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Section 1.2 Numbers and Measurements

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. 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 a degree of accuracy that depends on the instrument and environment. For a mathematician, a number is an exact entity—other numbers, regardless of how close they might be, are different. Consequently, a mathematical answer should always be left in an exact representation.

Mathematicians classify numbers according to the complexity of their definition. This section provides a brief review of the most important classifications of numbers. An appendix (Section A.1) reviews more details on the development of these different number sets. This section also reviews how basic formulas represent exact numerical values and how they can be simplified. Next, the section considers measurements. Because measurements are always inexact, many numbers will have values consistent with a given measurement. Thus, we introduce the ideas of precision and error.

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

\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*}

We often visualize numbers geometrically using a number line. On the number line, the integers are equally spaced by a unit length starting at an origin of 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.

Illustration of the real number line.
Figure 1.2.1 The number line graphically represents real numbers, both rational and irrational.

Every mathematical expression represents a specific point on the number line. A mathematical value represents that expression only when it refers to the very same point on the line. To replace an expression with a number that is not exactly the same would be to say that two different points are the same.

Calculators give decimal approximations for numbers. Many numbers of interest are irrational and can not be represented exactly using decimals. You should not replace an expression with a decimal value. You should also not rely on calculator features that rewrite decimal values as fractions. In applications where an answer represents a measurement, decimal approximations can be used, particularly when the calculations were based on other measurements.

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.

Example 1.2.2

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*}

In practice, we don't always have to find the prime factorization. Instead, we can find one common factor at a time until no common factors remain. For example, since 126 and 24 are both even, we would write

\begin{equation*} \frac{126}{24} = \frac{63}{12}. \end{equation*}

Then, we look at 63 and 12 and recognize that they are both divisible by 3, allowing us to rewrite the fraction as

\begin{equation*} \frac{126}{24} = \frac{63}{12} = \frac{21}{4}. \end{equation*}

Because \(21=3 \cdot 7\) and \(4=2^2\) do not have common factors, we know this is simplified.

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.

Example 1.2.3

The square root \(\sqrt{126}\) is not simplified. A square root is the inverse operation of squaring numbers (for non-negative numbers) so that \(\sqrt{3^2} = 3\text{.}\) 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*}
Example 1.2.4

The cube root \(\sqrt[3]{48}\) is not simplified. We start by factoring:

\begin{equation*} 48=2 \cdot 24 = 2 \cdot 3 \cdot 8 = 2^4 \cdot 3. \end{equation*}

A cube root is the inverse operation of cubing numbers so that for every perfect cube, we can simplify \(\sqrt[3]{a^3} = a\text{.}\) We have

\begin{equation*} \sqrt[3]{48} = \sqrt[3]{2^4 \cdot 3} = \sqrt[3]{2^3 \cdot 2 \cdot 3} = 2 \sqrt[3]{6}. \end{equation*}

Some rules of simplification you may have learned were created when fast calculators and computers were not accessible. 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.

Example 1.2.5

Simplify \(\displaystyle \frac{4}{3\sqrt{2}}\) by rationalizing the denominator.

Solution

A square root can simplify if there is a perfect square inside. The square root in this denominator \(\sqrt{2}\) would need another 2 inside to have a perfect square. Multiply numerator and denominator by the extra \(\sqrt{2}\) to get a square in the denominator.

\begin{equation*} \frac{4}{3\sqrt{2}} = \frac{4 \sqrt{2}}{3 \sqrt{2} \sqrt{2}} = \frac{4 \sqrt{2}}{3 \cdot 2} \end{equation*}

Now we can finish simplifying the fraction by canceling common factors:

\begin{equation*} \frac{4}{3\sqrt{2}} = \frac{4 \sqrt{2}}{6} = \frac{2 \sqrt{2}}{3}. \end{equation*}

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.

We close this subsection to illustrate that not all numbers are rational. 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. However, they 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.

Example 1.2.6

The basic argument is to consider a rational number that might represent \(\sqrt{2}\) and then proceed to show that such a representation doesn't make sense. The detailed argument is shown below.

Solution

Suppose that \(\sqrt{2}\) is a rational number. Then it can be written as the ratio of two integers \(\sqrt{2} = \frac{p}{q}\) in reduced form, meaning \(p\) and \(q\) do not have common factors. By definition of square roots, we must have \(\frac{p^2}{q^2} = 2\) which implies

\begin{equation*} p^2 = 2 q^2 \end{equation*}

so that \(p^2\) is an even number. The only way that \(p^2\) can be even is if \(p\) itself is even, since the product of two odd numbers is always odd.

