1.1.3 Approximation & Estimation

What are bounds?

• Bounds are the smallest (lower bound, LB) and largest (upper bound, UB) numbers that a rounded number can lie between
  • It simply means how low or high the number could have been before it was rounded
• The bounds for a number, , can be written as 
  • Note that the lower bound is included in the range of values could have taken but the upper bound is not

How do we find bounds?

• The basic rule is “half up, half down”
  • To find the upper bound add on half the degree of accuracy
  • To find the lower bound take off half the degree of accuracy
• Remember that the upper bound is the cut off point for the greatest value that the number could have been rounded from but will not actually round to the number itself

How do we calculate using bounds?

• Find bounds before carrying out the calculation and then use the rules:
  • To add UB = UB + UB and LB = LB + LB
  • To multiply UB = UB × UB and LB = LB × LB
  • To divide UB = UB / LB and LB = LB / UB
  • To subtract UB = UB - LB and LB = LB – UB
• Use logic to decide which bound to use within the calculation
  • For example if you are finding the maximum volume of a sphere with the radius given correct to 1 decimal place substitute the upper bound of the radius into your calculation for the volume

How do I know what to round my answer to?

• Unless otherwise told, always round your answers to 3 significant figures (3 s.f.)
  • The first non-zero digit is the first significant digit
  • The first digit after the third significant digit determines whether to 'round up' (5) or 'leave it alone' (<5)
    • where the ‘it’ we are rounding up or leaving alone is the third significant figure

• • Your final answer will have three significant digits and the rest will be zero
    • Any zero after the first significant digit is still significant
    • For large numbers be careful not to change the place value of the significant digits, you will have to fill in any zeros after the third significant figure
    • If your GDC is in scientific mode it may display unnecessary zeros after the decimal point, you do not need to copy these
• Look out for any questions that ask you to round your answer in a different way
  • Questions often ask for 2 decimal places (2 d.p.)
    • Your final answer will only have 2 digits after the decimal point
    • For 2 d.p. it is the third digit after the decimal place that determines whether to 'round up' (5) or 'leave it alone' (<5)
• If you are working with a currency you must choose the appropriate degree of accuracy
  • For most this will be a whole number
    • E.g. yen, yuan, peso
  • For others this will be to 2 decimal places
    • E.g. dollars, euro, pounds
  • It will be clear from the question which currency you are using and how you should round your answer
    • The question will state the name of the currency and the symbol you should use as a unit
    • E.g. YEN, ￥

Are there cases when I always have to round up?

• Yes - there are cases when it makes sense to always round up (or down)
• These normally involve finding the minimum or maximum number of objects
  • For example consider the scenario: There are 26 people and 5 people can fit in a single vehicle, how many vehicles are needed?
    • and normally we'd round to 5
    • However 5 vehicles wouldn't be enough as there would only be room for 25 people
    • In this case we would round up to find the minimum number needed
• These kind of problems can be solved by inequalities
  • For  take the smallest value of x at the appropriate degree of accuracy that is greater than k
    • For example: Using 3sf the smallest solution to x > 2.5731... is x = 2.58
  • For  take the biggest value of x at the appropriate degree of accuracy that is less than k
    • For example: The biggest integer solution to x < 10.901... is x = 10

What is percentage error?

• Percentage error is how far away from the actual value an estimated or rounded answer is
  • Percentage error can be calculated using the formula

• • • where  is the exact value and is the approximate value of 
    • The | | is the absolute value meaning if you get a negative value within these straight brackets, you should take the positive value
  • This formula is in the formula booklet so you do not need to remember it
• The further away the estimated answer is from the true answer the greater the percentage error
• If the exact value is given as a surd or a multiple of π make sure you enter it into the formula exactly as you see it
• Percentage error should always be a positive number

What are exact values?

• Exact values are forms that represent the full and precise value of a number
  • For example,is an exact value and 3.14 is an approximation using 3 significant figures
• If a number has an infinite number of non-zero digits after the decimal point then you can use three dots to signal that the decimal representation goes on for example
  • For example, √2 = 1.414...
• Exact values can involve
  • Fractions:   
  • Roots: ,  
  • Logarithms: , 
  • Mathematical constants: , 
• Your GDC might automatically give your answer as an exact answer
• If your GDC does not do this then you may need to evaluate parts of the expression separately and use algebra
  • For example: If  then your GDC will probably not give you the exact value of
  • You would insist evaluate it without a GDC to get the exact value:  

Why use estimation?

• We estimate to find approximate answers to difficult sums
• Or to check our answers are about the right size (order of magnitude)
  • For example, if the question is to find a length the answer cannot be negative
  • or if we are looking for the mean age of some people an answer of 150 must be incorrect
• Estimating an answer before carrying out a calculation will help you know what you are looking for and determine if your answer is likely to be correct or not
• In real life estimation skills are used every day in many activities

How do I choose the correct answer?

• Sometimes a mathematical argument will lead to more than one answer
  • This is common with problems involving quadratics, you will usually have two solutions
  • If you have more than one solution after you have solved a problem, always check to see if they are both valid
• Most of the time you can simply use logic to choose the correct answer
  • If the problem involves length or area and one of the answers is negative, the true solution will be the positive answer
• Occasionally you will need to see if an answer can be valid
  • If one of your answers is for example, will not give a true solution