GCSE Maths Practice: standard-form

Question 2 of 10

The population of a small town is about 100000. Express this in standard form.

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\( \begin{array}{l}\text{A town has a population of }100000.\text{ Write this in standard form.}\end{array} \)

Choose one option:

For large numbers, move the decimal left and use a positive exponent. The coefficient must stay between 1 and 10.

Understanding Large Numbers in Standard Form

Standard form helps us write large numbers more simply and clearly. When a number is bigger than ten, its decimal point moves to the left until the first number lies between 1 and 10. The number of moves becomes the power of ten, which is positive for large numbers. For example, 1.0 × 10⁵ represents one hundred thousand because 10⁵ = 100000.

Why Use Standard Form?

Working with long numbers such as 2,300,000 or 45,000,000 can lead to counting errors. Standard form shortens these numbers and makes them easier to compare and calculate. In GCSE Maths, standard form appears frequently in questions about population, distance, and scientific data.

Real-World Example

The small town of Oakridge has a population of about 100000 people. In standard form, this is written as 1.0 × 10⁵. Nearby, a larger city might have 2.5 × 10⁶ inhabitants (2,500,000). Expressing populations this way allows quick comparisons without writing all the zeros.

Step-by-Step Conversion

  1. Write the number in full, e.g. 100000.
  2. Move the decimal left until the number is between 1 and 10 → 1.0.
  3. Count the number of moves → 5.
  4. Write as 1.0 × 10⁵.

Worked Example 1

Convert 45000 into standard form.

  • Move decimal four places left → 4.5.
  • Power of ten = 4.
  • Answer: 4.5 × 10⁴.

Worked Example 2

Convert 7800000 into standard form.

  • Move decimal six places left → 7.8.
  • Exponent = 6.
  • Result: 7.8 × 10⁶.

Worked Example 3

Convert 52000000 into standard form.

  • Move decimal seven places left → 5.2.
  • Exponent = 7.
  • Result: 5.2 × 10⁷.

Common Mistakes

  • Moving the decimal the wrong way — for large numbers, move it left.
  • Forgetting that the coefficient must be between 1 and 10.
  • Writing 10⁻⁵ instead of 10⁵ for large numbers (sign error).

Applications in Science and Engineering

Standard form is vital when dealing with very large quantities. Astronomers express the distance from the Earth to the Sun as 1.5 × 10¹¹ metres. Engineers use it to describe the output of powerful machines, while computer scientists use it to record data measured in terabytes or gigabytes. It helps manage scale efficiently without confusing zeros.

FAQs

  • Why is the exponent positive? Because large numbers are formed by multiplying by powers of ten.
  • Can I write 10 × 10⁴ instead? No, because the first number must be between 1 and 10. The correct form is 1.0 × 10⁵.
  • What’s the easiest way to check? Multiply your standard-form number back out — it should return to 100000.

Study Tip

Always check whether your number is larger or smaller than one before choosing the exponent sign. For large numbers, move the decimal left and keep the power positive. With practice, you’ll recognise common values like 10⁵ = 100000 and 10⁶ = 1000000 instantly.