IB Mathematics AI HL Flashcards- Approximation decimal places

ROUNDING — This should be a simple process at the end of many IB questions, but can trip up many students still, so it must be mastered.

SIGNIFICANT FIGURES — Zeros before the first non-zero are not significant (leading zeros).
→ All other zeros and other digits are significant.

DECIMAL PLACES → Rounding to 1 dec. place is the same as rounding to the nearest tenth.
→ Rounding to 2 d.p. is like rounding to the nearest hundredth, etc.

[q]

When rounding to n sig. figs., if the (n+1)ᵗʰ figure is < 5, keep the nᵗʰ figure the same. If ≥ 5, then increase the nᵗʰ figure by one.

– E.G. 1: Round 0.00726 to 2 sig. figs.:
→ Go to ‘2’, check ‘6’, raise ‘2’ to ‘3’ → 0.0073.

– E.G. 2: Round 43.8037 to 2 dec. places:
→ 0 is 2ⁿᵈ d.p., 3 < 5, so keep ‘0’: 43.80.

[a]

UPPER & LOWER BOUND — This refers to the process of saying how low or high a number could have been before rounding. For example, 3.1 (to 2 sig. figs) could have been as low as 3.05, and as high (but not including) 3.15. i.e., 3.05 ≤ x < 3.15.

– E.G. 3: If x = 7860 to 3 sig. figs., give an upper & lower bound for x:
→ 7855 ≤ x < 7865.

[q]

You could have a more complex problem of this kind:

– E.G. 4: A circle has a radius of 3.1 cm, to 1 d.p., give an upper bound for the area:
→ With r = 3.1, area = 30.19 cm². U. Bound of radius = 3.15 cm, max area = 3.15²π = 31.7.

[a]

PERCENTAGE ERROR — As a percentage of the actual answer, how far off is your estimate/guess/rounded answer? This is what is called the percentage error.

FORMULA
\[
\text{ERROR} = \frac{| \text{Actual} – \text{Estimate} |}{\text{Actual}} \times 100\%
\]

– E.G. 5: Est = 3000, actual = 3203.5, find error:
\[
\frac{|3203.5 – 3000|}{3203.5} \times 100\% = 6.35\%
\]

You should round to 3 significant figures in exams unless told otherwise.

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