Standard Form (Scientific Notation)

Learn how to write, calculate, and simplify numbers using powers of ten

GCSE Number
Standard Form (Scientific Notation) – cover
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Introduction

Standard form, also called scientific notation, is a universal way of writing numbers that are either extremely large or extremely small. Instead of writing out long strings of digits, we compress the number into a compact format. This is especially useful in GCSE Mathematics and beyond, where efficiency and clarity are key.

The general structure is: \[ N = a \times 10^n \] where \(1 \leq a < 10\) and \(n\) is an integer. The number \(a\) is called the coefficient, and the integer \(n\) is the exponent (or power of ten).

For example:

  • The population of the Earth (~7,900,000,000) can be written as \(7.9 \times 10^9\).
  • The diameter of a red blood cell (~0.000007 metres) can be written as \(7 \times 10^{-6}\).
This format saves space and reduces mistakes, especially when performing calculations.

Historically, scientific notation became widely used with the growth of astronomy and physics, where numbers such as the size of galaxies or the mass of particles were too cumbersome to write in full. Today, it is a core skill in school mathematics and a tool that scientists, engineers, and computer scientists use every day.

Key Vocabulary

  • Coefficient: The number between 1 and 10 (not including 10). Example: in \(3.4 \times 10^5\), the coefficient is 3.4.
  • Base: Always 10 in standard form. It tells us we are working in powers of ten.
  • Exponent (Power): The integer above the 10, showing how many places the decimal point has moved. Example: \(2.1 \times 10^7\) has exponent 7.
  • Positive Exponent: Indicates a large number (10 or more). Example: \(5.6 \times 10^3 = 5600\).
  • Negative Exponent: Indicates a small number (less than 1). Example: \(4.2 \times 10^{-4} = 0.00042\).
  • Standard Form: The finished format \(a \times 10^n\).
  • Ordinary Number: The full written-out number, before conversion.

Why Use Standard Form?

There are several important reasons why standard form is so widely used:

  • Clarity: Numbers are easier to read and compare. For example, comparing \(3.1 \times 10^8\) and \(2.9 \times 10^9\) is far easier than reading 310,000,000 and 2,900,000,000.
  • Space-saving: Long numbers are shortened, reducing risk of transcription errors.
  • Efficiency in calculations: Multiplying and dividing powers of ten can be done quickly using index laws.
  • Universality: Standard form is used consistently across countries and scientific fields.

Real-world applications include:

  • Astronomy: Distances between stars and galaxies are written in standard form, such as \(4.2 \times 10^{16}\) metres for the nearest star.
  • Biology: Sizes of microscopic organisms, like bacteria (~\(1 \times 10^{-6}\) m), are written compactly.
  • Medicine: Drug doses and concentrations are often tiny and need clear notation.
  • Computing: File sizes, memory, and data transfer speeds often involve very large numbers.

Rules and Methods

To convert a large number into standard form:

  1. Find the first non-zero digit.
  2. Place the decimal point after it.
  3. Count how many places the decimal point moved. This is the positive exponent.
  4. Write the number as coefficient × 10n.

Example: 472000 → 4.72 × 105.

To convert a small decimal into standard form:

  1. Find the first non-zero digit.
  2. Place the decimal point after it.
  3. Count how many places the decimal point moved. This is the negative exponent.
  4. Write the number as coefficient × 10n.

Example: 0.00053 → 5.3 × 10-4.

Multiplication in standard form:

  • Multiply the coefficients.
  • Add the exponents.
  • Adjust if the coefficient is not between 1 and 10.

Division in standard form:

  • Divide the coefficients.
  • Subtract the exponents.
  • Adjust if needed.

Addition and subtraction:

Only possible when exponents are equal. Rewrite one number to match the other, then add/subtract the coefficients.

Worked Examples

Example 1: Large number
Write 75000000 in standard form.
Answer: 7.5 × 107.

Example 2: Small number
Write 0.00000095 in standard form.
Answer: 9.5 × 10-7.

Example 3: Multiplication
\((6.4 \times 10^5) \times (1.25 \times 10^2) = 8.0 \times 10^7\).

Example 4: Division
\((5.4 \times 10^{-3}) \div (9 \times 10^{-6}) = 0.6 \times 10^3 = 6.0 \times 10^2\).

Example 5: Addition
\((3.6 \times 10^2) + (8.4 \times 10^3)\).
Convert 3.6 × 102 to 0.36 × 103.
Add: (0.36 + 8.4) × 103 = 8.76 × 103.

Example 6: Subtraction
\((9.8 \times 10^6) – (1.3 \times 10^6)\).
= (9.8 – 1.3) × 106 = 8.5 × 106.

Example 7: Very small values
\((2.1 \times 10^{-8}) \times (5 \times 10^{-4})\).
= 10.5 × 10-12 = 1.05 × 10-11.

Example 8: Scientific context
Speed of light = 300,000,000 m/s = 3 × 108 m/s.
Planck’s constant = 0.000000000000000000000000000000000662 J·s = 6.62 × 10-34 J·s.

Common Mistakes

  • Writing coefficients outside 1 ≤ a < 10, e.g. 0.47 × 106 instead of 4.7 × 105.
  • Using the wrong sign on the exponent when converting small decimals.
  • Stopping at 15 × 109 instead of converting to 1.5 × 1010.
  • Adding numbers with different exponents without rewriting them.
  • Moving the decimal the wrong way when working with negative exponents.

Extra Practice with Answers

  1. Convert 0.00000062 → 6.2 × 10-7.
  2. (2.4 × 108) × (3.5 × 10-4) = 8.4 × 104.
  3. (9 × 1012) ÷ (3 × 105) = 3 × 107.
  4. (1.2 × 103) + (8.8 × 102) = 2.08 × 103.
  5. (7.5 × 10-5) × (4 × 10-3) = 3 × 10-7.

Quick Practice (no answers)

  1. Write 340000 in standard form.
  2. Convert 0.00045 into standard form.
  3. Simplify (4.2 × 104) × (2.5 × 103).
  4. Calculate (6 × 10-2) ÷ (2 × 10-5).
  5. (9.1 × 106) + (3.2 × 105).
  6. (1.8 × 10-4) – (6 × 10-5).
  7. Convert Avogadro’s constant 602000000000000000000000 into standard form.
  8. Express the thickness of human hair (~0.00008 m) in standard form.
  9. (7.2 × 1011) ÷ (1.8 × 108).
  10. (4.5 × 10-9) × (3 × 10-3).

Challenge Questions

  1. A spaceship travels 1.5 × 108 km in 5 × 103 seconds. Find its average speed in km/s.
  2. The radius of the Earth is about 6.37 × 106 m. Find the diameter in standard form.
  3. The mass of a proton is 1.67 × 10-27 kg. Find the total mass of 3 × 1023 protons.
  4. The wavelength of light is 4.8 × 10-7 m. Write the frequency if the speed of light is 3 × 108 m/s.

Conclusion

Standard form is a fundamental mathematical skill that bridges school work and real-world science. By learning how to convert, calculate, and simplify using powers of ten, you are preparing for both GCSE exams and further study in subjects like physics, chemistry, and computing. Always remember to check that your coefficient is between 1 and 10, and be careful with positive and negative exponents. With practice, you will find standard form not only easier but also essential for understanding the scale of our universe, from the tiniest particles to the largest galaxies.

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