Standard Form

2026 Syllabus Objectives

By the end of these notes, you should be able to:

  1. Use the standard form A × 10^n where n is a positive or negative integer and 1 ⩽ A < 10
  2. Convert numbers into and out of standard form
  3. Calculate with values in standard form
  4. Note: Core candidates are expected to calculate with standard form only on Paper 3

What is Standard Form?

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Standard form (also called scientific notation) is a way of writing very large or very small numbers in a more compact and manageable form.

Instead of writing out all the zeros in a number like 45,000,000 or 0.000032, we can write these numbers using powers of 10.

The Format

Standard form follows this pattern:

A × 10^n

Where:

  • A is a number between 1 and 10 (including 1, but not including 10). We write this as: 1 ⩽ A < 10
  • n is a whole number (called an integer), which can be positive, negative, or zero
  • 10^n means "10 to the power of n"

Important rules:

  • A must have exactly ONE non-zero digit before the decimal point
  • The power of 10 (the value of n) tells us how many places to move the decimal point

Converting Numbers INTO Standard Form

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Large Numbers (Greater than 1)

When converting a large number into standard form, follow these steps:

Step 1: Place the decimal point after the first non-zero digit to create a number between 1 and 10

Step 2: Count how many places the decimal point has moved to the left

Step 3: This number of places becomes your positive power of 10

Example 1: Write 3,500 in standard form

  • Place decimal after the 3: 3.5
  • The decimal point moved 3 places to the left (from 3500. to 3.5)
  • Answer: 3.5 × 10³

Example 2: Write 67,000,000 in standard form

  • Place decimal after the 6: 6.7
  • The decimal point moved 7 places to the left
  • Answer: 6.7 × 10⁷

Example 3: Write 420 in standard form

  • Place decimal after the 4: 4.2
  • The decimal point moved 2 places to the left
  • Answer: 4.2 × 10²

Small Numbers (Between 0 and 1)

When converting a small number (a decimal less than 1) into standard form:

Step 1: Place the decimal point after the first non-zero digit to create a number between 1 and 10

Step 2: Count how many places the decimal point has moved to the right

Step 3: This number of places becomes your negative power of 10

Example 4: Write 0.0056 in standard form

  • The first non-zero digit is 5, so we get: 5.6
  • The decimal point moved 3 places to the right (from 0.0056 to 5.6)
  • Answer: 5.6 × 10⁻³

Example 5: Write 0.000008 in standard form

  • Place decimal after the 8: 8
  • The decimal point moved 6 places to the right
  • Answer: 8 × 10⁻⁶

Example 6: Write 0.42 in standard form

  • Place decimal after the 4: 4.2
  • The decimal point moved 1 place to the right
  • Answer: 4.2 × 10⁻¹

Quick Check Method

To check if you've written a number correctly in standard form, ask yourself:

  • Is the first number (A) between 1 and 10? ✓
  • Is the power of 10 an integer (whole number)? ✓

If both answers are yes, your answer is in correct standard form.

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