GCSE · Physics · Edexcel · Spec 1PH0

Significant figures and standard form

Your calculator says 26 588.235 29. Is that the answer? Almost never — every digit you write is a promise you measured it.

Sort the numbers

Two questions about one number

For each number, tick every set it belongs to — one, both or neither. Two quick reminders: standard form looks like A × 10ⁿ, where A is at least 1 and less than 10. Significant figures are counted from the first non-zero digit.

A Written in standard form
B Given to 3 significant figures

3.00 × 10⁸
0.00452
1.6 × 10⁻¹⁹
45.2 × 10³
4.07 × 10⁵
0.0070
2.658823529 × 10⁴
0.52 × 10⁻³
5 × 10⁻⁷

Watch it done: from calculator display to answer

Problem

A space probe travels 4.52 × 10⁹ m in 1.7 × 10⁵ s. Calculate its average speed, using speed = distance ÷ time.

Check your thinking

Which of these would you sign your name to?

Here are four things students often say about writing numbers and answers.

Pick the one you agree with most — then say how sure you are.

Keeping every digit from the calculator makes my answer more accurate.In 0.0045 every digit counts, so it has 5 significant figures.45 × 10³ is already in standard form — it has the × 10 part.My answer can't be more precise than the least precise measurement I used.

How sure are you?

I'm sureI think soI'm guessing

Keeping every digit from the calculator makes my answer more accurate.

This is a common misconception

It feels that way, but the extra digits aren't information — they're noise. If a time was measured to 2 significant figures, the calculator's eighth digit is a guess dressed up as a fact. Accuracy comes from the measurements, not from the display. Round the final answer to match the least precise value.

In 0.0045 every digit counts, so it has 5 significant figures.

This is a common misconception

The zeros at the front are only placeholders — they show where the decimal point sits. Write the same number as 4.5 × 10⁻³ and they disappear without changing its value. So 0.0045 has 2 significant figures.

45 × 10³ is already in standard form — it has the × 10 part.

This is a common misconception

The power of ten is only half the rule. The number in front must be at least 1 and less than 10, and 45 isn't. Move the decimal point one place left and add one to the power: 4.5 × 10⁴. Same value, now in standard form.

My answer can't be more precise than the least precise measurement I used.

This matches the evidence

Exactly right. A chain is only as strong as its weakest link, and an answer is only as precise as its least precise value. That is why you keep the full value while working, then round once at the end to match it.

Before you write any final answer, ask two questions: is it very large or very small, so standard form helps? And how many significant figures can my data actually back up?

Your turn

Finish the method

A charge of 0.00385 C flows through a wire in 1.6 × 10³ s. Calculate the current, using current = charge ÷ time. Give your answer in standard form, to a sensible number of significant figures.

current = 0.00385 ÷ (1.6 × 10³)
Calculator display: 0.000 002 406 25
missing step

Which line is step 3?

2.406 25 × 10⁶2.406 25 × 10⁻⁶2.406 25 × 10⁻⁵24.0625 × 10⁻⁷

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Every digit you write is a promise that you measured it.

What you need to know

Standard form is A × 10ⁿ, where A is at least 1 and less than 10, and n is a whole number. Numbers of 10 or more get a positive power; numbers smaller than 1 get a negative power.
Significant figures are counted from the first non-zero digit. Zeros in front of it only hold the place, so they never count. Zeros between other digits do count, and so do zeros at the end of a decimal (3.0 has 2 significant figures).
A final answer should usually have the same number of significant figures as the least precise value used to calculate it. More claims precision you don't have; fewer throws away precision you do.
Standard form earns its place with very large or very small numbers, where writing out all the zeros is clumsy and easy to get wrong.

The big picture

Standard form writes a number as A × 10ⁿ, with A at least 1 and less than 10 — handy when a number is very large or very small. Significant figures show how precisely a number is known. In a calculation, keep the full calculator value while you work, then round the final answer to the same number of significant figures as the least precise value you started with.

Key points

1Standard form and significant figures answer different questions: how a number is written, and how precisely it is known.

2Changing a number into standard form never changes its value — only how it looks.

3Keep the full calculator value while you work, and round only the final answer.

4More digits do not mean more accuracy: the least precise measurement sets the limit.

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Worked example

Problem

A wave has a frequency of 4.6 × 10¹⁴ Hz and a wavelength of 6.52 × 10⁻⁷ m. Calculate its speed, using wave speed = frequency × wavelength.

⚠ Watch out

Counting the zeros after the decimal point to get the power of ten. In 0.0006 there are three zeros, but the decimal point has to move four places to reach the 6, so it is 6 × 10⁻⁴, not 6 × 10⁻³.

🧠

Memory hook

Every digit is a promise. Only write the digits your data can keep.

✓

Check yourself

No calculator: how many significant figures does 0.0306 have, and what is it in standard form? (3 — the 3, the middle 0 and the 6; 3.06 × 10⁻².)

Flashcards

(6)

What does a number in standard form look like?

A × 10ⁿ, where A is at least 1 and less than 10, and n is a whole number.

How can you tell from the power of ten whether a number in standard form is big or small?

A positive power means a number of 10 or more; a negative power means a number smaller than 1.

Where do you start counting significant figures?

At the first non-zero digit. Zeros in front of it only show where the decimal point is.

Which zeros DO count as significant figures?

Zeros between non-zero digits (4.07 has 3), and zeros at the end of a decimal (3.0 has 2).

How many significant figures should a calculated answer have?

Usually the same number as the least precise value used in the calculation.

When should you round during a calculation?

Only at the end. Keep the full calculator value through the working, then round the final answer once.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Two questions about one number

Which of these would you sign your name to?

Finish the method

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