# What are significant figures and why do they matter?

May 8, 2018

Significant figures tell us what amount of uncertainty we have in a reported value. The more digits you have, the more sure of yourself you are. That is why you should almost never report all the decimal places you see in your calculator.

The following is a reference for what counts as significant figures.

The following are rules for determining significant figures/digits:

NONZERO DIGITS

• All of them count, except if subscripted or past an underlined digit.

EX: $0.0 \textcolor{b l u e}{1} 0 \textcolor{b l u e}{3}$ has 2 significant nonzero digits.

EX: $0. \textcolor{b l u e}{102 \underline{4}} 5293$, or 0.color(blue)(1024)_(5293, is stated to only have 4 significant digits.

SCIENTIFIC NOTATION

• All digits here are significant. This is written so that the number to the left of $\times$ is between $1. \overline{00}$ and $9. \overline{99}$.

EX: $\textcolor{b l u e}{2.015000} \times {10}^{23}$ has 7 significant digits.

ZERO DIGITS

• Leading zeroes do NOT count.

EX: $\textcolor{red}{00} 7$ has two leading zeroes that do not matter. We could just say $7$ and it numerically says the same thing.

EX: $\textcolor{red}{0} . \textcolor{red}{0000} 23$ has 5 leading zeroes, none of which are significant.

• Trailing zeroes after a decimal point DO count.

EX: $2 \textcolor{b l u e}{0} . \textcolor{b l u e}{00}$ has 3 significant trailing zeroes (1 before, and 2 after the decimal point).

• Trailing zeroes in a number larger than $1$ that have a decimal point placed after them are still significant, but no decimal point would be ambiguous.

EX: $2 \textcolor{b l u e}{000} .$ has 3 significant zeroes, although it is better to write this as $2. \textcolor{b l u e}{000} \times {10}^{3}$, scientific notation.

NOTE: If we write it as $1000$, we might report it as 1 significant digit, unless it is part of a unit conversion and thus exact. So, $\text{1000 g/kg}$ does not affect significant figures in a calculation.

• Sandwiched zeroes DO count, except if no previous digits are nonzero.

EX: $2 \textcolor{b l u e}{00} 2$ has two significant zeroes, but $0.01 \textcolor{b l u e}{0} 3$ has only 1 significant zero.