Standard Form (CIE IGCSE Maths: Core)

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 calculations with very big or small numbers easier

How do we use standard form?

• Numbers written in standard form are always written as:

• The rules:
  • ( is between 1 and 10) so there is one non-zero digit before the decimal point
    • could be 3.56 or 1
    • could not be 10 or 0.56
  • ( is positive) for large numbers
    • 50 000 000 can be written as 5×10⁷
  • ( is negative) for small numbers
    
    • 0.00000005 can be written as 5×10^-8
• To write a large number such as 3 240 000 in standard form
  • Choose the value of a; 3.24
  • Find how many times you must multiply 3.24 by 10, to make 3 240 000 (counting how many places you would need to move the decimal point can help)
  • We would need to multiply by 10, 6 times
  • 3 240 000 = 3.24 × 10 × 10 × 10 × 10 × 10 × 10 = 3.24 × 10⁶
• To write a small number such as 0.000567 in standard form
  • Choose the value of a; 5.67
  • Find how many times you must divide 5.67 by 10, to make 0.000567 (counting how many places you would need to move the decimal point can help)
  • We would need to divide by 10, 4 times
  • Because we are dividing rather than multiplying, the power will be negative
  • 0.000567 = 5.67 ÷ 10 ÷ 10 ÷ 10 ÷ 10 = 5.67 × 10^-4

How do I multiply or divide two numbers in standard form?

• If you can, use a calculator!
• Otherwise multiply/divide the number parts first
  
  • If this answer is less than 1, or 10 or more, then you will need to write it in standard form again
    • e.g. 4 × 5 = 20 = 2 × 10¹
    • or 2 ÷ 4 = 0.5 = 5 × 10^-1
• Then multiply/divide the powers of 10 using the laws of indices
  • 
  • 
• Multiply the two parts together to get your answer in standard form
  • You might have to use the laws of indices once more
    • e.g. (4 × 10²) × (5 × 10⁷) = (4 × 5) × (10² × 10⁷) = 20 × (10⁹)
    • = 2 × 10¹ × 10⁹ = 2 × 10¹⁰

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

• If you can, use a calculator!
• If the two numbers have the same power of 10 then you can simply add/subtract the number parts
  • If the answer is less than 1 or 10 or more then you will have to rewrite in standard form
    • e.g. (7 × 10⁵) - (6.2 × 10⁵) =  0.8 × 10⁵ = 8 × 10^-1 × 10⁵ = 8 × 10⁴ 
• Otherwise convert both numbers so that they have the same power of 10 (choosing the larger power)
  • e.g. (7 × 10⁵) + (6 × 10⁴) = (7 × 10⁵) + (0.6 × 10⁵) = 7.6 × 10⁵
• If the powers of 10 are small then you might find it easier to convert both numbers to ordinary numbers before adding/subtracting
  • e.g.  (4 × 10³) - (5 × 10^-2) = 4000 - 0.05 = 3999.95
  • You can convert your answer back to standard form if needed

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

• If you can, use a calculator!
• As standard form is two terms multiplied together, the power is applied to both terms
  
  • We can use the law of indices  to help
  • 
  • e.g. (3 × 10⁴)⁵  =  3⁵ × (10⁴)⁵  =  3⁵ × 10²⁰  =  243 × 10²⁰
• Check to see whether you have to write your final answer in standard form
  • e.g. 243 × 10²⁰ = 2.43 × 10²²