1.10
Rules for Significant Figures
In any measurement, the precision of the measuring tool is an essential factor. An ordinary ruler, for example, can measure length to the closest mill…
The certainty of measurement depends on two factors: the number of digits in the measurement and the precision of the instrument used.
In a measured quantity, all of the digits, including the last uncertain digit, are called significant figures - and can be determined using specific rules.
Any non-zero digits and all captive zeros - which lie between two non-zero digits - are significant. For example, 28 has two significant figures, while 26.25 has four, and 208 has three.
Leading zeros are never significant, as they just locate the decimal point. For example, 0.00208 has three significant figures. Such quantities can be expressed using exponential notations. Thus, 0.00208 can be written as 2.08 × 10−3.
Trailing zeros are only significant in decimal formatted numbers. 2200 has two trailing zeros and two significant figures, whereas 2200.0 and 2200.1 both have 5 significant figures.
For quantities without decimal points, the significance of trailing zeros becomes ambiguous. Thus, 2200 can be written as 2.2 × 103 with two significant figures or 2.20 x 103 with three significant figures.
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Q1: What are significant figures and why do they matter in measurements?
Significant figures are all digits in a measured quantity, including the last uncertain digit, that reflect the precision of the measuring instrument. They represent how precisely a measurement was taken. The certainty of any measurement depends on both the number of digits recorded and the precision of the instrument used. More precise instruments yield measurements with more significant figures.
Q2: How do you count significant figures in a number?
Start from the first non-zero digit on the left and count all digits through the last digit written on the right. Non-zero digits and captive zeros between non-zero digits are always significant. Leading zeros never count as they only locate the decimal point. For example, 0.00208 has three significant figures, while 26.25 has four.
Q3: Why are leading zeros not considered significant figures?
Leading zeros serve only to locate the decimal point and do not represent measured precision. They are placeholders that indicate the magnitude of the number. For instance, 0.00208 and 2.08 × 10−3 represent the same value with identical significant figures. The zeros in front simply position the decimal without adding measurement information.
Q4: What is the difference between trailing zeros in whole numbers versus decimal numbers?
Trailing zeros in decimal-formatted numbers are always significant. However, trailing zeros in whole numbers without decimal points create ambiguity. For example, 2200 has two significant figures, but 2200.0 has five. Using exponential notation like 2.2 × 103 clarifies that only two figures are significant, while 2.20 × 103 indicates three.
Q5: How does measuring instrument precision affect the number of significant figures?
More precise measuring tools produce measurements with more significant figures. An ordinary ruler measures to the nearest millimeter, while a caliper measures to 0.01 mm, making it more precise. The last digit in any measurement is estimated by the person taking it, and this uncertainty is reflected in the count of significant figures recorded.
Q6: Why is the last digit in a measurement considered uncertain?
The last digit written in a measurement is the first digit with some uncertainty because it has been estimated by the person making the measurement. For example, when measuring a stick's length between 13.2 cm and 13.3 cm with a ruler, the observer must estimate the final decimal place. This estimation reflects the limit of the measuring instrument's precision.
Q7: How can exponential notation help clarify the number of significant figures?
Exponential notation eliminates ambiguity about trailing zeros in whole numbers. The number 2200 could have two or three significant figures, but writing it as 2.2 × 103 clearly shows two significant figures, while 2.20 × 103 shows three. This notation makes the precision of the measurement explicit and unambiguous.