A Modeling Approach to Calculus: MA2C

Subsection 1.2.1 Overview

Numbers play a fundamental role in science because they allow us to quantify observations. That is, instead of saying things are big or small, we can assign numbers to measurements. Science in large part has progressed because of our ability to determine mathematical relationships between different measurements that make prediction possible.

Mathematics admittedly views numbers themselves as objects worthy of study. A number is an exact entity—other numbers, regardless of how close they might be, are different. Consequently, a mathematical result has an exact value. Sometimes we can only approximate the value, but we should give an exact expression for that value when possible. Mathematicians classify numbers according to the complexity of their definition. Historical conventions often suggest ways in which we can simplify an expression, often so that the classification can be more easily recognized.

Measurements allow us to assign numerical values to physical attributes, such as length, temperature, mass, and speed. Instruments need to be designed so that repeated measurements will result in the same values using standard units. Measurement error results from the limited accuracy intrinsic in reading a measurement from an instrument's scale. Most often, we measure more than one attribute of an object or system under a given condition. The collection of all such measurements is called the state of the system. We generally wish to understand the relationships between these different quantities.

Subsection 1.2.2 Numbers in Mathematics

In mathematics, numbers have precise meanings and classifications. Here, we review the basic sets of numbers. The natural numbers are the positive integers

Including the number zero gives us all counting numbers

The set of all integers is written

The rational numbers are all numbers that can be represented as a ratio of integers

This set-builder notation indicates that \(\mathbb{Q}\) is a set of numbers that can be written in the form \(p/q\) where \(p\) is some integer and \(q\) is some natural number.

We often visualize numbers geometrically using a number line. First, the origin of the line is specified with a value of zero. The integers are then equally spaced by a unit length counting from zero. (See Figure A.1.1.) Subdividing the unit length into a whole number of equal parts generates additional points that are rational numbers. (See Figure A.1.2.) 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 \(\sqrt{3}\text{,}\) as well as transcendental numbers like \(\pi\) and \(e\text{.}\) The set of real numbers is written \(\mathbb{R}\) and consists of both rational and irrational numbers.

Every mathematical value represents a single point on the number line. Two values are equal only when they refer to the very same point on the line. Calculators give decimal approximations for numbers using a limited number of digits and so they can actually only represent a finite collection of the infinitely many possible numbers. In particular, irrational numbers can never be represented exactly using decimals. Thus, we usually represent mathematical values by their mathematical expression rather than decimals. When a decimal approximation is useful, we should indicate we are making an approximation using the approximation symbol (\(\approx\)) rather than an equals sign (\(=\)).

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 \(q \in \mathbb{N}\text{.}\) But there is only one representation where \(p\) and \(q\) have no common factors. Canceling any common factors to find this representation would be simplification. Other examples of simplification include simplifying a root or rationalizing a denominator.

A square root is not simplified if there is a factor inside the root that is a perfect square. Similarly, a cube root is not simplified if there is a factor inside that is a perfect cube. We use the factors of the value inside a root to determine if we can simplify it.

Some additional rules of simplification you may have learned were created before fast calculators and computers were available. These rules were taught so that scientists and engineers could express an answer that would be in a form where it would be faster to use the tables and slide rules available at the time. We no longer need such rules for efficiency, but they often illustrate important algebra rules.

One example of such a rule is the simplification of fractions with square roots, called rationalizing a fraction. It was much more costly to use a table or slide rule if the root was in the denominator. The practice was developed to rewrite such an answer so that the root was in the numerator. This could be accomplished by multiplying the fraction on top and bottom by a factor that would eliminate the undesired root.

In the previous example, the two expressions \(\displaystyle \frac{4}{3\sqrt{2}}\) and \(\displaystyle \frac{2 \sqrt{2}}{3}\) are equally simplified. The first expression has a rationalized numerator. The second expression has a rationalized denominator. You should ask your instructor whether they expect a preferred simplified form.

Although we usually use simplification for aesthetic reasons, having a standard way to write numbers can be useful to prove mathematical results. Thanks to Pythagorus, the ancient Greeks knew that \(\sqrt{2}\) was a number that represented the hypotenuse of an isosceles right triangle with legs of unit length. The Greeks also originally thought that all numbers would ultimately be rational numbers. Realizing that \(\sqrt{2}\) was irrational was so shocking that, according to legend, the discoverer of this fact was drowned at sea.

Subsection 1.2.3 Numbers as Measurements

In science, numbers often arise from measurements. When counting objects, measurements use integers and are exact. Most measurements, however, are not exact and require the use of a scale. An instrument for measurement provides a physical tool that allows us to identify a number of units associated with the physical quantity.

The most elementary physical measurement of this type is a measurement of length. The instrument of measurement, a ruler, uses a constructed number line such that the spacing between numbers on the ruler represents distance. The standard unit for the ruler, such as an inch or centimeter, sets the spacing between integer distances. The designer of a ruler also chooses the number of subdivisions per unit. For example, many rulers with inches use either 8 or 16 subdivisions per inch while metric rulers use 10 subdivisions per centimeter. You should see a similarity between the construction of the ruler and the development of rational numbers, except that the set of rational numbers allows for all possible integer number of subdivisions of the unit.

