Chapter 1, Problem 1.43PAE

1.43 Calculate the following to the correct number of significant figures. Assume that all these numbers are measurements.

(a) $x=17.2+65.18-2.4$

(b) $x=13.0217/17.10$

(c) $x=\left(0.0061020\right)\left(2.0092\right)\left(1200.00\right)$

(d) $x=0.0034+\frac{\sqrt{{\left(0.0034\right)}^2+4\left(1.000\right)\left(6.3\times {10}^{-1}\right)}}{\left(2\right)\left(1.000\right)}$

Interpretation Introduction

To perform: The calculations of $17.2+65.18-2.4$and express the result in proper number of significant figure.

Explanation of Solution

First, set the rules for “significant numbers”

• zeroes to the left are not significant
• zeroes to the right are not significant if not specified as “exact” zeroes
• all other digits other than zero are significant.
• zeroes between non-zero numbers are significant

Second, define the addition/subtraction and multiplication/division rules for significant numbers

• Addition/Subtraction ( Always keep the least significant number after the decimal point.
• Multiplication/Division ( Always keep the least significant number of the whole number.
• If addition/subtraction and Multiplication/division are included, separate the operation to simple operations and keep following the rules.

Now, for the given operation.$17.2+65.18-2.4$

$17.2+65.18-2.4=79.98$

We must keep only 1 significant figure after the decimal. Round if required$79.98\to 80.0$

Interpretation Introduction

To perform: The calculations of $x=13.0217/17.10$and express the result in proper number of significant figure.

Explanation of Solution

First, set the rules for “significant numbers”

• zeroes to the left are not significant
• zeroes to the right are not significant if not specified as “exact” zeroes
• all other digits other than zero are significant.
• zeroes between non-zero numbers are significant

Second, define the addition/subtraction and multiplication/division rules for significant numbers

• Addition/Subtraction ( Always keep the least significant number after the decimal point.
• Multiplication/Division ( Always keep the least significant number of the whole number.
• If addition/subtraction and Multiplication/division are included, separate the operation to simple operations and keep following the rules.

Now, for the given operation. Substitute$13.0217/17.10$

$0.7615029$

According to the rules, we must keep the number with the least sig. fig. This is 17.10, with only 4 significant figure then$0.7615029\to 0.7615$

Interpretation Introduction

To perform: The calculations of $(0.0061020)(2.0092)(1200.00)$and express the result in proper number of significant figure.

Explanation of Solution

First, set the rules for “significant numbers”

• zeroes to the left are not significant
• zeroes to the right are not significant if not specified as “exact” zeroes
• all other digits other than zero are significant.
• zeroes between non-zero numbers are significant

Second, define the addition/subtraction and multiplication/division rules for significant numbers

• Addition/Subtraction ( Always keep the least significant number after the decimal point.
• Multiplication/Division ( Always keep the least significant number of the whole number.
• If addition/subtraction and Multiplication/division are included, separate the operation to simple operations and keep following the rules.

Now, for the given operation. Perform calculation$\begin{array}{l}(0.0061020)(2.0092)(1200.00)\\ 14.712166\end{array}$

The significant figures are given as:$\begin{array}{l}0.0061020\to 5\\ 2.0092\to 5\\ 1200.00\to 6\end{array}$

We must keep the smallest amount of sig. fig. for multiplication, therefore, choose 5.$14.712166\to 14.712$

Interpretation Introduction

To perform: The calculations of $0.0034+\frac{\sqrt{{(0.0034)}^2+4(1.000)(6.3x{10}^{-4})}}{(2)(1.000)}$and express the result in proper number of significant figure.

Explanation of Solution

First, set the rules for “significant numbers”

• zeroes to the left are not significant
• zeroes to the right are not significant if not specified as “exact” zeroes
• all other digits other than zero are significant.
• zeroes between non-zero numbers are significant

Second, define the addition/subtraction and multiplication/division rules for significant numbers

• Addition/Subtraction ( Always keep the least significant number after the decimal point.
• Multiplication/Division ( Always keep the least significant number of the whole number.
• If addition/subtraction and Multiplication/division are included, separate the operation to simple operations and keep following the rules.

Now, for the given operation. Perform calculation$\begin{array}{l}0.0034+\frac{\sqrt{{(0.0034)}^2+4(1.000)(6.3x{10}^{-4})}}{(2)(1.000)}\\ 0.0034+\frac{\sqrt{0.000012+25.2x{10}^{-4}}}{2.000}\\ 0.0034+\frac{\sqrt{0.12x{10}^{-4}+25.2x{10}^{-4}}}{2.000}\end{array}$

For the inside addition, keep only 1 significant figure after the digit, since 0.12 has 2 significant figures after the digit and 25.2 has only 1.$0.0034+\frac{\sqrt{25.32x{10}^{-4}}}{2.000}\to 0.0034+\frac{\sqrt{25.3x{10}^{-4}}}{2.000}$

When performing the inside of the square root$0.0034+\frac{0.050299}{2.000}\to 0.0034+\frac{0.0503}{2.000}$

Then, continue with the division. Keep the least amount of significant figures in total.$0.0034+0.02515\to 0.0034+0.0252$

Now, add all.$0.0034+0.0252=0.0286$

Keep the least amount of significant figures after the decimal point. Which is 4 for both amounts.$0.0286$

Conclusion

Always ensure to keep the correct significant numbers when performing addition, subtraction, multiplication and division. It is relevant to keep track of accuracy in the measurements.