Significant figures in calculations

20 Indicate the significant zeros, if any, in each of the following 2.24 Identify the measurement in each of the following pairs that 

22 How many significant figures are in each of the following significant figures  

In the sciences, we measure many things the length of a bacterium, the volume of a gas sample, the temperature of a reaction mixture, or the mass of iron in a sample. The numbers obtained from these types of measurements are often used in calculations. The number of significant figures in the measured numbers determines the number of significant figures in the calculated answer. 

Using a calculator will help you perform calculations faster. However, calculators cannot think for you. It is up to you to enter the numbers correctly, press the correct function keys, and give an answer with the correct number of significant figures. 

Keeping track of the number of significant figures in calculations depends on the kind of calculation you are doing. 

Multiplication and division The product or quotient can have no more significant figures than the number with the smallest number of significant figures used in the calculation. 

Addition and subtraction The sum or difference can have no more places after the decimal than there are in the number with the smallest number of digits after the decimal. 

In addition and subtraction, those places after the decimal that are of unknown value ( ) negate the numerals in those same places in the other numbers used in the calculation. 

Example 3.45762 t— five places after the decimal are known 

For example, how should we report the answer in the problem below to the proper number of significant figures  

When adding, the answer should have the same number of decimal places as the addend with the smallest number of decimal places. The second number has only one decimal place. Therefore the answer can have only one decimal place and is correctly reported as 106.3 

Each number has four significant figures. Therefore the correct answer is 411.6 (four significant figures). 

Each time you use a calculator, it is important to look at the original measurements and determine the number of significant figures that can be used for the answer. You can use the following rules to round off the numbers shown in a calculator display. 

If the first digit to be dropped is 4 or less, then it and all following digits are simply 

The calculator shows 23.4225 cm, but you should report the answer as 23 cm, with two significant figures, because the length and width measurements determine the overall certainty, and they contain only two significant figures. 

For addition and subtraction. The answer has the same number of decimal places as there are in the measurement with the fewest decimal places. Suppose you measure 83.5 mL of water in a graduated cylinder and add 23.28 mL of protein solution from a buret. The total volume is 

Volume (mL) = 83.5 mL + 23.28 mL = 106.8 mL Here the calculator shows 106.78 mL, but you report the volume as 106.8 mL, with one decimal place, because the measurement with fewer decimal places (83.5 mL) has one decimal place. 

If the digit removed is more than 5, the preceding number is increased by 1 5.379 rounds to 5.38 if three significant figures are retained and to 5.4 if two significant figures are retained. 

If the digit removed is less than 5, the preceding number is unchanged  

The general rule for rounding is that the least certain measurement sets the limit on certainty for the entire calculation and determines the number of significant figures in the final answer. Suppose you want to find the density of a new ceramic. You measure the mass of a piece on a precise laboratory balance and obtain 3.8056 g you measure its volume as 2.5 mL by displacement of water in a graduated cylinder. The mass has five significant figures, but the volume has only two. Should you 

After numbers are obtained by a laboratory measurement, they are normally subjected to mathematical operations to get the desired final result. It is important that the answer have the correct number of significant figures. It should not have so few that accuracy is sacrificed or so many that an unjustified degree of accuracy 

Example Number Number of Significant Digits Rule 

397 5 1. Nonzax) digits in a number are always significant. The 1, 1,3,9, and 7 in fiiis numbm ate each significant 

039 6 2. Zeros between nonzero digits are significant The 1,4,0,0,3, and 9 in fiiis number are each significant 

00329 3 3. Zeros to the left of the first nonzero digit arc not significant because th are used only to locate the decimal point Only 3,2, and 9 in this number are significant 

When carrying measured quantities through calculations, the least certain measurement limits the certainty of the calculated quantity and thereby determines the number ofsign -cant figures in the final answer. The final answer should be reported with only one uncertain digit. To keep track of significant figures in calculations, we will make frequent use of two rules, one for addition and subtraction, and another for multiplication and division. 

We report the result as 104.8 because 83.1 has only one decimal place. 

Area = (6.221cm)(5.2cm) = 32.3492cm Ground off to 32 cm because 5.2 has two significant figures. 

The results of a calculation based on measurements cannot be least precise measurement. 

In calculations involving multiplication or division, the answer must contain the same number of significant figures as in the measurement that has the least number of significant figures. Consider the following examples  

The value 438.38 was obtained with a calculator. The answer should have two significant figures because 2.3, the number with the fewest significant figures, has only two significant figures. 

Move the decimal two places to the left to express in scientific notation. 

A researcher reports that the Spirit rover on the surface of Mars recently measured the temperature to be —25.49 °F. What is the actual temperature  

When we use measured quantities in calculations, the results of the calculation must reflect the precision of the measured quantities. We should not lose or gain precision during mathematical operations. 

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Laboratories use the concept of significant figures in manual calculations for standard and sample preparation, and in data reduction. The rule for determination of the number of significant figures in calculations is as follows When experimental quantities are multiplied and divided, the final result cannot be more accurate than the least precise measurement. 

The following rules determine how to use significant figures in calculations that involve measurements. 

However, your value can be no more accurate than that as long as you are using a meter stick. It is possible that you may obtain a more sophisticated device, like a sonic range-finder, that can provide you with a more accurate measurement. The important thing to remember is that your data can be no more accurate than the device with which you are measuring. This concept is the foundation for the use of significant figures in calculations. 

A second set of rules specifies how to handle significant figures in calculations. 

Scientific Notation Significant Figures in Calculation of Results Rounding Off Numbers... 

Determining Significant Digits Significant Figures in Calculations Precisbn and Accuracy... 

A second set of rules specifies how to handle significant figures in calculations. In addition and subtraction, the answer cannot have more digits to the right of the decimal point than either of the original numbers. Consider these examples ... 

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