Grade by Grade
B. Number · Number Sense: Scientific Notation, Real Numbers, Square Roots and Percents

Very Small Numbers and Negative Exponents

B1.1About 40 minutes
Learning goals
  • I can explain what a negative exponent on 10 means, like 10⁻³ = 0.001.
  • I can write a very small decimal number in scientific notation.
  • I can change a small number in scientific notation back to standard form.
  • I can connect powers of ten to metric units like millimetres and micrometres.

Words to know

negative exponent
On a power of 10, a negative exponent means dividing by 10 that many times. 10⁻³ = 1 ÷ 10³ = 0.001.
10⁰
Equal to 1. It is the step between 10¹ = 10 and 10⁻¹ = 0.1.
micrometre (µm)
One millionth of a metre: 0.000 001 m = 10⁻⁶ m.
nanometre (nm)
One billionth of a metre: 0.000 000 001 m = 10⁻⁹ m.

From huge to tiny

Last lesson you wrote huge numbers, like the distance to the Sun, in scientific notation. Science also needs tiny numbers. A human red blood cell is only about 0.000 007 5 m across (about 7.5 micrometres). A sheet of printer paper is about 0.000 1 m thick.

Counting all those zeros after the decimal point is just as tricky. Scientific notation works for tiny numbers too. We just need a new idea: negative exponents.

A microscope view of many round, pale-red blood cells, each lighter in the centre.
Each of these red blood cells is only about 0.000 007 5 m across. You need a microscope to see them, and scientific notation to write their size easily. AI-generated image

A pattern with powers of ten

Look at the table below. Each time the exponent goes down by 1, the value is divided by 10.

Keep the pattern going past 10¹ and you get:

  • 10⁰ = 1
  • 10⁻¹ = 1/10 = 0.1
  • 10⁻² = 1/100 = 0.01
  • 10⁻³ = 1/1000 = 0.001

A negative exponent on 10 does not make a negative number. It makes a small positive number: 10⁻ⁿ means 1 divided by 10ⁿ. The exponent tells how many places the 1 sits after the decimal point.

PowerFractionDecimal
10³1 0001 000
10²100100
10¹1010
10⁰11
10⁻¹1/100.1
10⁻²1/1000.01
10⁻³1/1 0000.001
10⁻⁶1/1 000 0000.000 001
Going down the table, each value is one tenth of the one above it.
ones10⁰0.tenths10⁻¹0hundredths10⁻²0thousandths10⁻³0ten-thousandths10⁻⁴4hundred-thousandths10⁻⁵5The first non-zero digit, 4, sits in the 10⁻⁴ place,so 0.000 45 = 4.5 × 10⁻⁴.
Each place after the decimal point matches a negative power of ten. In 0.000 45 the first non-zero digit is in the 10⁻⁴ place, so the exponent is −4.

Small numbers in scientific notation

The rule is the same as before: the coefficient must be at least 1 and less than 10.

  1. Move the decimal point to the right until it sits after the first non-zero digit.
  2. Count the places it moved. Because you moved right, the exponent is negative.

Quick check: numbers 10 or more have a positive exponent. Numbers from 1 up to 10 have exponent 0. Numbers between 0 and 1 have a negative exponent.

000045The decimal point hops 4 places to the right0.000 45 = 4.5 × 10⁻⁴
In 0.000 45 the decimal point starts after the first 0 (open circle). It hops 4 places to the right to sit after the 4 (filled circle). Moving right 4 places gives the exponent −4.
Worked example

Example 1: Small number → scientific notation

Write 0.000 38 in scientific notation.

  1. The first non-zero digit is 3. Move the decimal point to sit after it: 3.8.
  2. Count the hops: 0.000 38 → 0.003 8 → 0.038 → 0.38 → 3.8. That is 4 places to the right.
  3. Moving right means a negative exponent: −4. Check: 3.8 × 0.000 1 = 0.000 38. ✓

0.000 38 = 3.8 × 10⁻⁴

Worked example

Example 2: Scientific notation → standard form

Write 2.05 × 10⁻⁵ in standard form.

  1. The exponent −5 means divide by 10 five times. Move the decimal point 5 places to the left.
  2. 2.05 → 0.205 → 0.020 5 → 0.002 05 → 0.000 205 → 0.000 020 5.
  3. Check: the number should be very small, between 0 and 1. ✓

2.05 × 10⁻⁵ = 0.000 020 5

Worked example

Example 3: Measurements

A sheet of printer paper is about 0.1 mm thick. Write this thickness in metres, in scientific notation.

