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Section2.1Numbers and Measurements

Subsection2.1.1Overview

Numbers play a fundamental role in science because they allow us to quantify observations. Instead of saying things are big or small, we can assign numbers to measurements. Standards of measurement allow us to have confidence that repeated measurements will result in the same values, albeit with some variation due to measurement error. Science in large part has progressed because of our ability to determine mathematical relationships between different measurements that make prediction possible.

Mathematics admittedly takes a different view of numbers than just their ability to serve as measurements. A measurement can only be determined to some degree of accuracy. But for a mathematician, a number is an exact entity and other numbers, regardless of how close, are different. In addition, mathematicians are concerned with how a number is defined. So mathematicians classify different numbers according to the complexity of their definition.

An appendix (A.1) reviews more details on the development of different number sets. This section introduces basic ideas of number classification and how basic formulas represent exact numerical values. Once this introduction is complete, we look at measurement, which requires introducing ideas of precision and error.

Subsection2.1.2Numbers in Mathematics

In mathematics, numbers have precise meanings and classifications. The natural numbers are the positive integers \begin{equation*}\mathbb{N} = \{1, 2, 3, \ldots\}.\end{equation*} Including the number zero gives us all counting numbers \begin{equation*}\mathbb{N}_0 = \{0, 1, 2, 3, \ldots\}.\end{equation*} The set of all integers is written \begin{equation*}\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}.\end{equation*} The rational numbers are all numbers that can be represented as a ratio of integers \begin{equation*}\mathbb{Q} = \{ \frac{p}{q} : p \in \mathbb{Z}, q \in \mathbb{N} \}.\end{equation*} The set of all real numbers, \(\mathbb{R}\), includes rational and irrational numbers.

When we think of the real numbers geometrically as the number line, the integers are equally spaced by a unit length. Subdividing the unit length into a whole number of equal parts generates additional points that are rational numbers. However, even when all rational numbers are included, there are infinitely many points on the line that are never covered. These are the irrational numbers, which include algebraic numbers like \(\sqrt{2}\) or \(\sqrt{3}\) as well as transcendental numbers like \(\pi\).

Every mathematical expression represents a specific point on the number line. Mathematics expects exactness. To replace an expression with a number that is not exactly the same is to say that two different points are the same. Calculators give decimal approximations for numbers. Because many numbers of interest are irrational and can not be represented exactly using decimals, you should not replace an expression with a decimal value unless the application represents a measurement where an approximation would be adequate.

Simplification of numbers corresponds to finding a new representation of a number in a reduced form. For example, a rational number has many different representations of the form \(p/q\) with \(p \in \mathbb{Z}\) and \(q \in \mathbb{N}\), but only one where \(p\) and \(q\) have no common factors. Canceling the common factors would be simplification, as would be simplifying a root or rationalizing a denominator.

Example2.1.1

The fraction \(\frac{126}{24}\) is not simplified. If we find the prime factorization of the numerator and denominator, we find \begin{gather*} 126 = 6 \cdot 21 = 2 \cdot 3^2 \cdot 7, \\ 24 = 3 \cdot 8 = 2^3 \cdot 3. \end{gather*} The fraction simplifies by canceling all common factors: \begin{equation*}\frac{126}{24} = \frac{2 \cdot 3^2 \cdot 7}{2^3 \cdot 3} = \frac{3 \cdot 7}{2^2} = \frac{21}{4}.\end{equation*}

Example2.1.2

The square root \(\sqrt{126}\) is not simplified. A square root is the inverse operation of squaring numbers (for non-negative numbers) so that \(\sqrt{3^2} = 3\). Because \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) (for \(a,b \ge 0\)), we can simplify as \begin{equation*}\sqrt{126} = \sqrt{2 \cdot 3^2 \cdot 7} = \sqrt{9} \cdot \sqrt{14} = 3 \sqrt{14}.\end{equation*}

Example2.1.3

The expression \(3e^{2\ln(5)-1}\) represents a number. Because exponentials with base \(e\) and the natural logarithm are inverse operations (see Theorem A.2.30), we can rewrite this expression in a simpler form. However, we have to apply the properties of exponents and logarithms correctly. The formula in the power is subtraction, so using the properties of exponents A.2.28 we find \begin{equation*}3e^{2 \ln(5)-1} = 3 \cdot \frac{e^{2 \ln(5)}}{e^{1}} = \frac{3e^{2\ln(5)}}{e}.\end{equation*} (Do not cancel the ‘\(e\)’!) Now the formula in the power is a product. We can either use another property of exponents or the properties of logarithms A.2.33 to see that \(e^{2 \ln(5)} = (e^{\ln(5)})^2\) or \(e^{2 \ln(5)} = e^{\ln(5^2)}\), after which we use the inverse property to finish the simplification: \begin{equation*}3e^{2 \ln(5)-1} = \frac{3e^{2\ln(5)}}{e} = \frac{3 \cdot 5^2}{e} = \frac{75}{e}.\end{equation*}

When answering mathematical questions, leave your answers in mathematical form rather than use your calculator to get a decimal approximation. This is especially true if you use a calculator to find just part of an answer and then continue the problem with a decimal approximation. This introduces unnecessary errors in your calculation. Avoid the bad habit of relying on a calculator's feature that converts decimal values back to fractions.

