Bounds and error intervals · Revision guide

Learning objectives

For usual rounding of positive measurements, use half the rounding unit on each side.

v-\frac u2\le x<v+\frac u2

Positive values truncated to one decimal place keep the stated lower endpoint.

v\le x<v+0.1

For positive measurements, use opposite bounds; the denominator lower bound must be strictly positive.

\text{upper bound}=\frac{a_U}{b_L},\qquad\text{lower bound}=\frac{a_L}{b_U}

Keep intermediate bounds exact. An upper bound may not be attained.

Distance is 120 km to the nearest km; time is 2.4 hours to the nearest 0.1 hour. Find the speed upper bound.

119.5\le d<120.5\ \text{km}

2.35\le t<2.45\ \text{hours}

Use the distance upper bound and time lower bound.

v_U=\frac{120.5}{2.35}=51.276595\ldots\ \text{km/h}

v_U\approx51.28\ \text{km/h}

A length is 4.7 cm to the nearest 0.1 cm. Write the interval.

\frac{0.1}{2}=0.05\ \text{cm}

4.65\le l<4.75\ \text{cm}

Do not use the denominator upper bound for an upper quotient bound. Rounding and truncation give different intervals.