Standard Form (Scientific Notation)

What Is Standard Form?

Standard form is a way of writing very large or very small numbers compactly.

$$A \times 10^n$$

where $1 \leq A < 10$ and $n$ is an integer (positive, negative or zero).

Examples:

• $3{,}700{,}000 = 3.7 \times 10^6$
• $0.000052 = 5.2 \times 10^{-5}$
• $4.83 \times 10^4 = 48{,}300$

Converting to Standard Form

Large numbers (positive index): Count how many places the decimal point moves to the left.

$$450{,}000 \rightarrow 4.5 \times 10^5 \quad (\text{decimal moved 5 places left})$$

Small numbers (negative index): Count how many places the decimal point moves to the right.

$$0.0000307 \rightarrow 3.07 \times 10^{-5} \quad (\text{decimal moved 5 places right})$$

Converting from Standard Form

Multiply out: move the decimal point.

• $6.2 \times 10^3$: move decimal 3 places right → $6200$
• $9.1 \times 10^{-4}$: move decimal 4 places left → $0.00091$

Multiplying and Dividing in Standard Form

Multiply: Multiply the $A$ values and add the indices.

$$(3 \times 10^4) \times (2 \times 10^3) = 6 \times 10^7$$

If the result for $A$ falls outside $[1, 10)$, adjust:

$$(8 \times 10^5) \times (4 \times 10^3) = 32 \times 10^8 = 3.2 \times 10^9$$

Divide: Divide the $A$ values and subtract the indices.

$$\frac{9 \times 10^7}{3 \times 10^4} = 3 \times 10^3$$

Adding and Subtracting in Standard Form

Convert both numbers to the same power of 10 first, then add/subtract the $A$ values.

$$3.2 \times 10^5 + 4.7 \times 10^4 = 32 \times 10^4 + 4.7 \times 10^4 = 36.7 \times 10^4 = 3.67 \times 10^5$$

WJEC Exam Tips

• WJEC expects the calculator paper answer to be in proper standard form ($1 \leq A < 10$).
• On non-calculator papers you are expected to add/subtract by equalising the powers.
• Always write down any intermediate step to earn method marks.
• Watch out: $0.6 \times 10^4$ is NOT standard form — adjust to $6 \times 10^3$.
• Real-world contexts include: distances in astronomy (light-years, km), nanometres in chemistry, large populations.