Significant Figures in Measurement and Calculations
A successful chemistry student habitually labels all numbers, because the unit is important. Also of great importance is the number itself. Any number used in a calculation should contain only figures that are considered reliable; otherwise, time and effort are wasted. Figures that are considered reliable are called significant figures . Chemical calculations involve numbers representing actual measurements. In a measurement, significant figures in a number consist of:
Figures (digits) definitely known + One estimated figure (digit) In class you will hear this expressed as "all of the digits known for certain plus one that is a guess." Recording Measurements
When one reads an instrument (ruler, thermometer, graduate, buret, barometer, balance), he expresses the reading as one which is reasonably reliable. For example, in the accompanying illustration, note the reading marked A. This reading is definitely beyond
the 7 cm mark and also beyond the 0.8 cm mark. We
read the 7.8 with certainty. We further estimate that
the reading is five-tenths the distance from the 7.8
mark to the 7.9 mark. So, we estimate the length as
0.05 cm more than 7.8 cm. All of these have meaning
and are therefore significant. We express the reading as 7.85 cm, accurate to three significant figures. All of these figures, 7.85, can be used in calculations. In reading B we see that 9.2 cm is definitely known. We can include one estimated digit in our reading, and we estimate the next digit to be zero. Our reading is reported as 9.20 cm. It is accurate to three significant figures. Rules for Zeros
If a zero represents a measured quantity, it is a significant figure. If it merely locates the decimal point, it is not a significant figure.
Zero Within a Number. In reading the measurement 9.04 cm, the zero represents a measured quantity, just as 9 and 4, and is, therefore, a significant number. A zero between any of the other digits in a number is a significant figure.
Zero at the Front of a Number. In reading the measurement 0.46 cm, the zero does not represent a measured quantity, but merely locates the decimal point. It is not a significant figure. Also, in the measurement 0.07 kg, the zeros are used merely to locate the decimal point and are, therefore, not significant. Zeros at the first (left) of a number are not significant figures . Zero at the End of a Number. In reading the measurement 11.30 cm, the zero is an estimate and represents a measured quantity. It is therefore significant. Another way to look at this: The zero is not needed as a placeholder, and yet it was included by the person recording the measurement. It must have been recorded as a part of the measurement, making it significant. Zeros to the right of the decimal point, and at the end of the number, are significant figures .
Zeros at the End of a Whole Number. Zeros at the end of a whole number may or may not be significant. If a distance is reported as 1600 feet, one assumes two sig figs. Reporting measurements in scientific notation removes all doubt, since all numbers written in scientific notation are considered 3
1 600 feet
1.6 x10 feet
Two significant figures
1 600 feet
1.60 x 10 feet
Three significant figures
1 600 feet
1.600 x 10 feet
Four significant figures
Sample Problem #1: Underline the significant figures in the following numbers. (a) 0.0420 cm
answer = 0.0420 cm
(e) 2 403 ft.
answer = 2 403 ft.
(b) 5.320 in.
answer = 5.320 in.
(f) 80.5300 m
answer = 80.5300 m
(c) 10 lb.
answer = 10 lb.
(g) 200. g
answer = 200 g
(d) 0.020 ml
answer = 0.020 ml
(h) 2.4 x 10 kg
answer = 2.4 x 10 k g
Rounding Off Numbers
In reporting a numerical answer, one needs to know how to "round off" a number to include the correct number of significant figures. Even in a series of operations leading to the final answer, one must "round off" numbers. The rules are well accepted rules:
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