## Section1.7Basic Math Chapter Review

### Subsection1.7.1Arithmetic with Negative Numbers

##### Adding Real Numbers with the Same Sign

When adding two numbers with the same sign, we can ignore the signs, and simply add the numbers as if they were both positive.

###### Example1.7.1
1. $5+2=7$

2. $-5+(-2)=-7$

##### Adding Real Numbers with Opposite Signs

When adding two numbers with opposite signs, we find those two numbers' difference. The sum has the same sign as the number with the bigger value. If those two numbers have the same value, the sum is $0\text{.}$

###### Example1.7.2
1. $5+(-2)=3$

2. $(-5)+2=-3$

##### Subtracting a Positive Number

When subtracting a positive number, we can change the problem to adding the opposite number, and then apply the methods of adding numbers.

###### Example1.7.3
1. \begin{aligned}[t] 5-2\amp=5+(-2)\\ \amp=3 \end{aligned}

2. \begin{aligned}[t] 2-5\amp=2+(-5)\\ \amp=-3 \end{aligned}

3. \begin{aligned}[t] -5-2\amp=-5+(-2)\\ \amp=3 \end{aligned}

##### Subtracting a Negative Number

When subtracting a negative number, we can change those two negative signs to a positive sign, and then apply the methods of adding numbers.

###### Example1.7.4
1. \begin{aligned}[t] 5-(-2)\amp=5+2\\ \amp=7 \end{aligned}

2. \begin{aligned}[t] -5-(-2)\amp=-5+2\\ \amp=-3 \end{aligned}

3. \begin{aligned}[t] -2-(-5)\amp=-2+5\\ \amp=3 \end{aligned}

##### Multiplication and Division of Real Numbers

When multiplying and dividing real numbers, each pair of negative signs cancel out each other (becoming a positive sign). If there is still one negative sign left, the result is negative; otherwise the result is positive.

###### Example1.7.5
1. $(6)(-2)=-12$

2. $(-6)(2)=-12$

3. $(-6)(-2)=12$

4. $(-6)(-2)(-1)=-12$

5. $(-6)(-2)(-1)(-1)=12$

6. $\frac{12}{-2}=-6$

7. $\frac{-12}{2}=-6$

8. $\frac{-12}{-2}=6$

##### Powers

When we raise a negative number to a certain power, apply the rules of multiplying real numbers: each pair of negative signs cancel out each other.

###### Example1.7.6
1. \begin{aligned}[t] (-2)^2\amp=(-2)(-2)\\ \amp=4 \end{aligned}

2. \begin{aligned}[t] (-2)^3\amp=(-2)(-2)(-2)\\ \amp=-8 \end{aligned}

3. \begin{aligned}[t] (-2)^4\amp=(-2)(-2)(-2)(-2)\\ \amp=16 \end{aligned}

##### Difference between $(-a)^n$ and $-a^n$

For the exponent expression $2^3\text{,}$ the number $2$ is called the base, and the number $3$ is called the exponent. The base of $(-a)^n$ is $-a\text{,}$ while the base of $-a^n$ is $a\text{.}$ This makes a difference in the result when the power is an even number.

###### Example1.7.7
1. \begin{aligned}[t] (-4)^2\amp=(-4)(-4)\\ \amp=16 \end{aligned}

2. \begin{aligned}[t] -4^2\amp=-(4)(4)\\ \amp=-16 \end{aligned}

3. \begin{aligned}[t] (-4)^3\amp=(-4)(-4)(-4)\\ \amp=-64 \end{aligned}

4. \begin{aligned}[t] -4^3\amp=-(4)(4)(4)\\ \amp=-64 \end{aligned}

### Subsection1.7.2Fraction Arithmetic

###### Example1.7.8Multiplying Fractions

When multiplying two fractions, we simply multiply the numerators and denominators. To avoid big numbers, we should reduce fractions before multiplying. If one number is an integer, we can write it as a fraction with a denominator of $1\text{.}$ For example, $2=\frac{2}{1}\text{.}$

\begin{align*} \frac{1}{2}\cdot\frac{3}{4}\amp=\frac{1\cdot3}{2\cdot4}\\ \amp=\frac{3}{8} \end{align*}
###### Example1.7.9Dividing Fractions

