Upper and Lower Bounds

Bounds from Rounding

A value rounded to some degree of accuracy has bounds:

$\text{Lower bound} = \text{rounded value} - \frac{1}{2} \text{unit}$ $\text{Upper bound} = \text{rounded value} + \frac{1}{2} \text{unit}$

Example: 5.3 cm (to 1 d.p.): LB = 5.25, UB = 5.35.

Example: 200 (to nearest 10): LB = 195, UB = 205.

Error Interval

$5.25 \leq x < 5.35$

Note: lower bound is included, upper bound is excluded (since it would round up).

Bounds in Calculations

Operation	Maximum	Minimum
$a + b$	UB(a) + UB(b)	LB(a) + LB(b)
$a - b$	UB(a) - LB(b)	LB(a) - UB(b)
$a \times b$	UB(a) × UB(b)	LB(a) × LB(b)
$a \div b$	UB(a) ÷ LB(b)	LB(a) ÷ UB(b)