  1. 1 mm = 0.001 m, so 0.1 mm = 0.1 × 0.001 m = 0.000 1 m.
  2. Move the decimal point 4 places right to get 1. The exponent is −4.

About 1 × 10⁻⁴ m thick

Metric unitPart of a metrePower of ten
millimetre (mm)0.001 m10⁻³ m
micrometre (µm)0.000 001 m10⁻⁶ m
nanometre (nm)0.000 000 001 m10⁻⁹ m
Each unit in the table is 1 000 times smaller than the one above it.

Watch out!

  • 10⁻³ is not −1 000. It is 0.001, a small positive number.
  • When writing a small number like 0.000 6 in scientific notation, count the hops to reach the 6, not the zeros you see: 0.000 6 = 6 × 10⁻⁴ (4 hops).
Just for fun

Why was 10⁻³ so calm about getting a negative exponent?

Because it knew it was still positive (0.001)!

Hands-on activity

How thick is one sheet of paper?

You need: a ruler, and either a thick book or a stack of printer paper.

  1. Squeeze a stack of pages tightly and measure its thickness in millimetres. In a book, measure only the pages, not the covers.
  2. Count the sheets. In a book, each sheet (leaf) has two page numbers, so sheets = pages ÷ 2.
  3. Divide the thickness by the number of sheets to find the thickness of one sheet in millimetres.
  4. Change your answer to metres (divide by 1 000) and write it in scientific notation.

What to look for: printer paper is often about 0.1 mm, so 1 × 10⁻⁴ m. Book pages are often a bit thinner or thicker. For example, 300 pages = 150 sheets measuring 12 mm gives 12 ÷ 150 = 0.08 mm = 0.000 08 m = 8 × 10⁻⁵ m.

Math talk: "Why do we measure a big stack instead of one sheet?"

Your turn

1

Write 0.003 in scientific notation.

2

Write 0.000 07 in scientific notation.

3

Write 0.000 512 in scientific notation.

4

Write 4 × 10⁻² in standard form.

5

Write 9.3 × 10⁻⁶ in standard form.

6

Write 10⁻³ as a fraction and as a decimal.

7

Is 0.6 × 10⁻³ in scientific notation? If not, fix it.

More practice: warm-up

1

Write 10⁻² as a decimal.

2

Write 10⁻⁴ as a decimal.

3

Write 0.1 as a power of 10.

4

Write 0.008 in scientific notation.

5

Write 5 × 10⁻¹ in standard form.

More practice: on your own

1

Write 0.000 000 6 in scientific notation.

2

Write 0.020 4 in scientific notation.

3

Write 7.07 × 10⁻⁴ in standard form.

4

A nanometre is 10⁻⁹ m. Write this length in standard form.

5

A bead is 2.5 mm wide. Write its width in metres using scientific notation.

Challenge

1

A red blood cell is about 7.5 × 10⁻⁶ m across. About how many would fit side by side across 1 cm (which is 10⁻² m)?

2

Use the table pattern to explain why 10⁰ = 1.

3

A tiny bacterium is 2 × 10⁻⁶ m long. How long is it in millimetres? Write your answer in standard form.

Check your understanding

Quiz

1.

What is 0.000 6 in scientific notation?

2.

What is 3.4 × 10⁻³ in standard form?

3.

What is 10⁻² equal to?

4.

What is 10⁰ equal to?

5.

A number between 0 and 1 written in scientific notation has:

6.

A micrometre is 10⁻⁶ m. Which is equal to it?

For parents and teachers

Big idea: This lesson completes the scientific-notation part of B1.1 (very large and very small numbers). Grade 7 students used positive powers of ten only, so negative exponents are new here. The key message is that 10⁻ⁿ is a small positive number, not a negative number.

Common difficulties:

  • Thinking 10⁻³ is negative. Use the pattern table: each step down divides by 10.
  • Moving the decimal point the wrong way. Small numbers (between 0 and 1) always get negative exponents; check the answer is still small.
  • Off-by-one errors in counting hops. Have your child draw the hops, as in the diagram.

How to help: Do the paper-thickness activity together. Look at the ruler: 1 mm is 10⁻³ m. Ask "How many millimetres in a metre?" and "What fraction of a metre is a millimetre?" Persevering through careful counting builds A1 skills too.