When dividing two fractions, we “flip” the second number, and then do multiplication.

\begin{align*} \frac{1}{2}\div\frac{4}{3}\amp=\frac{1}{2}\cdot\frac{3}{4}\\ \amp=\frac{3}{8} \end{align*}

Before adding/subtracting fractions, we need to change each fraction's denominator to the same number, called the common denominator. Then, we add/subtract the numerators, and the denominator remains the same.

\begin{align*} \frac{1}{2}-\frac{1}{3}\amp=\frac{1}{2}\multiplyright{\frac{3}{3}}-\frac{1}{3}\multiplyright{\frac{2}{2}}\\ \amp=\frac{3}{6}-\frac{2}{6}\\ \amp=\frac{1}{6} \end{align*}

### Subsection1.7.3Absolute Value and Square Root

###### Example1.7.11Absolute Value

The absolute value of a number is the distance from that number to $0$ on the number line. An absolute value is always positive or $0\text{.}$

1. $\abs{2}=2$

2. $\abs{-\frac{1}{2}}=\frac{1}{2}$

3. $\abs{0}=0$

###### Example1.7.12Square Root

The symbol $\sqrt{b}$ has meaning when $b\geq0\text{.}$ It means the positive number that can be squared to result in $b\text{.}$

1. $\sqrt{9}=3$

2. $\sqrt{2}\approx1.414$

3. $\sqrt{\frac{9}{16}}=\frac{3}{4}$

4. $\sqrt{-1}\text{ is undefined}$

### Subsection1.7.4Order of Operations

###### Example1.7.13Order of Operations

When evaluating an expression with multiple operations, we must follow the order of operations:

1. (P)arentheses and other grouping symbols

2. (E)xponentiation

3. (M)ultiplication, (D)ivision, and Negation

4. (A)ddition and (S)ubtraction

\begin{align*} 4-2\left( 3-(2-4)^2 \right)\amp=4-2\left( 3-(\nextoperation{2-4})^2 \right)\\ \amp=4-2\left( 3-\nextoperation{(\highlight{-2})^2} \right)\\ \amp=4-2\left( \nextoperation{3-\highlight{4}} \right)\\ \amp=4-\nextoperation{2\left( \highlight{-1} \right)}\\ \amp=4-\highlight{(-2)}\\ \amp=6 \end{align*}

### Subsection1.7.5Set Notation and Types of Numbers

A set is an unordered collection of items. Braces, $\{{}\}\text{,}$ are used to show what items are in a set. For example, the set $\{1,2,\pi\}$ is a set with three items that contains the numbers $1\text{,}$ $2\text{,}$ and $\pi\text{.}$

##### Types of Numbers

Real numbers are categorized into the following sets: natural numbers, whole numbers, integers, rational numbers and irrational numbers.

###### Example1.7.14

Here are some examples of numbers from each set of numbers:

Natural Numbers

The natural numbers are all counting numbers larger $1$ and larger.

$1,251,3462$

Whole Numbers

The whole numbers are all counting numbers larger $0$ and larger.

$0,1,42,953$

Integers

The integers are all counting numbers both negative and positive.

$-263,-10,0,1,834$

Rational Numbers

The rational numbers are all possible fractions of integers.

$\frac{1}{3},-3,1.1,0,0.\overline{73}$

Irrational Numbers

The irrational numbers are all numbers that cannot be written as a fraction of integers.

$\pi,e,\sqrt{2}$

### Subsection1.7.6Comparison Symbols and Notation for Intervals

The following are symbols used to compare numbers.

The following are some examples of set-builder notation and interval notation.

 Graph Set-builder Notation Interval Notation $\left\{x\mid x\ge1\right\}$ $[1,\infty)$ $\left\{x\mid x\gt1\right\}$ $(1,\infty)$ $\left\{x\mid x\le1\right\}$ $(-\infty,1]$ $\left\{x\mid x\lt1\right\}$ $(-\infty,1)$

### Subsection1.7.7Exercises

###### 1

Perform the given addition and subtraction.

1. $\displaystyle{ {-19-8+\left(-2\right)}= }$

2. $\displaystyle{ {2-\left(-19\right)+\left(-14\right)}= }$

###### 2

Perform the given addition and subtraction.

1. $\displaystyle{ {-18-5+\left(-8\right)}= }$

2. $\displaystyle{ {9-\left(-19\right)+\left(-19\right)}= }$

###### 3

Multiply the following.

1. $\displaystyle{ (-2)\cdot(-6)\cdot(-3) = }$

2. $\displaystyle{ 5\cdot(-9)\cdot(-2)= }$

3. $\displaystyle{ (-99)\cdot(-60)\cdot0= }$

###### 4

Multiply the following.

1. $\displaystyle{ (-2)\cdot(-4)\cdot(-5) = }$

2. $\displaystyle{ 3\cdot(-9)\cdot(-5)= }$

3. $\displaystyle{ (-98)\cdot(-77)\cdot0= }$

###### 5

Evaluate the following.

1. $\displaystyle{ \frac{-25}{-5}= }$

2. $\displaystyle{ \frac{10}{-5}= }$

3. $\displaystyle{ \frac{-35}{5}= }$

###### 6

Evaluate the following.

1. $\displaystyle{ \frac{-8}{-4}= }$

2. $\displaystyle{ \frac{32}{-4}= }$

3. $\displaystyle{ \frac{-15}{5}= }$

###### 7

Evaluate the following.

1. $\displaystyle{ (-1)^{2}= }$

2. $\displaystyle{ -4^{2}= }$

###### 8

Evaluate the following.

1. $\displaystyle{ (-1)^{2}= }$

2. $\displaystyle{ -8^{2}= }$

###### 9

Evaluate the following.

1. $\displaystyle{ (-4)^{3}= }$

2. $\displaystyle{ -1^{3}= }$

###### 10

Evaluate the following.

1. $\displaystyle{ (-4)^{3}= }$

2. $\displaystyle{ -3^{3}= }$

###### 11

Add: $\displaystyle{-\frac{9}{10} + \frac{5}{6}}$

###### 12

Add: $\displaystyle{-\frac{1}{6} + \frac{7}{10}}$

###### 13

Subtract: $\displaystyle{-\frac{5}{6} - \left(-\frac{9}{10}\right)}$

###### 14

Subtract: $\displaystyle{-\frac{1}{10} - \left(-\frac{5}{6}\right)}$

###### 15

Subtract: $\displaystyle{ 2 - \frac{28}{9}}$

###### 16

Subtract: $\displaystyle{ 4 - \frac{25}{6}}$

###### 17

Multiply: $\displaystyle{-\frac{12}{13} \cdot \frac{7}{22}}$

###### 18

Multiply: $\displaystyle{-\frac{2}{13} \cdot \frac{5}{26}}$

###### 19

Multiply: $\displaystyle{-4\cdot \frac{5}{6} }$

###### 20

Multiply: $\displaystyle{-5\cdot \frac{9}{20} }$

###### 21

Divide: $\displaystyle{ \frac{7}{15} \div \left(-\frac{5}{12}\right) }$

###### 22

Divide: $\displaystyle{ \frac{1}{9} \div \left(-\frac{5}{12}\right) }$

###### 23

Divide: $\displaystyle{27 \div \frac{9}{4} }$

###### 24

Divide: $\displaystyle{9 \div \frac{9}{4} }$

###### 25

Evaluate the following.

1. $\displaystyle{ - \lvert 3-10 \rvert = }$

2. $\displaystyle{ \lvert -3-10 \rvert = }$

3. $\displaystyle{ -2 \lvert 10-3 \rvert = }$

###### 26

Evaluate the following.

1. $\displaystyle{ - \lvert 1-7 \rvert = }$

2. $\displaystyle{ \lvert -1-7 \rvert = }$

3. $\displaystyle{ -2 \lvert 7-1 \rvert = }$

###### 27

Evaluate the following.

1. $\displaystyle{ \sqrt{1} }$ =

2. $\displaystyle{ \sqrt{81} }$ =

3. $\displaystyle{ \sqrt{100} }$ =

###### 28

Evaluate the following.

1. $\displaystyle{ \sqrt{4} }$ =

2. $\displaystyle{ \sqrt{25} }$ =

3. $\displaystyle{ \sqrt{9} }$ =

###### 29

Evaluate the following.

1. $\displaystyle{ \sqrt{{{\frac{16}{49}}}} }$ =

2. $\displaystyle{ \sqrt{{-{\frac{25}{64}}}} }$ =

###### 30

Evaluate the following.

1. $\displaystyle{ \sqrt{{{\frac{25}{81}}}} }$ =

2. $\displaystyle{ \sqrt{{-{\frac{144}{49}}}} }$ =

###### 31

Evaluate the following.

$\displaystyle{ -6^{2}-5[ 4-( 6-4^{3} ) ] = }$

###### 32

Evaluate the following.

$\displaystyle{ -6^{2}-9[ 8-( 4-4^{3} ) ] = }$

###### 33

Evaluate the following.

$\displaystyle{ \frac{27-(-4)^{3}}{3-10} = }$

###### 34

Evaluate the following.

$\displaystyle{ \frac{27-(-2)^{3}}{7-12} = }$

###### 35

Evaluate the following.

$\displaystyle{ 10-8\left\lvert -9+(4-7)^{3}\right\rvert = }$

###### 36

Evaluate the following.

$\displaystyle{ 1-6\left\lvert -5+(3-6)^{3}\right\rvert = }$

###### 37

Compare the following integers:

1. $2$

• <

• >

• =

$-7$

2. $-2$

• <

• >

• =

$-7$

3. $-7$

• <

• >

• =

$0$

###### 38

Compare the following integers:

1. $3$

• <

• >

• =

$-6$

2. $-1$

• <

• >

• =

$-6$

3. $-6$

• <

• >

• =

$0$

###### 39

Determine the validity of each statement by selecting True or False.

1. The number $\sqrt{(-60)^2}$ is irrational

2. The number $\sqrt{\frac{9}{16}}$ is an integer, but not a whole number

3. The number $\sqrt{23}$ is rational

4. The number $60$ is an integer, but not a whole number

5. The number $0$ is a natural number

###### 40

Determine the validity of each statement by selecting True or False.

1. The number $\sqrt{\frac{25}{81}}$ is rational, but not an integer

2. The number $\frac{19}{43}$ is rational, but not an integer

3. The number $\sqrt{11}$ is a real number, but not an irrational number

4. The number $0.14404004000400004...$ is rational

5. The number $\sqrt{4}$ is a real number, but not a rational number

###### 41

A set is written using set-builder notation. Write it using interval notation.

$\displaystyle{ \{ x \mid {{x}} \gt 2 \} }$

###### 42

A set is written using set-builder notation. Write it using interval notation.

$\displaystyle{ \{ x \mid {{x}} \gt 4 \} }$

###### 43

For each interval expressed in the number lines, give the interval notation and set-builder notation.

1. In set-builder notation:

In interval notation:

2. In set-builder notation:

In interval notation:

3. In set-builder notation:

In interval notation:

###### 44

For each interval expressed in the number lines, give the interval notation and set-builder notation.

1. In set-builder notation:

In interval notation:

2. In set-builder notation:

In interval notation:

3. In set-builder notation:

In interval notation: