GCSE Maths Practice: standard-form

Working with Extremely Small Numbers in Standard Form

In GCSE Maths, numbers smaller than one can be written in standard form to make them clearer and easier to handle. Standard form represents numbers as a value between 1 and 10 multiplied by a power of ten. When a number is very small, this power becomes negative because the decimal point moves to the right to create a value within the range 1 ≤ n < 10. This form is widely used in scientific and engineering contexts where such tiny quantities are common.

Why Use Standard Form?

Writing 0.0000009 or similar numbers can be tedious and prone to mistakes. Standard form provides a quick, precise alternative that saves space and simplifies calculations. It also makes comparing very small or very large values much easier, as you can simply look at the powers of ten rather than counting zeros.

How to Convert a Small Decimal into Standard Form

1. Identify the current position of the decimal point.
2. Move the decimal to the right until the number becomes between 1 and 10.
3. Count the number of moves made — this count becomes the exponent.
4. Use a negative sign for the exponent since the original number was smaller than one.

Worked Example 1

Convert 0.000004 into standard form.

• Move the decimal six places right → 4
• Exponent = −6
• Result: 4 × 10⁻⁶

Worked Example 2

Convert 0.00000012 into standard form.

• Move the decimal seven places right → 1.2
• Exponent = −7
• Result: 1.2 × 10⁻⁷

Worked Example 3

Convert 0.000000045 into standard form.

• Move the decimal eight places right → 4.5
• Exponent = −8
• Answer: 4.5 × 10⁻⁸

Common Mistakes to Avoid

• Incorrect sign: Using a positive power for numbers smaller than one.
• Too many zeros: Double-counting zeros before the first significant digit.
• Wrong range: The number before the × symbol must always be between 1 and 10.

Real-World Uses

Scientists and engineers constantly use negative powers of ten. For example, a typical wavelength of light is around 5 × 10⁻⁷ metres, and the radius of a small virus might be 2 × 10⁻⁸ metres. Standard form enables clear and direct comparison between such values, which would otherwise require writing long strings of zeros. In computing, this same notation helps describe times measured in microseconds or voltages measured in microvolts.

FAQs

• Why is the exponent negative? Because each move of the decimal point to the right divides the number by ten.
• Can the coefficient include a trailing zero? Yes, such as 9.0 × 10⁻⁷ — this shows the number has two significant figures.
• How can I check my work? Multiply your standard-form result by the power of ten and ensure you get back the original decimal.

Study Tip

Always count the zeros carefully before you move the decimal point, especially when several appear in a row. Practise with both microscopic (10⁻⁶ to 10⁻⁹) and large (10³ to 10⁹) values so that you can move fluently between positive and negative exponents in your GCSE Maths exam.