GCSE Maths Practice: standard-form

Question 1 of 10

Learn to express small decimals in standard form using negative powers of ten.

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\( \begin{array}{l}\text{Write } 0.00012 \text{ in standard form.}\end{array} \)

Choose one option:

Shift the decimal right until the number is between 1 and 10. Use a negative power equal to the number of moves.

Writing Small Numbers in Standard Form

In GCSE Maths, standard form is used to represent numbers that are very large or very small in a more compact and readable way. When a number is less than one, it has several zeros after the decimal point. In this situation, standard form expresses the number as a value between 1 and 10 multiplied by a negative power of ten. This method is extremely useful for simplifying data and avoiding counting long strings of zeros.

Why It’s Useful

Standard form allows you to work efficiently with small quantities in scientific, mathematical, or everyday contexts. For example, engineers, physicists, and chemists frequently use numbers that are thousands or even millions of times smaller than one. Using standard form ensures calculations stay accurate and easier to manage when converting between units or comparing scales.

How to Convert a Small Decimal into Standard Form

  1. Identify the current position of the decimal point.
  2. Move the decimal point to the right until the number becomes between 1 and 10.
  3. Count how many places you moved the decimal.
  4. Use a negative exponent equal to that number of moves.

Worked Example 1

Convert 0.000008 into standard form.

  • Move the decimal six places right → 8
  • Exponent = −6
  • Result: 8 × 10⁻⁶

Worked Example 2

Convert 0.00034 into standard form.

  • Move the decimal four places right → 3.4
  • Exponent = −4
  • Result: 3.4 × 10⁻⁴

Worked Example 3

Convert 0.00000059 into standard form.

  • Move the decimal seven places right → 5.9
  • Exponent = −7
  • Answer: 5.9 × 10⁻⁷

Common Errors

  • Writing the first part of the number outside the range 1 ≤ n < 10.
  • Confusing positive and negative powers of ten — small numbers always use negative exponents.
  • Counting too many or too few decimal places when shifting.

Real-World Applications

Standard form is essential in science and technology. For example, a single bacterium might measure around 2 × 10⁻⁶ metres in length, while the thickness of human hair is roughly 7 × 10⁻⁵ metres. In electronics, microcurrents and nanosecond timings are written using standard form to ensure accuracy and readability. This way of writing numbers bridges maths with real-world scientific measurement.

FAQs

  • Why is the exponent negative? A negative exponent shows that the number is smaller than one. Each move of the decimal point to the right divides by ten.
  • Can I round in standard form? Yes — exam questions may ask for a certain number of significant figures. Just round the coefficient accordingly.
  • How can I check my work? Multiply the number by the power of ten to see if it returns to the original decimal.

Study Tip

When practising, remember this quick rule: small number → negative exponent, big number → positive exponent. Write out a few examples each day to train your eye to spot how many decimal places the point must move. This habit makes standard form questions much quicker in your GCSE Maths exam.