Subsection2.1.3Numbers as Measurements

In science, numbers often arise from measurements. The most elementary physical measurement is a measurement of length. In fact a ruler and a number line have many similar features. A ruler has a standard unit, such as an inch or centimeter, with fractional marks. However, a length can not be measured to an exact value. The observer must choose a length based on the existing rulings. There is always uncertainty in measurement. Consequently, numbers coming from measurements have an associated precision.

One way that precision is indicated for a measurement is with significant digits. Significant digits are those digits which represent the value measured up to the accuracy of the measurement. For example, consider an object with a length of 15.2772 cm, which is a measurement accurate to the nearest micron (micrometer). If that object was measured using a rule showing only centimeters, we would see that the length was between 15 and 16 cm but closer to 15. Our measurement would be written as 15 cm, and we would have two significant digits. If the ruler showed millimeters, then our measurement would be 15.3 cm with three significant digits.

Significant digits with zeros have some trickier rules. For example, in the previous example, if the length were reported in microns instead of centimeters, the two significant digit result would be 150,000 microns. The zeros following the 5 are only present as placeholders to maintain the appropriate magnitude. To avoid any confusion, scientific notation would be best, writing this as \(1.5 \times 10^5\) microns. Zeros following the decimal point are considered significant. So a measurement of \(2.530 \times 10^4\) would have four significant digits but \(2.53 \times 10^4\) would only have three significant digits.

An alternative to using significant digits is to explicitly state a margin of error. A measurement 15.3 cm with three significant digits means that the true value is somewhere between 15.25 cm and 15.35 cm. Using a margin of error, this would be written as \(15.3 \pm 0.05\) cm. A margin of error is more precise than significant digits. For example, if we wanted to say that a measurement was somewhere between 15.2 cm and 15.4 cm, then we would write \(15.3 \pm 0.1\) cm. The value 15.3 was used as the midpoint (or average) of the two extremes while the margin of error was identified as the distance from this midpoint to the extreme values.

We can write a statement about the margin of error using absolute values. (Refer to the Appendix A.2.5.) Suppose that a quantity \(Q\) is being measured and we have a margin of error measurement \(Q = 15.3 \pm 0.1\). This says that \(Q\) is within a margin of error 0.1 of the measurement 15.3. In the language of distance and absolute value A.2.43, we would say \begin{equation*}|Q-15.3| \le 0.1.\end{equation*}

Related to the margin of error are the concepts of absolute error, relative error and percentage error. Suppose that some measurable quantity is represented by \(Q\) and it is approximated (e.g., by measurement) by a value \(\widehat{Q}\). (In the example above, the measurement 15.3 would be \(\widehat{Q}\) while \(Q\) would be the exact (but unknown) value.) The error \(E\) (also called a residual) is that value which must be added to the approximation to recover the true value, \begin{equation*} Q = \widehat{Q} + E, \end{equation*} which we can solve for as the true value minus the approximation, \begin{equation*} E = Q - \widehat{Q}. \end{equation*} The absolute error concerns only the magnitude (or size) of the error and is the absolute value of the error, \begin{equation*} \hbox{absolute error} = |E| = |Q - \widehat{Q}|. \end{equation*}

The relative error concerns the size of the error relative to the size of the quantity being measured and is computed as a ratio, \begin{equation*} \hbox{relative error} = \frac{|E|}{|Q|} = \left| \frac{Q-\widehat{Q}}{Q} \right|. \end{equation*} A percentage is a ratio rescaled to 100 (percent literally means per hundred), so the percentage error is 100 times the relative error and indicated with a percent symbol, \begin{equation*} \hbox{percentage error} = 100\% \cdot \frac{|E|}{|Q|} = 100\% \cdot \left| \frac{Q-\widehat{Q}}{Q} \right|. \end{equation*} A measurement with a margin of error given by a percentage, such as \(15 \pm 5\%\) cm, indicates that the percentage error is \(5\%\) The range is computed by converting the percentage back to a standard ratio \(5\% = \frac{5}{100} = 0.05\) and multiplying by the measurement, \(15(0.05) = 0.75\). So a measurement \(15 \pm 5\%\) cm means that the length is between 14.25 cm and 15.75 cm, found by adding and subtracting the margin of error from the reported measurement.

Example2.1.4

Suppose that a swinging pendulum has a period (time to complete a full cycle) of 2.43 seconds, but you record it as 2.5 seconds. What is the error, the absolute error, and the percentage error?

Solution