4, check the size. 8.5 × 10−8 = 0.000 000 085 m. That is 85 billionths of a metre, which matches 85 nm.
Answer: 8.5 × 10−8 m
The mistake to watch for
The usual slip is to count zeros instead of places.
Mistaken working: 0.0000072 = 7.2 × 10−5
The student saw five zeros after the decimal point and used 5 as the power.
The correction: the decimal point has to move past the five zeros and the first digit’s position, so it moves 6 places. 0.0000072 = 7.2 × 10−6.
Always count how far the point moves to sit right after the first non-zero digit. Then check by asking whether 7.2 × 10−6 is about seven millionths. It is, so the answer is sensible.
Check yourself
Try these without a calculator, then open each answer.
1. Write 0.00081 in standard form.
Show answer
The point moves 4 places to sit after the 8: 8.1 × 10−4.
2. Write 6.02 × 10−3 as an ordinary number.
Show answer
Move the point 3 places to the left: 0.00602.
3. Write 12 × 10−5 in standard form.
Show answer
12 = 1.2 × 101, so 12 × 10−5 = 1.2 × 101−5 = 1.2 × 10−4.
Where this leads next
Once conversion is comfortable, go on to comparing quantities with different powers of ten. The module overview shows the full route. The non-calculator working trainer is useful for checking conversions by hand.
If you keep losing a mark to a power that is one out, a teacher can look at your working line by line in online one-to-one Mathematics tuition.