KS3 · Maths

Standard form for small numbers

A microchip is 0.000002 m wide. Standard form writes that as 2 × 10⁻⁶ m, and the minus sign does not make it negative. So what does it do?

Tiny numbers on one axis

Tap each chip, starting with the atom. Moving one tick to the left means two more × 1/10 steps, so the numbers shrink fast. Watch what the exponent counts: steps, not zeros and not a minus sign on the number.

Atom radius

about 1 × 10⁻¹⁰ m

The radius of an atom is approximately 1 × 10⁻¹⁰ metres. That is ten multiplications of 1/10, so it sits at the far small end of the axis. Ten steps down and the number is still positive.

Same A, opposite exponent

3.6 × 10⁴vs3.6 × 10⁻⁴

Both numbers use the digits 3.6 and an exponent of size 4. Only the sign of the exponent is different.

Focus

What the exponent counts

3.6 × 10⁴

Four multiplications of 10

3.6 × 10⁻⁴

Four multiplications of 1/10

The insight

The count is the same, 4. The sign says which way each step goes: × 10 steps climb, × 1/10 steps descend.

Bigger or smaller than 1?

3.6 × 10⁴

Large: greater than 1

3.6 × 10⁻⁴

Small: less than 1, but still positive

Where the 3 ends up

3.6 × 10⁴

Four places above the ones column

3.6 × 10⁻⁴

Four places below the ones column

As an ordinary number

3.6 × 10⁴

36 000

3.6 × 10⁻⁴

0.00036

Writing small numbers in standard form: two methods

Problem

Write 0.00098 in standard form using a place value chart. Then write 0.0045 in standard form without one.

Maths · Standard form

What does the −4 actually do?

A number is written as 2.5 × 10⁻⁴.

Which idea is closest to what you think right now?

It makes the number negative, so 2.5 × 10⁻⁴ is below zero.It counts four multiplications of 1/10, so the number is small but still positive: 0.00025.It tells me to write four zeros after the decimal point, so the number is 0.000025.

How sure are you?

I'm sureI think soI'm guessing

It makes the number negative, so 2.5 × 10⁻⁴ is below zero.

This is a common misconception

Not quite. The minus sign belongs to the exponent, not to the number. 2.5 × 10⁻⁴ is 0.00025, which is small and positive. Each × 1/10 makes the number smaller, but it never pushes it below zero.

It counts four multiplications of 1/10, so the number is small but still positive: 0.00025.

This matches the evidence

Yes. 10⁻⁴ means 1/10⁴, which is four multiplications of 1/10. That is what the exponent counts, and it gives a number between 0 and 1.

It tells me to write four zeros after the decimal point, so the number is 0.000025.

This is a common misconception

That shortcut is a trap. Four zeros after the point would give 0.000025, and that is 2.5 × 10⁻⁵, five multiplications. Count the steps instead: 2.5 → 0.25 → 0.025 → 0.0025 → 0.00025. Four steps, but only three zeros after the point, because the 2 started in the ones column.

The minus sign on the power measures smallness, not negativity. Count the multiplications of 1/10, never the zeros.

Maths · Standard form

Your turn: back to an ordinary number

A calculator gives 24 ÷ 250 000 as 9.6 × 10⁻⁵, because the answer is too small for its display. Write that as an ordinary number, one multiplication of 1/10 at a time.

The calculator answer is 9.6 × 10⁻⁵. The −5 means five multiplications of 1/10, starting from 9.6.
Once: 9.6 × 1/10 = 0.96
missing step

Which line is step 3?

Twice: 0.96 × 1/10 = 0.0096Twice: 0.96 × 1/10 = 0.096Twice: 0.96 × 1/10 = 9.6

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Why 10⁻⁴ is tiny but not negative

What you need to know

Standard form is A × 10ⁿ, where A is at least 1 and less than 10, and n is an integer.
10⁻ⁿ means n multiplications of 1/10. It is another way of writing 1/10ⁿ.
A negative exponent gives a number less than 1, which is small but still positive.
To write a small number in standard form, make A from its significant figures, count the multiplications of 1/10 to reach the first significant figure, and make that count negative.
To get back to an ordinary number, apply the multiplications of 1/10 one at a time, or use a place value chart.

The big picture

Standard form writes a very small number as A × 10ⁿ, where A is from 1 up to 10 and n is a negative integer. The negative exponent counts how many times you multiply by 1/10, so the number is small but still positive. You can find the exponent with a place value chart or by counting multiplications of 1/10, and you can reverse the process to get back to an ordinary number.

Key points

1A × 10ⁿ is standard form only when 1 ≤ A < 10 and the number is built with multiplication.

210⁻⁴ means four multiplications of 1/10, so it makes a number small, not negative.

3The first significant figure's column heading on a place value chart gives the power of 10.

4Without a chart, count the multiplications of 1/10, not the zeros.

5A calculator shows an answer that is too small for its display in standard form, so you need to convert it back.

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Worked example

Problem

Is 41 × 10⁻³ in standard form? If it is not, rewrite it so that it is.

⚠ Watch out

Counting zeros instead of multiplications. The number 0.0006 has three zeros after the decimal point, so it is tempting to write 6 × 10⁻³. But the 6 has to travel four columns down from the ones column, so it is 6 × 10⁻⁴. Count the steps, not the zeros.

🧠

Memory hook

The minus is about SIZE, not SIGN. Every unit of a negative exponent is one more × 1/10: one more step down.

✓

Check yourself

Write 0.00063 in standard form, then write 5.1 × 10⁻² as an ordinary number. (Answers: 6.3 × 10⁻⁴ and 0.051.)

Flashcards

(13)

What is standard form?

A number written as A × 10ⁿ, where A is at least 1 and less than 10, and n is an integer (positive, negative or zero).

Is 0.6 × 10⁻⁴ in standard form?

No. A = 0.6 is less than 1. A must be at least 1 and less than 10.

Why is 8.1 ÷ 10² not standard form?

Standard form uses multiplication, not division by a power of 10. It would be written 8.1 × 10⁻².

What does 10⁻⁴ mean?

It means 1/10⁴: four multiplications of 1/10.

Does a negative exponent make the number negative?

No. It makes the number small, between 0 and 1, but still positive. The minus sign is on the exponent, not on the number.

Positive exponent or negative exponent: which gives a number greater than 1?

A positive exponent gives a number greater than 1. A negative exponent gives a number less than 1.

On a place value chart with powers of 10 as headings, how do you find the exponent?

The first significant figure sits under a column heading, and that heading is the power of 10.

Without a chart, how do you find the exponent for a small number?

Build A from the significant figures. Count the multiplications of 1/10 needed to get the first significant figure from the ones column to its place. Make that count negative.

Why is "count the zeros" an unreliable way to find the exponent?

It is easy to miscount. 0.0045 has two zeros after the decimal point, but the exponent is −3. Count multiplications of 1/10 instead.

How do you write A × 10⁻ⁿ as an ordinary number?

Multiply A by 1/10, n times, one step at a time (or use a place value chart). Each step moves every digit one column down.

Why write a tiny number in standard form?

It is shorter and more concise, no accuracy is lost, and there is less chance of an error copying all the zeros. It is also the convention.

What does a calculator do with an answer too small for its display?

It converts the answer to standard form automatically, so you need to be able to convert it back to an ordinary number.

How do you enter a number like 4 × 10⁻⁷ on a calculator?

Use the 'power of' button with base 10, or the 'power of 10' button, and then type the negative exponent.

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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What does the −4 actually do?

Your turn: back to an ordinary number

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