Once we know \(p\) is even, we can factor out 2 and write \(p=2k\) where \(k\) is also an integer. Now \(p^2 = 4k^2\) which implies \(4k^2=2q^2\) or

\begin{equation*} q^2 = 2k^2. \end{equation*}

This means \(q\) would also be an even number. This is where the contradiction occurs—since \(p\) and \(q\) were to have had no common factors, they couldn't both be even. This means that \(\sqrt{2}\) can be written as a reduced fraction, which in turn means that \(\sqrt{2}\) is not a rational number.

Subsection 1.2.3 Numbers 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. Measurements that are on the actual units correspond to numbers that are integers. Measurements that are on exact subdivisions of the unit correspond to rational numbers.

However, there is always uncertainty in measurement. A length can not be measured to an exact value. Given a ruler, the observer must choose a length based on the existing rulings. Consequently, there is a distinction between the actual length and the measured length. The difference between these lengths is called an error.

Definition 1.2.7

Given any quantity with an actual value \(Q\) and a measured value \(\widehat{Q}\text{,}\) the error or residual is a value, say \(E\text{,}\) that measures the difference between the actual and measured values,

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

Equivalently, \(E\) is that quantity such that the actual value is equal to the measured value plus the error,

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

Subsubsection 1.2.3.1 Significant Digits

In science, the precision of a measurement is often described using 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). (We never really know exact lengths of physical objects.) If that object was measured using a ruler 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.

We can think of a measurement with its given precision as representing many possible exact lengths. These possible lengths correspond to intervals on the number line. On a ruler measuring in centimeters, all lengths between 14.5 and 15.5 centimeters would be represented by the measurement of 15 centimeters. Consequently, a measurement of 15 cm corresponds to the interval of possible lengths \([14.5,15.5)\) (centimeters). We include the left end-point because we round the digit 5 up, and an exact length of 15.5 would round up to 16. (We'd actually have a hard time knowing exactly where those lengths are because of measurement error; we should probably flip a coin if we can't decide which side is closer.)

Similarly, if the ruler showed millimeters, all lengths between 15.25 and 15.35 centimeters would be represented by the measurement of 15.3 centimeters. A measurement of 15.3 cm then corresponds to the interval \([15.25,15.35)\) (centimeters).

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.

Subsubsection 1.2.3.2 Margin of Error

An alternative to using significant digits is to state explicitly 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 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.

We can write a statement about the margin of error using absolute values. Suppose that a quantity \(Q\) is being measured and we have measurement with a margin of error \(Q = 15.3 \pm 0.1\text{.}\) This is not an ordinary mathematical equation. Rather, it is saying that we have an approximate measurement \(\widehat{Q}=15.3\) and that the margin of error is between \(\pm 0.1\text{.}\) Writing the error as \(Q-\widehat{Q} = Q-15.3\text{,}\) the margin of error statement can be written

\begin{equation*} -0.1 \lt Q - 15.3 \lt 0.1. \end{equation*}

This says that \(Q\) is within a margin of error 0.1 of the measurement 15.3. Absolute values measure the magnitude of numbers. A margin of error gives the largest magnitude of the error, so we would say

\begin{equation*} |Q-15.3| \lt 0.1. \end{equation*}
Example 1.2.8

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 \(7.15 \lt H \lt 7.25\text{.}\)

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| \lt 0.05\text{.} \end{equation*}

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

Next, we work with the range \(7.2 \lt H \lt 7.3\text{.}\) 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| \lt 0.05. \end{equation*}

Finally, we repeat this process to indicate that \(H\) is in the range \(7.2 \lt H \lt 7.25\text{.}\) 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| \lt 0.025. \end{equation*}

When measurements are given with margins of error, dependent quantities calculated from them also have margins of error. Those errors need to be computed based on the margins of error that we know. One of the outcomes from calculus is understanding the relationships between these margins of error.

Example 1.2.9

Suppose that we have a square whose length is side \(s\text{.}\) From measurement, we have \(s = 5.3 \pm 0.025\) cm. What are the possible values for the perimeter \(P\) and the area \(A\text{?}\) Write them using a margin of error.

Solution

The perimeter is the sum of the lengths of the sides. For a square, we have \(P = 4s\text{.}\) Because our measurement for \(s\) is \(\widehat{s} = 5.3\) cm, our estimate of the perimeter is \(\widehat{P} = 4(5.3) = 21.2\) cm.

