Question

Significant figures and unit conversion are important to scientist. Please elaborate/expand on why these skills are important. Examples?

Answer 1

When working with EXPERIMENTAL VALUES in SCIENCE and ENGINEERING, we are typically concerned with the PRECISION of VALUES used in CALCULATIONS.

SIGNIFICANT FIGURES are used to ACCOUNT for the UNAVOIDABLE UNCERTAINTY in the MEASUREMENT of VALUES.

When performing CALCULATIONS, you will want to take NOTE of the number of SIGNIFICANT FIGURES of each value, to maintain a consistent level of PRECISION in your CALCULATIONS.

There are FIVE RULES that govern the application of SIGNIFICANT FIGURES in calculations:

1. Non-zero digits are always SIGNIFICANT. For example, the number 4.8383 has 5 significant figures and the number 5.21 contains 3 significant figures.

2. Any zeros between two significant digits or nonzero digits are SIGNIFICANT. For example, the number 1.08 has 3 significant figures and the number 3.2808 has5 significant figures.

3. Zeros before the decimal point are placeholders and NOT SIGNIFICANT. For example, the number 0.000482 has three significant figures and the number 0.005011 has four significant figures

4. Zeros after the decimal and after figures are SIGNIFICANT. For example, the number 0.3210 has four significant figures, and the number 0.120 has three significant figures.

5. Exponential digits in scientific notation are NOT SIGNIFICANT. For example, the number 1.38 x 10^6 has three significant figures, and the number 0.1082 x 10^-23 has four significant figures.

As SIGNIFICANT FIGURES are used to reflect the AMOUNT of PRECISION in MEASUREMENTS, there are two rules that govern the ARITHMETIC OPERATIONS of SIGNIFICANT FIGURES in CALCULATIONS.

Outside of these TWO RULES, the normal ORDER of OPERATIONS should be followed for all CALCULATIONS.

Rule 1: Addition and Subtraction

When ADDING or SUBTRACTING values, the FINAL VALUE must have only as many DECIMALS as the LEAST PRECISE MEASUREMENT with the LEAST number of DECIMAL PLACES.

In ADDITION, or SUBTRACTION, the answer cannot have MORE DIGITS to the RIGHT of the decimal point than either of the original numbers.

For example, let’s CALCULATE the SUM of three MASS measurements given as 153 g, 1.8 g, and 9.16 g:

1.53 g + 1.8 g + 9.16 g =163.96 g

As the LEAST PRECISE MEASUREMENT is 153 g with 0 DECIMAL PLACES, the FINAL VALUE of the SUM is shown as:

164 g

Rule 2: Multiplication and Division

When MULTIPLYING or DIVIDING, the FINAL VALUE can only have as many SIGNIFICANT FIGURES as the LEAST PRECISE MEASUREMENT with the LEAST number of SIGNIFICANT FIGURES.

In MULTIPLICATION, and DIVISION, the number of SIGNIFICANT FIGURES in the PRODUCT or QUOTIENT, is determined by the LEAST PRECISE MEASUREMENT that has the FEWEST NUMBER of SIGNIFICANT FIGURES.

For example, let’s CALCULATE the VALUE requiring the MULTIPLICATION and DIVISION of three values as shown as below:

(34.78 x 11.7)/(0.17)

As the LEAST PRECISE MEASUREMENT is 0.17 with 2 SIGNIFICANT FIGURES, the final ANSWER must have TWO SIGNIFICANT FIGURES.

You take the 2393.682353 value from your CALCULATOR and round it to 2,400 and EXPRESS it in SCIENTIFIC NOTATION with 2 SIGNIFICANT FIGURES as:

2.4 x 10^2

• Examples of rounding to the correct number of significant figures with a 5 as the first non-significant figure
  
  • Round 4.7475 to 4 significant figures: 4.7475 becomes 4.748 because the first non-significant digit is 5, and we round the last significant figure up to 6 to make it even.
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  • Round 4.7465 to 4 significant figures: 4.7465 is 4.746 because the first non-significant digit is 5 and since the last significant digit is even, we leave it alone.
    
• An example of a calculation where you can "lose" significant figures doing an operation.
  
  The mass of ¹⁹F is 18.99840 u. How much mass is converted to energy when a ¹⁹F atom is assembled from its constituent protons, neutrons, and electrons?
  
  ¹⁹F 9 p⁺ + 9 e^- + 10 n⁰