Significant Digits in Calculation

When two or more numbers are added, subtracted, multiplied or divided, how to decide about the number of significant digits in the answer? For example, suppose the mass of a body \(A\) is measured to be \(12.0 \mathrm{~kg}\) and of another body \(B\) to be \(7.0 \mathrm{~kg}\). What is the ratio of the mass of \(A\) to the mass of \(B\)? Arithmetic will give this ratio as
\(
\frac{12 \cdot 0}{7 \cdot 0}=1 \cdot 714285 \ldots
\)
However, all the digits of this answer cannot be significant. The zero of \(12.0\) is a doubtful digit and the zero of \(7 \cdot 0\) is also doubtful. The quotient cannot have so many reliable digits. The rules for deciding the number of significant digits in an arithmetic calculation are listed below.

Case-1: In a multiplication or division of two or more quantities, the number of significant digits in the answer is equal to the number of significant digits in the quantity which has the minimum number of significant digits. Thus, \(\frac{12 \cdot 0}{7 \cdot 0}\) will have two significant digits only.

The insignificant digits are dropped from the result if they appear after the decimal point. They are replaced by zeros if they appear to the left of the decimal point. The least significant digit is rounded according to the rules given below.

If the digit next to the one rounded is more than 5, the digit to be rounded is increased by 1. If the digit next to the one rounded is less than 5, the digit to be rounded is left unchanged. If the digit next to the one rounded is 5, then the digit to be rounded is increased by 1 if it is odd and is left unchanged if it is even.

Case-2: For addition or subtraction write the numbers one below the other with all the decimal points in one line. Now locate the first column from the left that has a doubtful digit. All digits right to this column are dropped from all the numbers and rounding is done to this column. Addition or subtraction is now performed to get the answer.

Example 1: Round off the following numbers to three significant digits (a) 15462, (b) \(14 \cdot 745\), (c) \(14 \cdot 750\) and (d) \(14 \cdot 650\) \(\times 10^{12}\)

Solution:

(a) The third significant digit is 4. This digit is to be rounded. The digit next to it is 6 which is greater than 5. The third digit should, therefore, be increased by 1. The digits to be dropped should be replaced by zeros because they appear to the left of the decimal. Thus, 15462 becomes 15500 on rounding to three significant digits.

(b) The third significant digit in \(14 \cdot 745\) is 7. The number next to it is less than 5. So \(14 \cdot 745\) becomes \(14 \cdot 7\) on rounding to three significant digits.

(c) \(14 \cdot 750\) will become \(14 \cdot 8\) because the digit to be rounded is odd and the digit next to it is 5.

(d) \(14 \cdot 650 \times 10^{12}\) will become \(14 \cdot 6 \times 10^{12}\) because the digit to be rounded is even and the digit next to it is 5.

Example 2: Evaluate \(\frac{25 \cdot 2 \times 1374}{33 \cdot 3}\). All the digits in this expression are significant.

Solution:
We have \(\frac{25.2 \times 1374}{33.3}=1039 \cdot 7838 \ldots\)
Out of the three numbers given in the expression \(25.2\) and \(33.3\) have 3 significant digits and 1374 has four. The answer should have three significant digits. Rounding \(1039 \cdot 7838 \ldots\) to three significant digits, it becomes 1040. Thus, we write
\(\frac{25.2 \times 1374}{33.3}=1040.\)

Example 3: 24.36 + 0.0623 + 256.2.

Solution:

\(
\begin{gathered}
\quad 24 \cdot 36 \\
\quad \quad 0.0623 \\
256.2 \\
\hline
\end{gathered}
\)
Now the first column where a doubtful digit occurs is the one just next to the decimal point (256.2). All digits right to this column must be dropped after proper rounding. The table is rewritten and added below
\(
\begin{array}{r}
24 \cdot 4 \\
0 \cdot 1 \\
256 \cdot 2 \\
\hline
280 \cdot 7
\end{array}
\)
The sum is \(280.7\).