Standard form calculator (scientific notation)

1. Move the decimal point so there's exactly one non-zero digit in front of it: $3\,200\,000 \to 3.2$.
2. Count how many places it moved: 6.
3. Moving left gives a positive power; moving right gives a negative power.

So $3\,200\,000 = 3.2 \times 10^6$ and $0.00045 = 4.5 \times 10^{-4}$.

Move the decimal point the number of places shown by the power: right for a positive power, left for a negative one, filling gaps with zeros. $6.02 \times 10^{3} = 6020$; $7 \times 10^{-3} = 0.007$.

• Multiplying: multiply the numbers, add the powers. $(3 \times 10^4) \times (2 \times 10^5) = 6 \times 10^9$.
• Dividing: divide the numbers, subtract the powers. $(8 \times 10^6) \div (2 \times 10^2) = 4 \times 10^4$.
• Tidy up: if the front number isn't between 1 and 10, adjust: $12 \times 10^5 = 1.2 \times 10^6$.

More in our standard form guide and the laws of indices.

• Writing $32 \times 10^5$: the front number must be less than 10.
• Getting the sign of the power wrong for small numbers: numbers below 1 have negative powers.
• On a calculator, E or ×10^x means "times ten to the power", so 4.5E-4 is $4.5 \times 10^{-4}$.

Science uses standard form for very large and very small quantities:

Being comfortable converting and calculating with these numbers helps in physics and chemistry as much as in maths.

Compare the powers of 10 first: $4 \times 10^5$ is bigger than $9 \times 10^4$, because $10^5$ is ten times $10^4$. Only when the powers are equal do you compare the front numbers. For negative powers, remember $10^{-3}$ is bigger than $10^{-5}$.