GCSE Maths Practice: standard-form

Question 4 of 10

The population of a small city is 500000. Express this in standard form.

‹ ›

\( \begin{array}{l}\text{A city has a population of }500000.\\\text{ Write this in standard form.}\end{array} \)

Choose one option:

For large numbers, move the decimal left. Each move increases the power of ten by one, and the exponent stays positive.

Converting Large Numbers into Standard Form

Standard form is a useful way to express very large numbers clearly and compactly. Instead of writing out all the zeros, we show the number as a value between 1 and 10 multiplied by a power of ten. For large numbers, this power of ten is positive because we are multiplying by tens, hundreds, thousands, or more.

Real-World Example

Imagine a small city with a population of 500,000 people. Writing that figure with all the zeros takes up space and increases the chance of errors when comparing with other populations. Instead, it can be written in standard form as 5.0 × 10⁵. This expresses the same number but in a more compact, precise, and professional way, which is especially helpful when analysing data or writing scientific reports.

Step-by-Step Method

  1. Write the full number (e.g., 500000).
  2. Move the decimal point to make the first number between 1 and 10.
  3. Count how many places the decimal moved—this becomes the exponent.
  4. Because the number is greater than one, the exponent is positive.

Worked Example 1

Convert 34000 into standard form.

  • Move decimal four places left → 3.4.
  • Exponent = +4.
  • Result: 3.4 × 10⁴.

Worked Example 2

Convert 7200000 into standard form.

  • Move decimal six places left → 7.2.
  • Exponent = +6.
  • Result: 7.2 × 10⁶.

Worked Example 3

Convert 450000000 into standard form.

  • Move decimal eight places left → 4.5.
  • Exponent = +8.
  • Result: 4.5 × 10⁸.

Common Mistakes

  • Moving the decimal the wrong way. For large numbers, always move left.
  • Forgetting that the coefficient must be between 1 and 10.
  • Using a negative power for a large number.

Practical Uses

Standard form is widely used in statistics, economics, and science. Scientists use it to express distances in astronomy, engineers use it for energy output, and demographers use it to describe population figures. For instance, the number of stars in the Milky Way is estimated at about 2 × 10¹¹, while Earth’s population is approximately 8 × 10⁹. These values are much easier to read and compare when expressed in standard form.

FAQs

  • Why is the exponent positive? Because large numbers are created by multiplying by ten, not dividing.
  • Can I write 50 × 10⁴ instead? No, because the first number must be between 1 and 10. The correct form is 5.0 × 10⁵.
  • What’s a quick way to check? Multiply your result by 10⁵ to make sure you return to 500000.

Study Tip

Before moving the decimal, quickly estimate whether your number is greater or smaller than one. For large numbers, move left and keep the exponent positive. Remember key powers like 10³ = 1000 and 10⁶ = 1,000,000 to speed up conversions during your GCSE Maths exam.