To find the bounds of a rounded number, identify its rounding unit, halve that unit, then subtract and add the half-unit to the rounded value. The lower bound is included, while the upper bound is excluded.
This is SL 1.6, Approximating and estimating in IB Mathematics: Applications and Interpretation. It applies to both SL and HL and may be tested on any GDC-active paper.
Suppose a value rounds to to the nearest unit . Its bounds are:
Therefore, the original value satisfies:
The lower endpoint is included because it rounds to . The upper endpoint is excluded because that exact value rounds to the next possible rounded number.
| Rounded value | Rounding unit | Bounds |
|---|---|---|
| to the nearest | ||
| to the nearest | ||
| to two significant figures |
For significant figures, determine the place value of the final significant digit. In to two significant figures, the final significant digit is in the hundreds place, so the rounding unit is .
For example, a length recorded as to the nearest has half-unit . Hence:
Bounds are also used when a rounded measurement enters a later calculation. For example, if a rectangle's length and width are rounded independently, the greatest possible area is found by multiplying their upper bounds, while the least possible area uses their lower bounds. However, division needs more care: the greatest positive quotient uses the numerator's upper bound and denominator's lower bound. This is why each quantity's role must be considered.
A common misconception is to add and subtract the full rounding unit. This is incorrect: always use half the rounding unit.
State the rounding unit before calculating. Use at the lower endpoint and at the upper endpoint. For maximum or minimum calculations, select the combination of bounds that produces the required extreme; do not automatically use every upper bound or every lower bound. Keep full precision until the final answer.
