Back to National 5 Mathematics

SQA National 5 Mathematics

Scientific notation

Writing very large and very small numbers in standard form.

Formula referenceMixed revision

Formula reference

Check the National 5 rules and formulae linked to this topic.

Which method?

Match exam clues to a suitable method.

Before you start

  • Know place value for decimals and large whole numbers.
  • Remember that standard form is a number from 1 up to 10 multiplied by a power of 10.
  • Check whether a number is large or small before deciding the sign of the power.
National 5 Mathematics lesson

Explanation

Scientific notation, also called standard form, writes a number as a × 10n, where 1 ≤ a < 10 and n is an integer.

Large numbers use positive powers of 10. For example, 320000 = 3.2 × 10⁵ because the decimal point moves five places.

Small decimals use negative powers of 10. For example, 0.0047 = 4.7 × 10−3 because the decimal point moves three places to make 4.7.

Calculator displays may use E notation, such as 3.2E5 or 4.7E-3. That is calculator/input notation for 3.2 × 10⁵ and 4.7 × 10−3.

Key formulae and rules

  • Standard form: a × 10n, where 1 ≤ a < 10
  • Large numbers: n is positive
  • Small decimals: n is negative

Watch out

Writing the first number outside the range 1 ≤ a < 10

Check

Compare your answer with the size you expected from the question.

Exam tip

For standard form, the first number must be at least 1 and less than 10. Then decide whether the original value was large or small.

Calculator tip

Use brackets for fractions, powers and square roots, then round only at the final line.

Worked examples

Worked example 1

Convert a large number

Write 58000000 in standard form.

  1. Move the decimal point to make a number between 1 and 10: 5.8.
  2. The decimal point moved 7 places.
  3. Use a positive power because the original number is large.

Answer: 58000000 = 5.8 × 10⁷

Worked example 2

Convert a small decimal

Write 0.00092 in standard form.

  1. Move the decimal point to make 9.2.
  2. The decimal point moved 4 places.
  3. Use a negative power because the original number is less than 1.

So: 0.00092 = 9.2 × 10−4

Worked example 3

Interpret standard form

Write 4.7 × 10−3 as an ordinary decimal

  1. The power is negative, so the number will be small.
  2. Move the decimal point 3 places left.
  3. Fill empty places with zeros.

Answer: 4.7 × 10−3 = 0.0047

Watch out

  • Writing the first number outside the range 1 ≤ a < 10
  • Using a positive power for a small decimal.
  • Thinking calculator E notation is the final written standard form.