Estimating and Bounds with Significant Figures Cheat Sheet

Key Facts

• The first significant figure is the first nonzero digit, so in $0.00472$ the first significant figure is $4$.
• To round to $n$ significant figures, keep $n$ significant digits and use the next digit to decide whether to round up or stay the same.
• Zeros between nonzero digits are significant, so $506$ has $3$ significant figures.
• Leading zeros are not significant, so $0.0038$ has $2$ significant figures.
• Trailing zeros after a decimal point are significant, so $2.40$ has $3$ significant figures.
• The percentage error is $\frac{|\text{approximate} - \text{exact}|}{\text{exact}} \times 100\%$.
• If a value is rounded to the nearest unit, its lower bound is $x - 0.5$ and its upper bound is $x + 0.5$.
• If a value is rounded to the nearest $0.1$, its bounds are $x - 0.05 \leq \text{exact value} < x + 0.05$.

Vocabulary

Significant figure

A significant figure is a digit that shows the meaningful precision of a number, starting with the first nonzero digit.

Estimate

An estimate is an approximate answer found by rounding numbers to make a calculation easier.

Upper bound

An upper bound is the largest possible value that could round to a given rounded number.

Lower bound

A lower bound is the smallest possible value that could round to a given rounded number.

Interval

An interval is a range of possible values, often written as $a \leq x < b$.

Percentage error

Percentage error compares the size of an error with the exact value using $\frac{|\text{error}|}{\text{exact}} \times 100\%$.