1.1: Numbers and Operations
- Classify a real number as a natural, whole, integer, rational, or irrational number.
- Perform calculations using order of operations.
- Round decimals
It is often said that mathematics is the language of science. If this is true, then the language of mathematics is numbers. The earliest use of numbers occurred \(100\) centuries ago in the Middle East to count, or enumerate items. Farmers, cattlemen, and tradesmen used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.
Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.
But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century A.D. in India that zero was added to the number system and used as a numeral in calculations.
Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century A.D., negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.
Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.
Classifying a Real Number
The numbers we use for counting, or enumerating items, are the natural numbers : \(1, 2, 3, 4, 5\) and so on. We describe them in set notation as \(\{1,2,3,...\}\) where the ellipsis \((\cdots)\) indicates that the numbers continue to follow the pattern. The natural numbers are, of course, also called the counting numbers . Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero: \(\{0,1,2,3,...\}\).
The set of integers adds the opposites of the natural numbers to the set of whole numbers: \(\{\cdots,-3,-2,-1,0,1,2,3,\cdots\}\).It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.
\[\begin{array}{ccc} \\[4pt] \text{negative integers}&\text{zero}&\text{positive integers}\\[4pt] \\[4pt] \cdots ,-3,-2,-1&0&1,2,3,\cdots \\[4pt] \end{array}\]
The set of rational numbers is written as \(\{\frac{m}{n} | \text{m and n are integers and } n \neq 0\}\). Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never \(0\). We can also see that every natural number, whole number, and integer is a rational number with a denominator of \(1\).
Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:
- a terminating decimal: \(\frac{15}{8} =1.875\), or
- a repeating decimal: \(\frac{4}{11} =0.36363636\cdots = 0.\overline{36}\)
We use a line drawn over the repeating block of numbers instead of writing the group multiple times.
Write each of the following as a rational number. Write a fraction with the integer in the numerator and \(1\) in the denominator.
- \(7\)
- \(0\)
- \(-8\)
Solution
- \(7= \frac{7}{1}\)
- \(0= \frac{0}{1}\)
- \(-8= \frac{-8}{1}\)
Write each of the following as a rational number.
- \(11\)
- \(3\)
- \(-4\)
- Answer
-
- \(\frac{11}{1}\)
- \(\frac{3}{1}\)
- \(-\frac{4}{1}\)
Write each of the following rational numbers as either a terminating or repeating decimal.
- \(-\frac{5}{7}\)
- \(\frac{15}{5}\)
- \(\frac{13}{25}\)
Solution
- \(-\frac{5}{7} =-0.714285714285\cdots = -0.\overline{714285}\), a repeating decimal
- \(\frac{15}{5} = 3\)(or \(3.0\)), a terminating decimal
- \(\frac{13}{25} =0.52\), a terminating decimal
Write each of the following rational numbers as either a terminating or repeating decimal.
- \(\frac{68}{17}\)
- \(\frac{8}{13}\)
- \(-\frac{13}{25}\)
- Answer
-
- \(4\) (or \(4.0\)), terminating
- \(0.\overline{615384}\), repeating
- \(-0.85\), terminating
Irrational Numbers
At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not \(2\) or even \(32\),but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than \(3\), but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.
\[{h\mid \text {h is not a rational number}}\]
Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.
- \(\sqrt{25}\)
- \(\frac{33}{9}\)
- \(\sqrt{11}\)
- \(\frac{17}{34}\)
- \(0.3033033303333…\)
Solution
- \(\sqrt{25}\): This can be simplified as \(\sqrt{25} = 5\).Therefore,\(\sqrt{25}\)is rational.
- \(\frac{33}{9}\): Because it is a fraction,\(\frac{33}{9}\)is a rational number. Next, simplify and divide. \[\frac{33}{9}=\cancel{\frac{33}{9}} \nonumber\] So,\(\frac{33}{9}\) is rational and a repeating decimal.
- \(\sqrt{11}\): This cannot be simplified any further. Therefore,\(\sqrt{11}\)is an irrational number.
- \(\frac{17}{34}\): Because it is a fraction,\(\frac{17}{34}\)is a rational number. Simplify and divide. \[\frac{17}{34} = 0.5 \nonumber\] So,\(\frac{17}{34}\) is rational and a terminating decimal.
- \(0.3033033303333…\) is not a terminating decimal. Also note that there is no repeating pattern because the group of \(3s\) increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.
Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.
- \(\frac{7}{77}\)
- \(\sqrt{81}\)
- \(4.27027002700027…\)
- \(\frac{91}{13}\)
- \(\sqrt{39}\)
- Answer
-
- rational and terminating;
- rational and repeating;
- irrational
Real Numbers
Given any number \(n\), we know that \(n\) is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.
The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as \(0\), with negative numbers to the left of \(0\) and positive numbers to the right of \(0\). A fixed unit distance is then used to mark off each integer (or other basic value) on either side of \(0\). Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line as shown in Figure(\(\PageIndex{1}\))
Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of \(0\) on the number line?
- \(-\frac{10}{3}\)
- \(-\sqrt{5}\)
- \(-6π\)
- \(0.615384615384…\)
Solution
- \(-\frac{10}{3}\)is negative and rational. It lies to the left of \(0\) on the number line.
- \(-\sqrt{5}\)is positive and irrational. It lies to the right of \(0\).
- \(-\sqrt{289} = -\sqrt{17^2} = -17\) is negative and rational. It lies to the left of \(0\).
- \(-6π\) is negative and irrational. It lies to the left of \(0\).
- \(0.615384615384…\) is a repeating decimal so it is rational and positive. It lies to the right of \(0\).
Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of \(0\) on the number line?
- \(\sqrt{73}\)
- \(-11.411411411…\)
- \(\frac{47}{19}\)
- \(-\frac{\sqrt{5}}{2}\)
- \(6.210735\)
- Answer
-
- positive, irrational
- right negative, rational
- left positive, rational
- right negative, irrational
- left positive, rational; right
Sets of Numbers as Subsets
Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as Figure(\(\PageIndex{2}\)).
The set of natural numbers includes the numbers used for counting: \(\{1,2,3,...\}\).
The set of whole numbers is the set of natural numbers plus zero: \(\{0,1,2,3,...\}\).
The set of integers adds the negative natural numbers to the set of whole numbers: \(\{...,-3,-2,-1,0,1,2,3,...\}\).
The set of rational numbers includes fractions written as \(\{mn\parallel \text{m and n are integers and }n eq 0\}\).
The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: \(\{h\parallel \text{h is not a rational number}\}\).
Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).
- \(\sqrt{36}\)
- \(\frac{8}{3}\)
- \(\sqrt{73}\)
- \(-6\)
- \(3.2121121112…\)
Solution
| N | W | I | Q | Q' | |
|---|---|---|---|---|---|
| a. \(\sqrt{36} = 6\) | X | X | X | X | |
| b. \(\frac{8}{3} =2.\overline{6}\) | X | ||||
| c. \(\sqrt{73}\) | X | ||||
| d. \(-6\) | X | X | |||
| e. \(3.2121121112...\) | X |
Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).
- \(-\frac{35}{7}\)
- \(0\)
- \(\sqrt{169}\)
- \(\sqrt{24}\)
- \(4.763763763...\)
- Answer
-
N W I Q Q' a. \(-\frac{35}{7}\) X X b. \(0\) X X X c. \(\sqrt{169}\) X X X X d. \(\sqrt{24}\) X e. \(4.763763763...\) X
Interactive Exercise \(\PageIndex{5}\)
Performing Calculations Using the Order of Operations
When we multiply a number by itself, we square it or raise it to a power of \(2\). For example, \(4^2 =4\times4=16\). We can raise any number to any power. In general, the exponential notation an means that the number or variable \(a\) is used as a factor \(n\) times.
\[a^n=a\cdot a\cdot a\cdots a \qquad \text{ n factors} \nonumber \]
In this notation, \(a^n\) is read as the \(n^{th}\) power of \(a\), where \(a\) is called the base and \(n\) is called the exponent. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, \(24+6 \times \dfrac{2}{3} − 4^2\) is a mathematical expression.
To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.
Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.
The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.
Let’s take a look at the expression provided.
\[24+6 \times \dfrac{2}{3} − 4^2 \nonumber\]
There are no grouping symbols, so we move on to exponents or radicals. The number \(4\) is raised to a power of \(2\), so simplify \(4^2\) as \(16\).
\[24+6 \times \dfrac{2}{3} − 4^2 \nonumber \]
\[24+6 \times \dfrac{2}{3} − 16 \nonumber\]
Next, perform multiplication or division, left to right.
\[24+6 \times \dfrac{2}{3} − 16 \nonumber\]
\[24+4-16 \nonumber\]
Lastly, perform addition or subtraction, left to right.
\[24+4−16 \nonumber\]
\[28−16 \nonumber\]
\[12 \nonumber\]
Therefore,
\[24+6 \times \dfrac{2}{3} − 4^2 =12 \nonumber\]
For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.
Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS :
- P (arentheses)
- E (xponents)
- M( ultiplication) and D (ivision)
- A (ddition) and S (ubtraction)
- Simplify any expressions within grouping symbols.
- Simplify any expressions containing exponents or radicals.
- Perform any multiplication and division in order, from left to right.
- Perform any addition and subtraction in order, from left to right.
Use the order of operations to evaluate each of the following expressions.
- \((3\times2)^2-4\times(6+2)\)
- \(\dfrac{5^2-4}{7}- \sqrt{11-2}\)
- \(6-\mid 5-8\mid +3\times(4-1)\)
- \(\dfrac{14-3 \times2}{2 \times5-3^2}\)
- \(7\times(5\times3)−2\times[(6−3)−4^2]+1\)
Solution
- \[\begin{align*} (3\times2)^2-4\times(6+2)&=(6)^2-4\times(8) && \qquad \text{Simplify parentheses}\\ &=36-4\times8 && \qquad \text{Simplify exponent}\\ &=36-32 && \qquad \text{Simplify multiplication}\\ &=4 && \qquad \text{Simplify subtraction}\\ \end{align*}\]
- \[\begin{align*} \dfrac{5^2-4}{7}- \sqrt{11-2}&= \dfrac{5^2-4}{7}-\sqrt{9} && \qquad \text{Simplify grouping symbols (radical)}\\ &=\dfrac{5^2-4}{7}-3 && \qquad \text{Simplify radical}\\ &=\dfrac{25-4}{7}-3 && \qquad \text{Simplify exponent}\\ &=\dfrac{21}{7}-3 && \qquad \text{Simplify subtraction in numerator}\\ &=3-3 && \qquad \text{Simplify division}\\ &=0 && \qquad \text{Simplify subtraction} \end{align*}\]
Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.
- \[\begin{align*} 6-\mid 5-8\mid +3\times(4-1)&=6-|-3|+3\times3 && \qquad \text{Simplify inside grouping symbols}\\ &=6-3+3\times3 && \qquad \text{Simplify absolute value}\\ &=6-3+9 && \qquad \text{Simplify multiplication}\\ &=3+9 && \qquad \text{Simplify subtraction}\\ &=12 && \qquad \text{Simplify addition}\\ \end{align*}\]
- \[\begin{align*} \dfrac{14-3 \times2}{2 \times5-3^2}&=\dfrac{14-3 \times2}{2 \times5-9} && \qquad \text{Simplify exponent}\\ &=\dfrac{14-6}{10-9} && \qquad \text{Simplify products}\\ &=\dfrac{8}{1} && \qquad \text{Simplify differences}\\ &=8 && \qquad \text{Simplify quotient}\\ \end{align*}\]
In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.
- \[\begin{align*} 7\times(5\times3)-2\times[(6-3)-4^2]+1&=7\times(15)-2\times[(3)-4^2]+1 && \qquad \text{Simplify inside parentheses}\\ &=7\times(15)-2\times(3-16)+1 && \qquad \text{Simplify exponent}\\ &=7\times(15)-2\times(-13)+1 && \qquad \text{Subtract}\\ &=105+26+1 && \qquad \text{Multiply}\\ &=132 && \qquad \text{Add} \end{align*}\]
Use the order of operations to evaluate each of the following expressions.
- \(\sqrt{5^2-4^2}+7\times(5-4)^2\)
- \(1+\dfrac{7\times5-8\times4}{9-6}\)
- \(|1.8-4.3|+0.4\times\sqrt{15+10}\)
- \(\dfrac{1}{2}\times[5\times3^2-7^2]+\dfrac{1}{3}\times9^2\)
- \([(3-8^2)-4]-(3-8)\)
- Answer
-
- \(10\)
- \(2\)
- \(4.5\)
- \(25\)
- \(-60\)
Interactive Exercise \(\PageIndex{6}\)
Fred earns $40 mowing lawns. He spends $10 on mp3s, puts a third of what is left in a savings account, and gets another $5 for washing his neighbor’s car.
- Write the numerical expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.
- How much money does Fred keep?
Solution
- \(\frac{2}{3}\times(40−10)+5\)
- $25
Computing Numerical Expressions with Technology
Luckily, when it comes to computing numerical expressions, we can trust technology with performing this task. Instead of following the order of operations and computing by hand, let's discuss how technology can be used.
In Desmos Scientific Calculator or Desmos Graphing Calculator , enter the expression then press Enter. For example, computing the numerical expression from part (a) of Example \(\PageIndex{6}\) is shown below:
In WolframAlpha , enter the expression then press Enter. For example, computing the numerical expression from part (a) of Example \(\PageIndex{6}\) is shown below:
Compute each numerical expression.
- \(\dfrac{5+3}{5-3}\)
- \(7-2(-2)\)
- \(\dfrac{1}{3}\pi \cdot11^2\)
- \(((-2)^2 \cdot 3)^3\)
- \(4(\dfrac{2}{3}-\dfrac{1}{3})-5(\dfrac{1}{3}-\dfrac{2}{3})\)
- Answer
-
- \(4\)
- \(11\)
- \(\dfrac{121}{3}\pi\)
- \(1728\)
- \(3\)
Rounding Decimals
Decimals are another way of writing fractions whose denominators are powers of ten.
\[\begin{array}{rcll} 0.1 & = & \dfrac{1}{10} & \text{is “one tenth”} \\ 0.01 & = & \dfrac{1}{100} & \text{is “one hundredth”} \\ 0.001 & = & \dfrac{1}{1000} & \text{is “one thousandth”} \\ 0.0001 & = & \dfrac{1}{10,000} & \text{is “one ten-thousandth”} \end{array}\]
Just as in whole numbers, each digit of a decimal corresponds to the place value based on the powers of ten. Figure shows the names of the place values to the left and right of the decimal point.
Figure 1.
When we work with decimals, it is often necessary to round the number to the nearest required place value. We summarize the steps for rounding a decimal here.
- Locate the given place value and mark it with an arrow.
- Underline the digit to the right of the place value.
-
Is the underlined digit greater than or equal to 5?
- Yes: add 1 to the digit in the given place value.
- No: do not change the digit in the given place value
- Rewrite the number, deleting all digits to the right of the rounding digit.
Round \(18.379\) to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.
Solution
Round \(18.379.\)
ⓐ to the nearest hundredth
| Locate the hundredths place with an arrow. | |
| Underline the digit to the right of the given place value. | |
| Because 9 is greater than or equal to 5, add 1 to the 7. | |
| Rewrite the number, deleting all digits to the right of the rounding digit. | |
| Notice that the deleted digits were NOT replaced with zeros. |
ⓑ to the nearest tenth
| Locate the tenths place with an arrow. | |
| Underline the digit to the right of the given place value. | |
| Because 7 is greater than or equal to 5, add 1 to the 3. | |
| Rewrite the number, deleting all digits to the right of the rounding digit. | |
| Notice that the deleted digits were NOT replaced with zeros. |
ⓒ to the nearest whole number
| Locate the ones place with an arrow. | |
| Underline the digit to the right of the given place value. | |
| Since 3 is not greater than or equal to 5, do not add 1 to the 8. | |
| Rewrite the number, deleting all digits to the right of the rounding digit. | |
Round \(6.582\) to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.
- Answer
-
ⓐ \(6.58\) ⓑ \(6.6\) ⓒ \(7\)
While rounding is a straightforward process feel free to check your work with any online calculator such as this one . For example, to round 18.379 to the nearest hundredth from part (1) of Example \(\PageIndex{9}\), enter 18.379 and choose hundredths in the precision dropdown.