Standard form | Mathematics

Unit 1 · Number

4 Standard Form

The Sun is 150 000 000 km away. A water molecule is 0.000 000 000 27 m wide. Writing all those zeros is slow and easy to get wrong — so mathematicians invented a shorthand. This is it.

🎯 By the end of this you can…

✓Spot instantly whether a number is written in standard form.
✓Turn any big or small number into standard form.
✓Turn standard form back into an ordinary number.

💡 The idea

Standard form is a × 10ⁿ — with one strict condition

Every number in standard form looks like this:

a \times 1 0^{n}

where $n$ is a whole number — and here is the part everyone forgets:

$1 \leq a < 10$

In plain words: the first part must have exactly one digit before the decimal point, and that digit cannot be zero. So 6.3 ✅, 9.99 ✅, but 15 ❌ and 0.7 ❌.

🗂️ Is it really in standard form?

0 / 6 correct

Tap a card, then tap the box you think it belongs in.

📋 Ordinary number → standard form

1
Put a decimal point so there is exactly one non-zero digit in front of it. That gives you $a$ .
2
Count the jumps the decimal point made to get there.
3
Started with a BIG number (bigger than 10)? The power is positive.
4
Started with a SMALL number (less than 1)? The power is negative.
5
Check: does your answer feel the right size? A positive power means big; a negative power means tiny.

✏️ Worked example

Write 6 300 000 in standard form.

1
Where does the point go? One non-zero digit in front: 6.3
2
Count the jumps from 6.300000 out to 6300000. — that is 6 jumps.
3
6 300 000 is a big number, so the power is positive.
4
Check: $1 0^{6}$ is a million, and 6.3 million is exactly what we started with ✓

✅ Answer: $6.3 \times 1 0^{6}$

✏️ Worked example

Write 0.000 006 21 in standard form.

1
One non-zero digit in front of the point: 6.21
2
Count the jumps from 0.00000621 across to 6.21 — that is 6 jumps.
3
The number is tiny (less than 1), so the power is negative.
4
Check: $1 0^{- 6}$ is a millionth — and our number is about six millionths ✓

✅ Answer: $6.21 \times 1 0^{- 6}$

⚠️ The classic trap

Sophie’s mistake. She converted 0.00027 and wrote $27 \times 1 0^{- 5}$ .

Work it out and her number is correct: 27 × 0.00001 really is 0.00027. But it is not in standard form, because 27 is not between 1 and 10.

The correct answer is $2.7 \times 1 0^{- 4}$ .

Notice what happened to the power: the first part got smaller (27 → 2.7), so the power got bigger (−5 → −4). They always move opposite ways.

🎮 Big numbers → standard form

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3200

760

4730

9300

12600

83460

48455

2500100

43002

66010000

812000

700000000

50005

fifty-two thousand

two million

0 of 15 finished

🎮 Small numbers → standard form

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0.003

0.00056

0.0000006

0.0348

0.000042

0.00005401

0.00521

0.00000326

0.56

0.07

0.00004802

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📋 Standard form → ordinary number

1
Look at the power. Positive means the number is big; negative means it is small.
2
Move the decimal point that many jumps — right for a positive power, left for a negative one.
3
Fill any empty spaces with zeros.
4
Check: a positive power must give an answer bigger than 10; a negative power must give one smaller than 1.

🎮 Standard form → ordinary number

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6.1 \times 1 0^{3}

8 \times 1 0^{- 5}

9.8 \times 1 0^{1}

3.33 \times 1 0^{0}

7.034 \times 1 0^{6}

1.1 \times 1 0^{- 4}

2 \times 1 0^{3}

1.835 \times 1 0^{5}

3.0012 \times 1 0^{4}

5.8302 \times 1 0^{7}

1.276 \times 1 0^{- 3}

5.1 \times 1 0^{- 2}

2.74 \times 1 0^{- 5}

5.001 \times 1 0^{- 4}

0 of 14 finished

✏️ Worked example

A water molecule is 0.000 000 000 27 m wide. DNA is 0.000 000 002 m wide. Which is wider?

1
Water: one non-zero digit in front gives 2.7, and the jumps count to 10.
2
So water is $2.7 \times 1 0^{- 10}$ m.
3
DNA: gives 2, with 9 jumps, so $2 \times 1 0^{- 9}$ m.
4
Now compare the powers first: −9 is bigger than −10, so DNA is the wider one. Only if the powers were equal would you compare the front numbers.

✅ Answer: DNA is wider — 2 × 10⁻⁹ m against 2.7 × 10⁻¹⁰ m

🤔 Which is the larger number: 3.1 × 10⁻⁴ or 9.8 × 10⁻⁶?

🏅

You finished this lesson!

Standard form is $a \times 1 0^{n}$ with one digit before the point. Big numbers get a positive power, small numbers get a negative one — and you can always check by asking whether your answer should be bigger or smaller than 1.

Next: the power rules that make all of this work.

Keep reading — this lesson marks itself done automatically.