Question

Indicate the number of significant figures in each number. If the number is an exact number, indicate an unlimited number of significant figures. a. 325,365,189 (July 4,2017 U.S. population) b. \(2.54 \mathrm{~cm}=1\) in c. \(11.4 \mathrm{~g} / \mathrm{cm}^{3}\) (density of lead) d. \(12=1\) dozen

Short answer

a. 9 significant figures b. Unlimited significant figures (exact number) c. 3 significant figures d. Unlimited significant figures (exact number)

Step-by-step solution

Evaluating the number of significant figures in 325,365,189

Since 325,365,189 is given as an example of population, a counted number which is an estimate rather than an exact count, it has significant figures. To determine the number of significant figures, we simply count all the digits since there are no leading or trailing zeros in this integer. This gives us a total of 9 significant figures.

Recognizing an exact number

The conversion factor of 2.54 cm equals 1 inch is an exact number because it is defined that way. Exact numbers have an infinite or unlimited number of significant figures because their value is known with absolute precision.

Determining significant figures in a decimal number

For the density of lead, which is 11.4 g/cm^3, we identify the significant figures by counting all digits since there are no leading zeros and the trailing zero would only appear if the value was written as 11.40. This gives us a total of 3 significant figures.

Evaluating the number of significant figures in a defined number

Since 12 equals 1 dozen, this is a relationship by definition. Like the conversion factor in step 2, defined numbers are considered exact and have an unlimited number of significant figures.

Key concepts

Scientific Notation

Scientific notation is an efficient way to express very large or very small numbers that we often encounter in science and mathematics. The essence of scientific notation is to represent a numerical value as a product of two factors: a coefficient and a power of 10. The coefficient should be a number greater than or equal to 1 and less than 10, and it's followed by 'x 10' raised to an exponent. This exponent indicates how many places the decimal point moves to the right, for positive values, or to the left, for negative values.

For example, the number 2,000 can be written in scientific notation as \(2 \times 10^3\). Similarly, 0.005 would be \(5 \times 10^{-3}\). The benefit of this method is that it simplifies working with cumbersome numbers and allows for easier comparison of orders of magnitude. Significantly, in scientific notation, all numerals in the coefficient are considered significant figures which directly relates to the precision of the value.

Measurement Precision

Measurement precision refers to the extent to which repeated measurements under unchanged conditions show the same results. It is about the consistency or repeatability of measurements, not necessarily how close they are to the actual or true value (which is more about accuracy).

In the context of significant figures, measurement precision is reflected in the number of digits reported. More significant figures mean higher precision, indicating that a measurement was taken with a finer scale or more sensitive instrument. For instance, if we have a length measured as 3.20 cm as opposed to 3.2 cm, the former suggests a precision up to hundredths of a centimeter, whereas the latter only to tenths.

Understanding precision is crucial when performing calculations in sciences; it ensures we do not overstate the precision of our results. The general rule is that the outcome of a calculation cannot be more precise than the least precise measurement from which it was derived.

Exact Numbers

Exact numbers hold a special place in measurements and calculations since they are considered to have perfect precision. These numbers are either defined quantities, or counts of objects that are not subject to measurement uncertainty.

Defined quantities, such as the 2.54 cm in one inch, are always exact since they stem from a definition. Similarly, counted numbers such as the number of students in a classroom are exact, provided that there is no uncertainty in the count. Consequently, exact numbers are said to have an infinite number of significant figures, as their value is known precisely, without any room for variability.

When using exact numbers in calculations, they do not limit the significant figures in the result. This attributes a unique property to exact numbers – in the midst of inexact measurements, they maintain their precision without affecting the overall precision of a calculation.