GCSE Maths Practice: standard-form

Question 10 of 10

Learn how to represent very small decimal numbers in standard form using negative powers of ten.

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\( \begin{array}{l}\text{Write } 0.00000045 \text{ in standard form.}\end{array} \)

Choose one option:

When numbers are less than one, move the decimal right and apply a negative power of ten equal to the number of moves.

Understanding Very Small Numbers in Standard Form

In GCSE Maths, extremely small numbers are often expressed in standard form to make them easier to read, write, and compare. Standard form allows us to show how many times ten has been divided rather than multiplied. When a number is smaller than one, the power of ten is negative, showing that the decimal point has moved to the right to make the first part of the number fall between 1 and 10.

Why It’s Important

Numbers like 0.00000045 can be difficult to interpret quickly. Using standard form simplifies these by converting them into a more readable pattern. This technique is essential in scientific and technical work where precision is required — for example, when describing atomic diameters or microscopic measurements. Being confident with negative powers of ten helps students make sense of data that spans many magnitudes.

How to Convert to Standard Form

  1. Locate the decimal point in the original number.
  2. Move the decimal point so that the new number lies between 1 and 10.
  3. Count the number of places moved. Each movement represents a power of ten.
  4. Because the original number is less than one, assign a negative sign to the exponent.

Worked Example 1

Convert 0.0000031 into standard form.

  • Move the decimal six places right → 3.1
  • Exponent = −6
  • Result: 3.1 × 10⁻⁶

Worked Example 2

Convert 0.00000082 into standard form.

  • Move the decimal seven places right → 8.2
  • Exponent = −7
  • Result: 8.2 × 10⁻⁷

Worked Example 3

Convert 0.000045 into standard form.

  • Move the decimal five places right → 4.5
  • Exponent = −5
  • Answer: 4.5 × 10⁻⁵

Common Mistakes

  • Forgetting the negative sign when the number is smaller than one.
  • Moving the decimal the wrong direction — to the left instead of the right.
  • Including too many or too few zeros when counting the decimal places.

Real-Life Examples

Standard form is vital in science. A human hair’s thickness is roughly 7 × 10⁻⁵ metres, while the wavelength of visible light is around 5 × 10⁻⁷ metres. These quantities are hard to read in ordinary form but easy in standard form. In computing, memory units such as nanoseconds or microvolts are also expressed using powers of ten. Learning this notation connects classroom maths to real-world technology and measurement.

FAQs

  • Why does a smaller number have a negative power? Because each movement to the right divides by ten, creating a smaller value.
  • Can the coefficient ever be zero? No. The coefficient (the first number) must be between 1 and 10, never zero.
  • How can I check my result? Multiply the number in standard form back out to see if it matches the original value.

Study Tip

When working with very small decimals, count slowly and clearly as you move the decimal point. Saying the moves aloud (‘one, two, three…’) can help prevent miscounting. Remember that each move toward the right makes the power more negative. Regularly practise with both large and tiny numbers so that converting between forms feels natural and automatic during your GCSE Maths exams.