# What is the measurement 111.009 mm rounded off to four significant digits?

Jun 14, 2016

First, you may want to review significant figures (sig figs).

The (snippeted) relevant rules are:

PROMINENT DEFINITIONS

• All nonzero digits are significant. EX: $\textcolor{b l u e}{21} 0$
• A zero is sandwiched when there is at least one nonzero digit to the left AND right. All zeros in between two nonzero digits, even if multiple zeros are adjacent, are significant. EX: $5 \textcolor{b l u e}{00} 5$

SPECIAL CONDITIONS

• All zeroes AFTER an implicit or explicit decimal point are subject to the following conditions:
• If the number is GREATER than $1$ (if the units digit is $1$ or greater), AND the decimal point directly precedes the zero(es), the zero(es) are significant. Otherwise, the zeroes are NOT significant. EX: $1 \textcolor{b l u e}{.0} 50$ vs. $0 \textcolor{red}{.0} 50$
• If it is sandwiched between two nonzero digits, even if the decimal point directly precedes the zero, it is significant. EX: $5 \textcolor{b l u e}{.0} 5$

Looking at $\text{111.009 mm}$, it currently has $6$ sig figs:

• $4$ nonzero digits
• $2$ sandwiched zeroes

For $4$ sig figs only, we examine the $5$th sig fig to see how we should round.

$\text{111.0"color(red)(ul(0))"9 mm}$

1. Since it is less than $5$, we round it and everything to its right, downwards, as per normal algebra rounding rules.
2. Then we get rid of every digit past the $5$th digit from the left, including the $5$th digit.

$\text{111.0} \cancel{\textcolor{red}{0}} \cancel{\textcolor{red}{9}}$ $\text{mm}$

$\implies$ $\textcolor{b l u e}{\text{111.0 mm}}$

The remaining sig figs are:

• $3$ nonzero digits
• $1$ trailing zero