Standard Form | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Converting To & From Standard Form

What is standard form?

• Standard Form (sometimes called Standard Index Form) is a way of writing very big and very small numbers using powers of 10

Why do we use standard form?

Writing big (and small) numbers in Standard Form allows us to:

• write them more neatly
• compare them more easily
• make things easier when doing calculations

How do we use standard form?

• Numbers in standard form are always written in the form
  a x 10ⁿ
• The rules:
  • 1 ≤ a < 10 so there is one non-zero digit before the decimal point
  • n > 0 for LARGE numbers – how many times a is multiplied by 10
  • n < 0 for SMALL numbers – how many times a is divided by 10
• Calculations involving standard form can be done using a calculator, if permitted.

Operations with Standard Form

• When multiplying or dividing two numbers in standard form:
  • If feasible, utilize a calculator for accuracy.
  • Otherwise, begin by multiplying or dividing the number parts.
  • If the result is less than 1 or greater than 10, express it in standard form.
  • For instance:
    • 4 × 5 = 20 = 2 × 10¹
    • 2 ÷ 4 = 0.5 = 5 × 10^-1
  • Proceed by multiplying or dividing the powers of 10 as per the laws of indices.
  • Multiply the two parts together to obtain the final answer in standard form.
  • Further application of the laws of indices may be necessary, for example:
    • 4 × 10² × 5 × 10⁷ = 2 × 10¹ × 10⁹ = 2 × 10¹⁰

How do I add or subtract two numbers in standard form?

• If you can, use a calculator!
• If you need to add or subtract numbers in standard form, ensure they have the same power of 10.
• If the powers of 10 are equal, you can directly add or subtract the numerical parts.
• If the result is not between 1 and 10, convert it back to standard form. 
  • For example, 7 × 10⁵ - 6.2 × 10⁵ = 0.8 × 10⁵ = 8 × 10⁴.
• If the powers of 10 differ, adjust the numbers to have the same power by choosing the larger one.
  • For instance, 7 × 10⁵ + 6 × 10⁴ = 7 × 10⁵ + 0.6 × 10⁵ = 7.6 × 10⁵
• If dealing with small powers of 10, it might be easier to convert numbers to regular form before performing operations.
  • Use a calculator whenever possible for accuracy and efficiency.

How do I find powers or roots of a number in standard form?

• Use a calculator whenever possible for complex calculations.
• Standard form involves two terms multiplied together; split the power or root for easier computation.
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• Always check if the final answer needs to be in standard form.