GCSE Maths Practice: standard-form

Question 9 of 10

Compare numbers in standard form and identify which represents the largest value.

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\( \begin{array}{l}\text{Which of the following is the largest number?}\end{array} \)

Choose one option:

When comparing small numbers, focus on the power of ten first. The greater (less negative) the power, the larger the number.

Comparing Numbers in Standard Form

In GCSE Maths, it’s not enough to simply convert numbers into standard form—you must also understand how to compare them. Numbers written with negative powers of ten can be confusing because the larger the negative power, the smaller the number. For example, 10⁻² is bigger than 10⁻³ because dividing by 100 gives a larger result than dividing by 1000.

How to Compare Numbers in Standard Form

  1. Look first at the power of ten. The greater the exponent, the larger the number (even if negative, closer to zero is bigger).
  2. If the powers are the same, compare the coefficients (the numbers before the × sign).
  3. Remember: −2 is greater than −3, so 10⁻² gives a larger number than 10⁻³.

Worked Example 1

Which is larger: 4.5 × 10⁻³ or 4.5 × 10⁻⁴?

  • Compare exponents: −3 and −4.
  • −3 is greater, so 4.5 × 10⁻³ is larger.

Worked Example 2

Compare 2.3 × 10⁻² and 2.9 × 10⁻³.

  • First, check powers: −2 vs −3 → −2 is greater.
  • Therefore, 2.3 × 10⁻² is larger, even though its coefficient is smaller.

Worked Example 3

Arrange in order: 1.7 × 10⁻², 1.7 × 10⁻³, 1.7 × 10⁻⁴.

  • As powers decrease, numbers get smaller.
  • Order: 1.7 × 10⁻² > 1.7 × 10⁻³ > 1.7 × 10⁻⁴.

Common Mistakes

  • Thinking a larger negative power makes a larger number—it’s the opposite.
  • Comparing only the coefficients without checking the exponents.
  • Forgetting that −2 is greater than −3, so 10⁻² represents a bigger value.

Real-World Context

Understanding this is vital in science and technology. For instance, 3 × 10⁻³ metres (3 mm) is larger than 3 × 10⁻⁴ metres (0.3 mm). In chemistry, concentrations written in standard form follow the same principle. The smaller the negative exponent, the higher the concentration.

FAQs

  • Does a smaller exponent always mean a smaller number? No—this only applies when exponents are negative. For positive exponents, larger powers give larger numbers.
  • Can I convert them to decimals to check? Yes, but that’s slower. It’s better to reason directly with powers of ten.
  • What’s the quick trick? Remember: for negative powers, ‘less negative’ means ‘bigger’.

Study Tip

When comparing small numbers in standard form, say the powers aloud: ‘to the minus two’ is larger than ‘to the minus three’. This helps you avoid reversing the logic under exam pressure. Regularly practise ordering three or four numbers by size to strengthen this skill for your GCSE exam.