Introduction to Approximation
In real-world mathematics, exact values are often impractical, unnecessary, or even impossible to obtain. Approximation allows us to work with numbers at a manageable level of precision — whether you're analysing a dataset, reporting a measurement, or presenting a final answer in an exam.
The process of finding a value that is close enough to the correct answer for a given purpose, usually by rounding to a specified degree of accuracy.
In IB Math AI, you will regularly be asked to:
- Round answers to a specific number of decimal places (d.p.) or significant figures (s.f.)
- Find upper and lower bounds for rounded values
- Calculate percentage error to measure how close an approximation is to the true value
- Use estimation to make reasonable approximations without precise calculation
The IB mark scheme often awards an accuracy mark only if the answer is given to 3 significant figures (unless the question specifies otherwise). Always check what precision is required.
Rounding to Decimal Places
The number of digits written after the decimal point in a number.
To round to decimal places:
- Identify the digit in the th decimal place.
- Look at the next digit (the th decimal place).
- If that digit is 5 or more, round up (increase the th digit by 1). If it is 4 or less, round down (leave the th digit unchanged).
Round 3.14159 to 2 decimal places.
- The 2nd decimal place digit is 4.
- The next digit is 1 (less than 5), so we round down.
- Answer:
Round 0.00567 to 3 decimal places.
- The 3rd decimal place digit is 5.
- The next digit is 6 (), so we round up: the 5 in the 3rd decimal place increases by 1 to become 6.
- Answer:
Be careful with zeros after the decimal point — they are significant when writing to a specified number of decimal places. For example, to 2 d.p. must be written as , not , to show the required precision.