Theory — Significant figures and mixed-operation rounding
Significant figures are the digits in a measured quantity that carry information about precision. They tell the reader how
carefully a value was measured, not just what the value is numerically. This matters in science because a reported answer
should never look more precise than the measurements that produced it. The basic purpose of significant-figure rules is
therefore to keep the final reported result honest.
The first step is to decide which digits are significant and which are only placeholder zeros. All non-zero digits are always
significant. Zeros between non-zero digits are also significant because they lie inside the measured part of the number.
Leading zeros are not significant because they only locate the decimal point. Trailing zeros are significant when a decimal
point is shown, because the decimal point tells you those zeros were intentionally reported. For example, 0.00450
has three significant figures, while 450. has three significant figures and 450 is normally read as only two.
Examples of counting.
\[
\begin{aligned}
0.00450 &\rightarrow 3\ \text{significant figures} \\
1002 &\rightarrow 4\ \text{significant figures} \\
450. &\rightarrow 3\ \text{significant figures} \\
2.80\times 10^{3} &\rightarrow 3\ \text{significant figures}
\end{aligned}
\]
Addition and subtraction
For addition and subtraction, the key idea is place value, not the total number of significant figures. The answer must be
rounded to the least precise decimal place among the terms being added or subtracted. In other words, if one value is only
known to tenths, then the final sum or difference cannot honestly claim hundredths or thousandths.
Addition/subtraction rule.
\[
\begin{aligned}
\text{Report the final value to the least precise place among the terms.}
\end{aligned}
\]
Example: in 7.250 - 2.04, the first number reaches thousandths, but the second only reaches hundredths. The difference
must therefore be reported to hundredths.
Worked subtraction example.
\[
\begin{aligned}
7.250 - 2.04 &= 5.210 \\
&\rightarrow 5.21
\end{aligned}
\]
Multiplication and division
For multiplication and division, the rule changes. Here the final answer is limited by the fewest significant figures among
the factors or quotients. If one quantity has only two significant figures, then the product or quotient cannot be reported
with three or four reliable figures.
Multiplication/division rule.
\[
\begin{aligned}
\text{Report the final value with the fewest significant figures among the operands.}
\end{aligned}
\]
Example: in 12.7 / 0.0045, the numerator has three significant figures but the denominator has only two, so the quotient
must be reported with two significant figures.
Worked division example.
\[
\begin{aligned}
\frac{12.7}{0.0045} &= 2822.222\ldots \\
&\rightarrow 2.8\times 10^{3}
\end{aligned}
\]
What to do in a mixed expression
Mixed expressions combine both ideas. You still follow the normal order of operations, so multiplication and division are
evaluated before addition and subtraction unless parentheses change the order. However, the precision information from each
intermediate step must be carried forward. A useful classroom habit is to keep guard digits internally so you do not create
extra rounding error too early, while still remembering which place or how many significant figures the intermediate result
is allowed to claim.
Consider the expression 12.345 + 1.2 × 3.45. The multiplication comes first. The product is numerically 4.14, but
because the factors have two and three significant figures, the product may only be reported with two significant figures,
namely 4.1. When that term is later added to 12.345, the addition must be rounded to the least precise place among the
terms being combined, which is the tenths place.
Worked mixed example.
\[
\begin{aligned}
1.2\cdot 3.45 &= 4.14 \\
&\rightarrow 4.1\ \text{(2 s.f.)}
\end{aligned}
\]
\[
\begin{aligned}
12.345 + 4.14 &= 16.485 \\
&\rightarrow 16.5\ \text{(tenths place)}
\end{aligned}
\]
That is why a good calculator does not simply count significant figures once at the start and once at the end. It must track
the precision of each typed quantity, apply the appropriate rule at each operation, and then explain why the last reported
digit sits where it does.
Scientific notation
Scientific notation is especially useful because it makes significant figures explicit. The number 2.80e3 clearly means
2.80 × 103 and therefore has three significant figures. This avoids ambiguity for large or small values. It is often
the cleanest way to report multiplication or division results such as 2.8 × 103, where the significant-figure count would
be harder to read in plain decimal form.
Scientific-notation interpretation.
\[
\begin{aligned}
2.80\times 10^{3} &\rightarrow 3\ \text{significant figures} \\
4.50\times 10^{-3} &\rightarrow 3\ \text{significant figures}
\end{aligned}
\]
In laboratory work, engineering reports, and chemistry or physics homework, these rules matter because the reported answer
should reflect the quality of the measured data. Significant figures are therefore not cosmetic formatting. They are part of
the meaning of the number itself.
