Scientific Notation (Standard Form) Explained with Examples

Q: When is the power of ten negative?

When the number is between 0 and 1. For example 0.003 = 3 × 10^-3.

How to write huge and tiny numbers compactly with powers of ten, how to convert in both directions, and how to multiply and divide numbers in scientific notation.

3 min read Supports 3 lessons

The Sun is about $150\,000\,000$ km away. A bacterium is about $0.000002$ m wide. Numbers like these are hard to read and easy to miscount. Scientific notation writes them as a manageable number times a power of ten.

150\,000\,000 = 1.5 \times 10^{8} \qquad\qquad 0.000002 = 2 \times 10^{-6}

Scientific notation

a \times 10^n, \qquad 1 \le a < 10,\; n \text{ a whole number}

The first part has exactly one non-zero digit before the decimal point.

Big numbers

Move the decimal point left until one non-zero digit is in front of it. The number of places you moved it is the power of ten.

$384\,000$
Average distance to the Moon in km.

1/3

Small numbers

For numbers less than $1$ , move the point to the right instead. Moving right makes the power negative, because the original number was small.

$0.00042$
Move the point right until one non-zero digit is in front.

1/3

Key idea

Big numbers get positive powers; small numbers (between $0$ and $1$ ) get negative powers. Use that as a quick sense check.

Converting back

Read the power as instructions: $10^5$ means move the point $5$ places to the right, $10^{-3}$ means $3$ places to the left. $7.1 \times 10^{-3} = 0.0071$ .

Multiplying and dividing

Multiply the front numbers, and multiply the powers of ten by adding their exponents. If the front number ends up $10$ or more, adjust it.

Worked example

Compute $(3 \times 10^5)(4 \times 10^3)$ .

1
Multiply the numbers and add the exponents.
$12 \times 10^{8}$

Worked example

Compute $\dfrac{6 \times 10^{2}}{8 \times 10^{-3}}$ .

1
Divide the numbers and subtract the exponents.
$0.75 \times 10^{2 - (-3)} = 0.75 \times 10^{5}$

Watch out

$12 \times 10^8$ is a correct value but not in scientific notation. The front number must be at least $1$ and less than $10$ .

Tip

Adding and subtracting are easier if you first rewrite both numbers with the same power of ten, or convert to ordinary numbers.

Calculators often show scientific notation with an E: `4.2E-4` means $4.2 \times 10^{-4}$ .

Common questions

Is standard form the same as scientific notation?: Yes, in the UK "standard form" usually means scientific notation: a number from $1$ up to (but not including) $10$ , multiplied by a power of ten.
When is the power of ten negative?: When the number is between $0$ and $1$ . For example $0.003 = 3 \times 10^{-3}$ .

Practise scientific notation

The Scientific Notation lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

Is standard form the same as scientific notation?

Yes, in the UK "standard form" usually means scientific notation: a number from $1$ up to (but not including) $10$ , multiplied by a power of ten.

When is the power of ten negative?

When the number is between $0$ and $1$ . For example $0.003 = 3 \times 10^{-3}$ .