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.
Section 1.7 Basic Math Chapter Review
¶Subsection 1.7.1 Arithmetic with Negative Numbers
Example 1.7.1
\(5+2=7\)
\(-5+(-2)=-7\)
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 \(0\text{.}\)
Example 1.7.2
\(5+(-2)=3\)
\((-5)+2=-3\)
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.
Example 1.7.3
\(\begin{aligned}[t] 5-2\amp=5+(-2)\\ \amp=3 \end{aligned}\)
\(\begin{aligned}[t] 2-5\amp=2+(-5)\\ \amp=-3 \end{aligned}\)
\(\begin{aligned}[t] -5-2\amp=-5+(-2)\\ \amp=3 \end{aligned}\)
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.
Example 1.7.4
\(\begin{aligned}[t] 5-(-2)\amp=5+2\\ \amp=7 \end{aligned}\)
\(\begin{aligned}[t] -5-(-2)\amp=-5+2\\ \amp=-3 \end{aligned}\)
\(\begin{aligned}[t] -2-(-5)\amp=-2+5\\ \amp=3 \end{aligned}\)
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.
Example 1.7.5
\((6)(-2)=-12\)
\((-6)(2)=-12\)
\((-6)(-2)=12\)
\((-6)(-2)(-1)=-12\)
\((-6)(-2)(-1)(-1)=12\)
\(\frac{12}{-2}=-6\)
\(\frac{-12}{2}=-6\)
\(\frac{-12}{-2}=6\)
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.
Example 1.7.6
\(\begin{aligned}[t] (-2)^2\amp=(-2)(-2)\\ \amp=4 \end{aligned}\)
\(\begin{aligned}[t] (-2)^3\amp=(-2)(-2)(-2)\\ \amp=-8 \end{aligned}\)
\(\begin{aligned}[t] (-2)^4\amp=(-2)(-2)(-2)(-2)\\ \amp=16 \end{aligned}\)
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.
Example 1.7.7
\(\begin{aligned}[t] (-4)^2\amp=(-4)(-4)\\ \amp=16 \end{aligned}\)
\(\begin{aligned}[t] -4^2\amp=-(4)(4)\\ \amp=-16 \end{aligned}\)
\(\begin{aligned}[t] (-4)^3\amp=(-4)(-4)(-4)\\ \amp=-64 \end{aligned}\)
\(\begin{aligned}[t] -4^3\amp=-(4)(4)(4)\\ \amp=-64 \end{aligned}\)
Subsection 1.7.2 Fraction Arithmetic
Example 1.7.8 Multiplying 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*}
Example 1.7.9 Dividing 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*}
Example 1.7.10 Adding/Subtracting Fractions
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*}
Subsection 1.7.3 Absolute Value and Square Root
Example 1.7.11 Absolute 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 \(0\text{.}\)
\(\abs{2}=2\)
\(\abs{-\frac{1}{2}}=\frac{1}{2}\)
\(\abs{0}=0\)
Example 1.7.12 Square Root
The symbol \(\sqrt{b}\) has meaning when \(b\geq0\text{.}\) It means the positive number that can be squared to result in \(b\text{.}\)
\(\sqrt{9}=3\)
\(\sqrt{2}\approx1.414\)
\(\sqrt{\frac{9}{16}}=\frac{3}{4}\)
\(\sqrt{-1}\text{ is undefined}\)
Subsection 1.7.4 Order of Operations
Example 1.7.13 Order of Operations
When evaluating an expression with multiple operations, we must follow the order of operations:
(P)arentheses and other grouping symbols
(E)xponentiation
(M)ultiplication, (D)ivision, and Negation
(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*}
Subsection 1.7.5 Set 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.
Example 1.7.14
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.
\(\pi,e,\sqrt{2}\)
Subsection 1.7.6 Comparison Symbols and Notation for Intervals
The following are symbols used to compare numbers.
| Symbol | Meaning | Examples | |
| \(=\) | equals | \(13=13\qquad\) | \(\frac{5}{4}=1.25\) |
| \(\gt\) | is greater than | \(13\gt11\) | \(\pi\gt3\) |
| \(\geq\) | is greater than or equal to | \(13\geq11\) | \(3\geq3\) |
| \(\lt\) | is less than | \(-3\lt8\) | \(\frac{1}{2}\lt\frac{2}{3}\) |
| \(\leq\) | is less than or equal to | \(-3\leq8\) | \(3\leq3\) |
| \(\neq\) | is not equal to | \(10\neq20\) | \(\frac{1}{2}\neq1.2\) |
The following are some examples of set-builder notation and interval notation.
