18.3 Writing Large Numbers in Standard Form

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: powers of 10, writing large numbers as A × 10ⁿ, 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 the powers of 10 and what they mean in place value
• Write a large number in standard form A × 10ⁿ with 1 ≤ A < 10
• Change a number in standard form back to an ordinary number
• Compare and order numbers written in standard form

Key words

• Standard form: A way of writing a number as A × 10ⁿ, where A is at least 1 but less than 10, and n is a whole number.
• Power of 10: 10, 100, 1000, … written as 10¹, 10², 10³, …
• Ordinary number: A number written out in full, such as 4 500 000.
• Place value: The value of a digit because of its position: units, tens, hundreds, thousands, …

Powers of 10

Each time you multiply by 10, the digits move one place to the left. The power of 10 tells you how many places.

• 10; 10² 100; 10³ 1000; 10⁴ 10 000; 10⁵ 100 000; 10⁶ 1 000 000; 10⁹ 1 000 000 000

10⁶ is one million and 10⁹ is one billion.

Worked example: multiplying by a power of 10

(a) 6 × 10³ = 6 × 1000 = 6000
(b) 2.7 × 10⁴ = 2.7 × 10 000 = 27 000 (the digits move 4 places left)

Answer: (a) 6000 (b) 27 000

Writing a large number in standard form

Standard form is a short way of writing very big numbers. Step 1: put the decimal point after the first non-zero digit to get A. Step 2: count how many places the digits must move to get back to the original number. That is n.

Definition: A × 10ⁿ with 1 ≤ A < 10 and n a whole number

Worked example: converting

(a) 3 050 000 → A = 3.05. The digits move 6 places. So 3.05 × 10⁶.
(b) 480 000 → A = 4.8. The digits move 5 places. So 4.8 × 10⁵.
(c) 72 000 000 000 → A = 7.2, 10 places. So 7.2 × 10¹⁰.

Answer: (a) 3.05 × 10⁶ (b) 4.8 × 10⁵ (c) 7.2 × 10¹⁰

Shortcut: n is the number of digits after the first digit: 480 000 has 5 digits after the 4, so n = 5.

Changing back to an ordinary number

Multiply A by 10ⁿ: move the digits n places to the left, filling empty places with zeros.

Worked example: back to ordinary numbers

(a) 5.6 × 10⁴ = 56 000 (b) 9.03 × 10⁷ = 90 300 000 (c) 1 × 10⁵ = 100 000

Answer: (a) 56 000 (b) 90 300 000 (c) 100 000

Worked example: is it in standard form?

34 × 10⁵ is not in standard form because 34 is not less than 10.
34 × 10⁵ = 3 400 000 = 3.4 × 10⁶. Also 0.8 × 10³ = 800 = 8 × 10².

Answer: 3.4 × 10⁶ and 8 × 10²

Comparing and ordering

First compare the powers of 10: the bigger power gives the bigger number. If the powers are the same, compare the A values.

Worked example: planet distances

Order these distances from the Sun (km), smallest first: 2.28 × 10⁸, 1.5 × 10⁸, 7.78 × 10⁸, 5.8 × 10⁷.
10⁷ is the smallest power, so 5.8 × 10⁷ is smallest. The others all have 10⁸: compare 1.5, 2.28, 7.78.

Answer: 5.8 × 10⁷, 1.5 × 10⁸, 2.28 × 10⁸, 7.78 × 10⁸

A calculator shows standard form as, for example, 3.2E9 or 3.2 ×10⁹. Always write it properly as 3.2 × 10⁹.

Common mistakes

• Writing A as 10 or more, for example 25 × 10⁴. It must be at least 1 and less than 10.
• Counting the zeros only, instead of the places moved: 3 050 000 has five zeros, but it is 3.05 × 10⁶, not 3.05 × 10⁵.
• Ordering by the A values only. 9 × 10⁴ is smaller than 1.2 × 10⁵.
• Writing calculator notation such as 3.2E9 as the final answer.

Revision questions

Try these, then check your answers below each one.

Question 1. Write 5 600 000 in standard form. (Choose from: 5.6 × 10⁶, 56 × 10⁵, 5.6 × 10⁵, 0.56 × 10⁷.)

Answer: 5.6 × 10⁶

Working: 5 600 000 = 5.6 × 10⁶ (the digits move 6 places).

Question 2. Write 4.03 × 10⁵ as an ordinary number.

Answer: 403000

Working: 4.03 × 100 000 = 403 000.

Question 3. True or false? 12 × 10⁴ is written in standard form.

Answer: False

Working: False. 12 is not less than 10. 12 × 10⁴ = 1.2 × 10⁵.

Related lessons

• Previous lesson: 18.2 The Laws of Indices
• Next lesson: 18.4 Writing Small Numbers in Standard Form
• In this chapter: 18.1 Multiplying and Dividing Fractions
• In this chapter: 18.5 Calculating with Standard Form

Continue in the full course

This lesson is section 18.3 (EGCMF1803) 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.