Measured quantities generally occur between two values that an instrument can measure. Given a ruler, the observer must choose a length based on the existing rulings. Even a length that appears to be exactly on a ruling might be found to be slightly off when examined under magnification. The observer might also make judgment errors in reading off a measurement. Consequently, a measurement represents an approximation of the value of a quantity. The difference between the true and measured values of a quantity is called an error.

In science, the error in a measurement is not known precisely but is represented by a bound. There are a variety of techniques used to indicate the bound for an error. We will briefly discuss how the use of significant digits or a margin of error represent error bounds.

One way that the accuracy of measurements are described is using a number of significant digits. The idea is that the last digit reported represents the smallest subdivision the instrument can distinguish.

An alternative to using significant digits is to state explicitly a margin of error. The margin of error is equivalent to providing a bound for the error. We saw that a measurement 15.3 cm with three significant digits corresponds to an inequality

such that the absolute error is bounded by 0.05,

Using a margin of error, we write \(Q = 15.3 \pm 0.05\) cm. The margin of error \(\pm 0.05\) is interpreted as \(-0.05 \le Q - \widehat{Q} \le 0.05\text{.}\)

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 a central value and the margin of error gives a distance in either direction to reach the extreme values. The true value must be between the extremes.

Example 1.2.11.

The length of the hypotenuse of a right triangle with legs of lengths 4 cm and 6 cm is \(H = \sqrt{4^2+6^2} = \sqrt{52} = 2 \sqrt{13}\) cm. A calculator shows the decimal approximation is \(H \approx 7.211103\) cm.

Now, suppose we use a ruler using centimeters but showing millimeters to measure the length. Different ways of describing the measurement with a margin of error give different information about the length.

• Write \(H\) using a margin of error to state that the measurement to the nearest millimeter is 7.2 cm.
• Write \(H\) using a margin of error to state that the measurement is between 7.2 and 7.3 cm.
• Write \(H\) using a margin of error to state that the measurement is between 7.2 and 7.25 cm.

Solution

First, we consider the nearest tick mark on the ruler. The measurement \(\widehat{H} = 7.2\) cm will be the nearest value for any actual length satisfying \(7.15 \le H \le 7.25\text{.}\)

The spacing from \(\widehat{H}\) and the edge of this interval is

\begin{equation*} \epsilon = |7.25-7.2| = |7.15-7.2| = 0.05. \end{equation*}

This value \(\epsilon\) is the largest margin of error so that

\begin{equation*} |H-7.2| \le 0.05\text{.} \end{equation*}

We write \(H = 7.2 \pm 0.05\) cm.

Next, we work with the range \(7.2 \le H \le 7.3\text{.}\) We find the mid-point of this interval as our recorded measurement

\begin{equation*} \widehat{H} = \frac{7.2+7.3}{2} = 7.25. \end{equation*}

Then we measure the distance from the center to the edge,

\begin{equation*} \epsilon = |7.3-7.25| = |7.2-7.25| = 0.05 \end{equation*}

to find the margin of error 0.05 cm. We can then express our measurement with a margin of error as \(H = 7.25 \pm 0.05\) cm corresponding to a bounded error

\begin{equation*} |H-7.25| \le 0.05. \end{equation*}

Finally, we repeat this process to indicate that \(H\) is in the range \(7.2 \le H \le 7.25\text{.}\) The mid-point gives

\begin{equation*} \widehat{H} = \frac{7.2+7.25}{2} = 7.225. \end{equation*}

The margin of error is 0.025 so that our new approximate measurement with a margin of error is written \(H = 7.25 \pm 0.025\) cm. Using an inequality involving absolute values, we could write

\begin{equation*} |H-7.225| \le 0.025. \end{equation*}

In general, a margin of error establishes a symmetric interval of possible values around the measurement. If we symbolically represent the margin of error by \(\epsilon > 0\) (the Greek letter epsilon), then the statement \(Q = \widehat{Q} \pm \epsilon\) is a statement that the absolute error is bounded by \(\epsilon\text{,}\)

In other words, we know from the measurement that the true value \(Q\) is in the interval

Subsection 1.2.4 Summary

• In mathematics, numbers represent specific points on the number line. Real numbers (\(\mathbb{R}\)) can be classified as natural numbers (\(\mathbb{N}\)), integers (\(\mathbb{Z}\)), rational numbers (\(\mathbb{Q}\)), and irrational numbers.

• To simplify an expression is to find an expression representing the same value in a simpler form. For fractions, there should be no common factors. For roots, the power of prime factors inside should be less than the root.

• In a physical context, numbers represent measurements that have limited precision. This precision might be characterized by significant digits or by a margin of error.

• The error of approximation \(E\) for a quantity \(Q\) and an approximation \(\widehat{Q}\) is defined by

  \begin{equation*} E = Q-\widehat{Q}. \end{equation*}

  A symmetrical error bound \(-\epsilon \le E \le \epsilon\) corresponds to the absolute value inequality \(|Q-\widehat{Q}| \le \epsilon\) for a range of values

  \begin{equation*} \widehat{Q}-\epsilon \le Q \le \widehat{Q}+\epsilon. \end{equation*}