Scottish National 5 Mathematics

National 5 Maths Scientific notation

Write, compare and calculate with numbers in scientific notation.

Formula referenceMixed revisionPrintable worksheet

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

Estimate first, then check the sign, order of magnitude and requested accuracy of the final value.

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.

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.