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.
Section1.8Chapter Review
¶Subsection1.8.1Review of Arithmetic with Negative Numbers
Examples
\(5+2=7\)
\(-5+(-2)=-7\)
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 \(0\text{.}\)
Examples
\(5+(-2)=3\)
\((-5)+2=-3\)
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.
Examples
- \begin{align*} 5-2\amp=5+(-2)\\ \amp=3 \end{align*}
- \begin{align*} 2-5\amp=2+(-5)\\ \amp=-3 \end{align*}
- \begin{align*} -5-2\amp=-5+(-2)\\ \amp=-7 \end{align*}
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.
Examples
- \begin{align*} 5-(-2)\amp=5+2\\ \amp=7 \end{align*}
- \begin{align*} -5-(-2)\amp=-5+2\\ \amp=-3 \end{align*}
- \begin{align*} -2-(-5)\amp=-2+5\\ \amp=3 \end{align*}
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.
Examples
\((6)(-2)=-12\)
\((-6)(2)=-12\)
\((-6)(-2)=12\)
\((-6)(-2)(-1)=-12\)
\((-6)(-2)(-1)(-1)=12\)
\(\frac{12}{-2}=-6\)
\(\frac{-12}{2}=-6\)
\(\frac{-12}{-2}=6\)
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.
Examples
- \begin{align*} (-2)^2\amp=(-2)(-2)\\ \amp=4 \end{align*}
- \begin{align*} (-2)^3\amp=(-2)(-2)(-2)\\ \amp=-8 \end{align*}
- \begin{align*} (-2)^4\amp=(-2)(-2)(-2)(-2)\\ \amp=16 \end{align*}
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.
Examples
- \begin{align*} (-4)^2\amp=(-4)(-4)\\ \amp=16 \end{align*}
- \begin{align*} -4^2\amp=-(4)(4)\\ \amp=-16 \end{align*}
- \begin{align*} (-4)^3\amp=(-4)(-4)(-4)\\ \amp=-64 \end{align*}
- \begin{align*} -4^3\amp=-(4)(4)(4)\\ \amp=-64 \end{align*}
Subsection1.8.2Fraction Arithmetic Review
Multiplying Fractions
Example
When multiplying two fractions, we simply multiply their numerators and denominators. To avoid big numbers, we should reduce fractions before multiplying. If one number is an integer, we can change the interger to 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*}
Dividing Fractions
Example
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*}
Adding/Subtracting Fractions
Example
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\highlight{\cdot3}}{2\highlight{\cdot3}}-\frac{1\highlight{\cdot2}}{3\highlight{\cdot2}}\\
\amp=\frac{3}{6}-\frac{2}{6}\\
\amp=\frac{1}{6}
\end{align*}
Subsection1.8.3Absolute Value and Square Root Review
Absolute Value
Examples
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 \(0\text{.}\)
\(\abs{2}=2\)
\(\abs{-\frac{1}{2}}=\frac{1}{2}\)
\(\abs{0}=0\)
Square Root
Examples
The symbol \(\sqrt{b}\) has meaning when \(b\geq0\text{.}\) It means the positive number that can be squared to result in \(b\text{.}\)
\(\sqrt{9}=3\)
\(\sqrt{2}=1.414\ldots\)
\(\sqrt{\frac{9}{16}}=\frac{3}{4}\)
\(\sqrt{-1}\text{ is undefined}\)
Subsection1.8.4Order of Operations Review
Order of Operations
Example
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-(\overbrace{2-4})^2 \right)\\
\amp=4-2\left( 3-\overbrace{(\highlight{-2})^2} \right)\\
\amp=4-2\left( \overbrace{3-\highlight{4}} \right)\\
\amp=4-\overbrace{2\left( \highlight{-1} \right)}\\
\amp=4-\highlight{(-2)}\\
\amp=6
\end{align*}
Subsection1.8.5Types of Numbers Review
Types of Numbers
Real numbers are categorized into the following sets: natural numbers, whole numbers, integers, rational numbers and irrational numbers.
Examples
Here are some examples of numbers from each set of numbers:
- Natural Numbers
\(1,251,3462\)
- Whole Numbers
\(0,1,42,953\)
- Integers
\(-263,-10,0,1,834\)
- Rational Numbers
\(\frac{1}{3},-3,1.1,0,0.\overline{73}\)
- Irrational Numbers
\(\pi,e,\sqrt{2}\)
Subsection1.8.6Comparison Symbols Review
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 |
\(10\neq20\) | \(\frac{1}{2}\neq1.2\) |
Subsection1.8.7Notation for Intervals Review
The following are some examples of set-builder notation and interval notation.
| Graph | Set-builder Notation | Interval Notation |
| \(\left\{x\mid x\ge1\right\}\) | \([1,\infty)\) | |
| \(\left\{x\mid x\gt1\right\}\) | \((1,\infty)\) | |
| \(\left\{x\mid x\le1\right\}\) | \((-\infty,1]\) | |
| \(\left\{x\mid x\lt1\right\}\) | \((-\infty,1)\) |