Subsection 1.7.7 Exercises
1
Perform the given addition and subtraction.
\(\displaystyle{ {-19-8+\left(-2\right)}= }\)
\(\displaystyle{ {2-\left(-19\right)+\left(-14\right)}= }\)
2
Perform the given addition and subtraction.
\(\displaystyle{ {-18-5+\left(-8\right)}= }\)
\(\displaystyle{ {9-\left(-19\right)+\left(-19\right)}= }\)
3
Multiply the following.
\(\displaystyle{ (-2)\cdot(-6)\cdot(-3) = }\)
\(\displaystyle{ 5\cdot(-9)\cdot(-2)= }\)
\(\displaystyle{ (-99)\cdot(-60)\cdot0= }\)
4
Multiply the following.
\(\displaystyle{ (-2)\cdot(-4)\cdot(-5) = }\)
\(\displaystyle{ 3\cdot(-9)\cdot(-5)= }\)
\(\displaystyle{ (-98)\cdot(-77)\cdot0= }\)
5
Evaluate the following.
\(\displaystyle{ \frac{-25}{-5}= }\)
\(\displaystyle{ \frac{10}{-5}= }\)
\(\displaystyle{ \frac{-35}{5}= }\)
6
Evaluate the following.
\(\displaystyle{ \frac{-8}{-4}= }\)
\(\displaystyle{ \frac{32}{-4}= }\)
\(\displaystyle{ \frac{-15}{5}= }\)
7
Evaluate the following.
\(\displaystyle{ (-1)^{2}= }\)
\(\displaystyle{ -4^{2}= }\)
8
Evaluate the following.
\(\displaystyle{ (-1)^{2}= }\)
\(\displaystyle{ -8^{2}= }\)
9
Evaluate the following.
\(\displaystyle{ (-4)^{3}= }\)
\(\displaystyle{ -1^{3}= }\)
10
Evaluate the following.
\(\displaystyle{ (-4)^{3}= }\)
\(\displaystyle{ -3^{3}= }\)
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.
\(\displaystyle{ - \lvert 3-10 \rvert = }\)
\(\displaystyle{ \lvert -3-10 \rvert = }\)
\(\displaystyle{ -2 \lvert 10-3 \rvert = }\)
26
Evaluate the following.
\(\displaystyle{ - \lvert 1-7 \rvert = }\)
\(\displaystyle{ \lvert -1-7 \rvert = }\)
\(\displaystyle{ -2 \lvert 7-1 \rvert = }\)
27
Evaluate the following.
\(\displaystyle{ \sqrt{1} }\) =
\(\displaystyle{ \sqrt{81} }\) =
\(\displaystyle{ \sqrt{100} }\) =
28
Evaluate the following.
\(\displaystyle{ \sqrt{4} }\) =
\(\displaystyle{ \sqrt{25} }\) =
\(\displaystyle{ \sqrt{9} }\) =
29
Evaluate the following.
\(\displaystyle{ \sqrt{{{\frac{16}{49}}}} }\) =
\(\displaystyle{ \sqrt{{-{\frac{25}{64}}}} }\) =
30
Evaluate the following.
\(\displaystyle{ \sqrt{{{\frac{25}{81}}}} }\) =
\(\displaystyle{ \sqrt{{-{\frac{144}{49}}}} }\) =
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:
\(2\)
<
>
=
\(-2\)
<
>
=
\(-7\)
<
>
=
38
Compare the following integers:
\(3\)
<
>
=
\(-1\)
<
>
=
\(-6\)
<
>
=
39
Determine the validity of each statement by selecting True or False.
The number \(\sqrt{(-60)^2}\) is irrational
The number \(\sqrt{\frac{9}{16}}\) is an integer, but not a whole number
The number \(\sqrt{23}\) is rational
The number \(60\) is an integer, but not a whole number
The number \(0\) is a natural number
40
Determine the validity of each statement by selecting True or False.
The number \(\sqrt{\frac{25}{81}}\) is rational, but not an integer
The number \(\frac{19}{43}\) is rational, but not an integer
The number \(\sqrt{11}\) is a real number, but not an irrational number
The number \(0.14404004000400004...\) is rational
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.
-
In set-builder notation:
In interval notation:
-
In set-builder notation:
In interval notation:
-
In set-builder notation:
In interval notation:
44
For each interval expressed in the number lines, give the interval notation and set-builder notation.
-
In set-builder notation:
In interval notation:
-
In set-builder notation:
In interval notation:
-
In set-builder notation:
In interval notation:
