# 1.1 Real numbers: algebra essentials  (Page 4/35)

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## Differentiating the sets of numbers

Classify each number as being a natural number ( N ), whole number ( W ), integer ( I ), rational number ( Q ), and/or irrational number ( Q′ ).

1. $\sqrt{36}$
2. $\frac{8}{3}$
3. $\sqrt{73}$
4. $-6$
5. $3.2121121112\dots$
N W I Q Q′
a. $\text{\hspace{0.17em}}\sqrt{36}=6$ X X X X
b. $\text{\hspace{0.17em}}\frac{8}{3}=2.\overline{6}$ X
c. $\text{\hspace{0.17em}}\sqrt{73}$ 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′ ).

1. $-\frac{35}{7}$
2. $0$
3. $\sqrt{169}$
4. $\sqrt{24}$
5. $4.763763763\dots$
N W I Q Q'
a. $\text{\hspace{0.17em}}-\frac{35}{7}$ X X
b. 0 X X X
c. $\text{\hspace{0.17em}}\sqrt{169}$ X X X X
d. $\text{\hspace{0.17em}}\sqrt{24}$ X
e. 4.763763763... X

## 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, $\text{\hspace{0.17em}}{4}^{2}=4\cdot 4=16.\text{\hspace{0.17em}}$ We can raise any number to any power. In general, the exponential notation     $\text{\hspace{0.17em}}{a}^{n}\text{\hspace{0.17em}}$ means that the number or variable $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is used as a factor $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ times.

In this notation, $\text{\hspace{0.17em}}{a}^{n}\text{\hspace{0.17em}}$ is read as the n th power of $\text{\hspace{0.17em}}a,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is called the base    and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ 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, $\text{\hspace{0.17em}}24+6\cdot \frac{2}{3}-{4}^{2}\text{\hspace{0.17em}}$ 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.

$24+6\cdot \frac{2}{3}-{4}^{2}$

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 $\text{\hspace{0.17em}}{4}^{2}\text{\hspace{0.17em}}$ as 16.

$\begin{array}{l}\hfill \\ \begin{array}{l}24+6\cdot \frac{2}{3}-{4}^{2}\hfill \\ 24+6\cdot \frac{2}{3}-16\hfill \end{array}\hfill \end{array}$

Next, perform multiplication or division, left to right.

$\begin{array}{l}\hfill \\ \begin{array}{l}24+6\cdot \frac{2}{3}-16\hfill \\ 24+4-16\hfill \end{array}\hfill \end{array}$

Lastly, perform addition or subtraction, left to right.

Therefore, $\text{\hspace{0.17em}}24+6\cdot \frac{2}{3}-{4}^{2}=12.$

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.

## Order of operations

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)

Given a mathematical expression, simplify it using the order of operations.

1. Simplify any expressions within grouping symbols.
2. Simplify any expressions containing exponents or radicals.
3. Perform any multiplication and division in order, from left to right.
4. Perform any addition and subtraction in order, from left to right.

#### Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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