This does not capture the margin of error. A common mistake is to use the same margin of error as we had for \(s\text{.}\) A better approach is to think about the entire range of possible values. The smallest possible value for \(s\) would be \(s=5.3-0.025 = 5.275\) cm. Using that value, the perimeter would be \(P=4(5.275) = 21.1\) cm. Similarly, the largest possible value for \(s\) would be \(s=5.3+0.025 = 5.325\) cm. Using this value, the perimeter would be \(P=4(5.325) = 21.3\) cm. The true perimeter must be a value in the range \(21.1 \lt P \lt 21.3\text{.}\)

Using the estimate \(\widehat{P} = 21.2\text{,}\) we measure the distance to each edge of the possible range. Both edges are a distance 0.1 away from the estimate, so we can write \(P = 21.2 \pm 0.1\) cm. What do you notice about the margin of error for \(P\) in comparison to the margin of error for \(s\text{?}\)

The area of a square is the square of the length of its side, \(A = s^2\text{.}\) Using the measurement \(\widehat{s} = 5.3\) cm, we obtain an estimate \(\widehat{A} = (5.3)^2 = 28.09\) cm2. To find the margin of error, we first need to consider the range of possible values. Using the smallest possible side \(s = 5.275\) cm, we obtain an area \(A = 27.825625\) cm2. The largest square gives \(A = 28.355625\) cm2. We now know \(27.825625 \lt A \lt 28.355625\text{.}\)

To consider the margin of error, we need to measure the distance from the estimate \(\widehat{A} = 28.09\) to each edge of the range.

\begin{gather*} | 27.825625 - 28.09 | = 0.264375 \\ | 28.355625 - 28.09 | = 0.265625 \end{gather*}

We can see that the range of possible values is not symmetric around the estimate. The margin of error will be the largest distance to an edge:

\begin{equation*} | A - 28.09 | \lt 0.265625\text{.} \end{equation*}

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 \(-\epsilon \lt E \lt \epsilon\) corresponds to the absolute value inequality \(|Q-\widehat{Q}| \lt \epsilon\) for a range of values

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

Subsection 1.2.5 Exercises

Simplify the following values.

1

\(\frac{42}{12}\)

2

\(\frac{210}{28}\)

3

\(\sqrt{75}\)

4

\(\sqrt{160}\)

5

\(\sqrt[3]{160}\)

6

\(\sqrt[4]{160}\)

7

\(\frac{\sqrt{72}}{4}\)

8

\(\frac{\sqrt{864}}{15}\)

Simplify the following values by rationalizing the denominator.

9

\(\displaystyle \frac{6}{5\sqrt{3}}\)

10

\(\displaystyle \frac{10}{\sqrt[3]{2}}\)

11

\(\displaystyle \frac{4\sqrt{2}}{\sqrt{3}}\)

Simplify the following values by rationalizing the numerator.

12

\(\displaystyle \frac{4\sqrt{2}}{\sqrt{3}}\)

13

\(\displaystyle \frac{5 \sqrt[3]{4}}{\sqrt[3]{3}}\)

14

A right triangle is formed with legs measuring 5 cm and 8 cm. Express the length of the hypotenuse \(H\) to the nearest tenth of a centimeter, stating the margin of error based on a ruler showing millimeters. Rewrite your statement about margin of error as an inequality involving absolute values.

15

A right triangle is formed with one leg measuring 10 cm and the hypotenuse measuring 18 cm. Express the length of the other leg \(L\) to the nearest tenth of a centimeter, stating the margin of error based on a ruler showing millimeters. Rewrite your statement about margin of error as an inequality involving absolute values.

16

The circumference of a tree trunk, with margin of error, is recorded as \(C = 138.4 \pm 0.2\) cm. What is the corresponding estimate for the diameter of the tree \(D\) with its corresponding margin of error? (Round the estimate and the margin of error to the nearest thousandth of a centimeter.)

17

A rectangle is formed with a width \(W=20 \pm 0.1\) cm and a height \(H=15 \pm 0.2\) cm. What range of areas are possible for this rectangle? Using the approximations \(\widehat{W}=20\) cm and \(\widehat{H}=15\) cm, the approximated area would be \(\widehat{A}=300\) cm2. Can the range of possible areas be expressed using a symmetric margin of error from this approximated value, \(|A-\widehat{A}| \lt \epsilon\text{?}\) Explain.