2.7 Summary of key concepts

Summary of key concepts

Variables and constants ( [link] )

Binary operation ( [link] )

Grouping symbols ( [link] )

$\begin{array}{ll}\text{Parentheses}:\hfill & (\begin{array}{cc}& \end{array})\hfill \\ \text{Brackets}:\hfill & [\begin{array}{cc}& \end{array}]\hfill \\ \text{Braces}:\hfill & \left\{\begin{array}{cc}& \end{array}\right\}\hfill \\ \text{Bar}:\hfill & \underset{¯}{\begin{array}{cc}& \end{array}}\hfill \end{array}$

Order of operations ( [link] , [link] )

The real number line ( [link] )

Coordinate and graph ( [link] )

Real number ( [link] )

Types of real numbers ( [link] )

• the natural numbers $\{1,2,3,\dots \}$
• the whole numbers $\{0,1,2,3,\dots \}$
• the integers $\{\dots ,-3,-2,-1,0,1,2,3,\dots \}$
• the rational numbers {all numbers that can be expressed as the quotient of two integers}
• the irrational numbers {all numbers that have nonending and nonrepeating decimal representations}

Properties of real numbers ( [link] )

• Closure If $a$ and $b$ are real numbers, then $a+b$ and $a\cdot b$ are unique real numbers.
• Commutative $a+b=b+a$ and $a\cdot b=b\cdot a$
• Associative $a+(b+c)=(a+b)+c$ and $a\cdot (b\cdot c)=(a\cdot b)\cdot c$
• Distributive $a(b+c)=a\cdot b+a\cdot c$
• Additive identity 0 is the additive identity. $a+0=a$ and $0+a=a$ .
• Multiplicative identity 1 is the multiplicative identity. $a\cdot 1=a$ and $1\cdot a=a$ .
• Additive inverse For each real number $a$ there is exactly one number $\mathrm{-a}$ such that $a+\left(\mathrm{-a}\right)=0$ and $\left(\mathrm{-a}\right)+a=0$ .
• Multiplicative inverse For each nonzero real number $a$ there is exactly one nonzero real number $\frac{1}{a}$ such that $a\cdot \frac{1}{a}=1$ and $\frac{1}{a}\cdot a=1$ .

Exponents ( [link] )

$\underset{n\text{factors}\text{of}x}{\underbrace{x\cdot x\cdot x\cdot ...\cdot x}}=x^n$

Rules of exponents ( [link] , [link] )

• $x^n\cdot x^m=x^{n+m}$
• $\frac{x^n}{x^m}=x^{n-m}$ , $x\ne 0$
• $x^0=1$ , $x\ne 0$
• ${(x^n)}^m=x^{n\cdot m}$
• ${\left(\frac{x}{y}\right)}^n=\frac{x^n}{y^n}$ , $y\ne 0$