18.4 Writing Small Numbers in Standard Form

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: negative powers of 10, writing small numbers as A × 10−ⁿ, changing back, and ordering. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Know that 10−¹ = 0.1, 10−² = 0.01 and so on
• Write a small number (between 0 and 1) in standard form
• Change a small number in standard form back to an ordinary number
• Compare and order numbers in standard form, including negative powers

Key words

• Standard form: A × 10ⁿ with 1 ≤ A < 10. For numbers less than 1, n is negative.
• Negative power of 10: 10−ⁿ = 1/10ⁿ. For example 10−³ = 1/1000 = 0.001.
• Decimal place: A position after the decimal point: tenths, hundredths, thousandths, …
• Leading zeros: The zeros at the start of a small decimal, such as the zeros in 0.004.

Negative powers of 10

Dividing by 10 moves the digits one place to the right. Negative powers of 10 are tenths, hundredths, thousandths, …

• 1; 10−¹ 0.1; 10−² 0.01; 10−³ 0.001; 10−⁴ 0.0001; 10−⁶ 0.000 001
• 1; 10−¹ 1/10; 10−² 1/100; 10−³ 1/1000; 10−⁴ 1/10 000; 10−⁶ 1/1 000 000

Worked example: multiplying by a negative power of 10

(a) 7 × 10−² = 7 × 0.01 = 0.07
(b) 4.5 × 10−³ = 4.5 ÷ 1000 = 0.0045 (the digits move 3 places right)

Answer: (a) 0.07 (b) 0.0045

Writing a small number in standard form

Find the first non-zero digit and put the decimal point after it to get A. Count how many places the digits move to the left to get from the number to A. The power is minus that number.

Worked example: converting

(a) 0.000 62 → A = 6.2. The digits move 4 places. So 6.2 × 10−⁴.
(b) 0.035 → A = 3.5. The digits move 2 places. So 3.5 × 10−².
(c) 0.000 000 8 → A = 8. The digits move 7 places. So 8 × 10−⁷.

Answer: (a) 6.2 × 10−⁴ (b) 3.5 × 10−² (c) 8 × 10−⁷

Shortcut: For a number less than 1, the power is minus the position of the first non-zero digit after the point. In 0.000 62 the 6 is 4th, so the power is −4.

Changing back to an ordinary number

For 10−ⁿ, move the digits n places to the right (the decimal point moves n places left). Fill gaps with zeros.

Worked example: back to ordinary numbers

(a) 2.9 × 10−³ = 0.0029 (b) 1.05 × 10−⁵ = 0.000 010 5 (c) 6 × 10−¹ = 0.6

Answer: (a) 0.0029 (b) 0.000 010 5 (c) 0.6

Worked example: tiny things

A grain of sand is about 0.000 25 m across. Write this in standard form.
A = 2.5, the 2 is in the 4th decimal place.

Answer: 2.5 × 10−⁴ m

Comparing and ordering

With negative powers, the power closer to zero gives the bigger number. 10−² = 0.01 is bigger than 10−⁵ = 0.000 01. If the powers are the same, compare the A values.

Worked example: ordering

Order, smallest first: 4.5 × 10−³, 6 × 10−⁴, 2.1 × 10−³, 9 × 10−⁶.
Powers from smallest: 10−⁶, then 10−⁴, then the two with 10−³ (2.1 before 4.5).

Answer: 9 × 10−⁶, 6 × 10−⁴, 2.1 × 10−³, 4.5 × 10−³

Common mistakes

• Using a positive power for a small number: 0.0035 is 3.5 × 10−³, not 3.5 × 10³.
• Moving the digits the wrong way when changing back: 2.9 × 10−³ is 0.0029, not 2900.
• Thinking 10−⁶ is bigger than 10−² because 6 > 2. It is much smaller.
• Leaving A outside the range, for example 35 × 10−⁴. It should be 3.5 × 10−³.

Revision questions

Try these, then check your answers below each one.

Question 1. Write 0.000 72 in standard form. (Choose from: 7.2 × 10⁻⁴, 7.2 × 10⁻⁵, 72 × 10⁻⁵, 7.2 × 10⁴.)

Answer: 7.2 × 10⁻⁴

Working: 0.000 72 = 7.2 × 10⁻⁴.

Question 2. Write 3.6 × 10⁻³ as a decimal.

Answer: 0.0036

Working: 3.6 ÷ 1000 = 0.0036.

Question 3. True or false? 10⁻² is bigger than 10⁻⁵.

Answer: True

Working: True. 10⁻² = 0.01 and 10⁻⁵ = 0.000 01.

Related lessons

• Previous lesson: 18.3 Writing Large Numbers in Standard Form
• Next lesson: 18.5 Calculating with Standard Form
• In this chapter: 18.1 Multiplying and Dividing Fractions
• In this chapter: 18.2 The Laws of Indices

Continue in the full course

This lesson is section 18.4 (EGCMF1804) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.