Standard Form

La méthode, étape par étape

1. 1

   Place the decimal point after the first non-zero digit

   4,780,000 → 4.78

   A must be at least 1 and under 10, so there is exactly one non-zero digit before the point. This is the rule students most often break, writing 47.8 × 10⁵ — which is a correct value in an incorrect form, and exam boards mark it wrong.

2. 2

   Count how far the point moved

   4780000. → 4.78 — the point moved 6 places left → × 10⁶

   Moving the point left means the number was large, so the power is positive. Counting places rather than counting zeros matters, because numbers like 4,780,000 have fewer zeros than the power suggests.

3. 3

   Small numbers take a negative power

   0.00032 → 3.2 × 10⁻⁴

   The point moved four places right, so the power is negative four. The negative index is not making the number negative — 3.2 × 10⁻⁴ is a small positive number, and confusing those is the topic's most common misreading.

4. 4

   Multiply by handling numbers and powers separately

   (3 × 10⁵) × (2 × 10³) = 6 × 10⁸

   Multiply the A parts, add the powers. If the resulting A falls outside 1 to 10, adjust: 5 × 10³ times 4 × 10² is 20 × 10⁵, which must be rewritten as 2 × 10⁶.

5. 5

   Divide by dividing and subtracting

   (8 × 10⁷) ÷ (2 × 10³) = 4 × 10⁴

   Divide the A parts and subtract the powers. The same adjustment applies — if the division gives an A below 1, shift the point right and reduce the power by one to bring it back into range.

What standard form is for

Some quantities are simply not writable in ordinary notation. The mass of an electron has thirty zeros after the decimal point before a significant digit appears, and nobody can read that, count it reliably, or compare two of them by eye. Standard form makes the size of a number visible in a single exponent.

It also makes comparison instant. Deciding whether 3.1 × 10⁸ or 2.9 × 10⁹ is larger takes a glance at the powers, where the ordinary forms would require counting digits carefully. This is why every science subject uses it constantly.

• Form: A × 10ⁿ with 1 ≤ A < 10
• Large numbers → positive power
• Small numbers → negative power
• Multiply: multiply A parts, add powers
• Divide: divide A parts, subtract powers

A must be between 1 and 10

Writing 47.8 × 10⁵ or 0.478 × 10⁷ gives the right value in the wrong form, and both lose marks. The constraint exists so that every number has exactly one standard form, which is what makes two numbers directly comparable by their exponents.

The fix after a multiplication is mechanical: if A is 10 or more, move the point one place left and add one to the power; if A is below 1, move it right and subtract one. Students who check the range every time stop losing these marks entirely.

Adding requires the same power first

Multiplication and division work on the parts separately; addition and subtraction do not. To add 3 × 10⁵ and 4 × 10⁴, both must first be expressed with the same power — 3 × 10⁵ and 0.4 × 10⁵ — giving 3.4 × 10⁵.

This mirrors fractions, where multiplication needs no common denominator and addition does. Pointing out the parallel helps, because students who have accepted it once for fractions accept it faster the second time.

How we teach standard form

We teach it through quantities students find genuinely striking — the distance to the sun, the size of a virus, the national debt — because the notation only justifies itself when the ordinary form is visibly unusable.

We then drill the range check as a separate reflex. Most marks lost in this topic are lost to a correct value written outside the permitted form, which is an entirely avoidable way to lose a mark on a question you could do.