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Wize High School Algebra I Textbook (Common Core) > Foundations of Algebra

📝Review: Working With Numbers

📝Review: Integers, DecimalsNext Section
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Numbers


Numbers are used to represent quantities in everyday life.

Examples
  • The temperature on a hot day is 26 °C26\ \degree\text{C} or 79°F79\degree\text{F}
  • The temperature on a cold day is −20°C-20\degree\text{C} or −4°F-4\degree\text{F}
  • A coffee costs $1.45
  • 25\frac{2}{5} of the class wears glasses

A number line is a very helpful tool for organizing numbers


We say that two numbers are opposite if they have the same value but one is negative and one is positive.
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Types of Numbers


  • Natural Numbers - You count them, starting at 1
  • Examples: 1, 2, 3,...1,\ 2,\ 3,...
  • Whole Numbers - You count them, starting at 0
  • Examples: 0, 1, 2, 3,...0,\ 1,\ 2,\ 3,...
  • Integers - Positive and negative whole numbers
  • Examples: ..., −3, −2, −1, 0, 1, 2, 3,......,\ -3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,...
  • Rational Numbers - Any number that can be written as a quotient (dividion) ab\displaystyle \frac{a}{b} where aa and bb are integers and b≠0b\neq 0
  • Examples: 25, −113, 312, 0.5, −1.23, 3.3‾\displaystyle \frac{2}{5},\ -\frac{11}{3},\ 3\frac{1}{2},\ 0.5,\ -1.23,\ 3.\overline{3}
  • Irrational Numbers - Any number that cannot be written as a quotient of integers
  • Examples: 2\sqrt{2}, π\pi, 3\sqrt{3}, ...
  • Real Numbers - Any number that can be written on the number line

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Order of Operations - BEDMAS

When numbers are being added, subtracted, multiplied, and divided, we need to be careful with the order we do our calculations.

*Some teachers call this PEDMAS, where P is "Parentheses" , which is same as Brackets
  1. We always start by simplifying (calculating) whatever is inside brackets
  2. Then, we simplify any exponents AND SQUARE ROOTS (\sqrt{\boxed{}})
  3. Next, we go from left to right, and simplify anything that is being divided or multiplied
  4. Finally, we go from left to right, and simplify anything that is being added or subtracted
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Example
Simplify 15−3×4+2(3+2)2+6÷215-3\times4+2\left(3+2\right)^2+6\div2.

15−3×4+2(3+2)⏟Brackets2+6÷215-3\times4+2{\underbrace{\bco{\left(3+2\right)}}_{\text{Brackets}}}^2 +6\div2

=15−3×4+2(5)2+6÷2=15-3\times4+2(\bco{5})^2+6\div2

=15−3×4+2(5)2⏟Exponents+6÷2=15-3\times 4+2\underbrace{\bct{(5)^2}}_\text{Exponents}+6\div2

=15−3×4+2(25)+6÷2=15-3\times 4+2(\bct{25})+6\div2

=15−3×4⏟Multiplication+2(25)⏟Multiplication+6÷2⏟Division=15-\underbrace{\bcth{3\times4}}_\text{Multiplication}+\underbrace{\bcth{2(25)}}_\text{Multiplication}+\underbrace{\bcth{6\div2}}_\text{Division}

=15−12+50+3=15-\bcth{12}+\bcth{50}+\bcth{3}

=15−12+50+3⏟Do the Addition & Subtraction from left to right=\underbrace{\bcfi{15-12+50+3}}_\text{Do the Addition \& Subtraction from left to right}

=56=\boxed{56}
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Example: Order of Operations

Simplify 5(2)+402−(4−1)(2×3−22)35(2)+\dfrac{40}{2}-(\sqrt4-1)(2\times3-2^2)^3.

5(2)+402−(4−1)⏟Brackets(2×3−22)3⏟Bracket5(2)+\dfrac{40}{2}-\underbrace{\bco{(\sqrt4-1)}}_\text{Brackets}{{\underbrace{\bco{(2\times 3-2^2)}^3}_\text{Bracket}}}

=5(2)+402−(4−1)⏟BracketsDeal with Square Roots first(2×3−22)3⏟BracketsDeal with Exponents first=5(2)+\dfrac{40}{2}-\underbrace{\bco{(\colorbox{yellow}{$\sqrt4$}-1)}}_{\begin{array}{c}\scriptsize\text{Brackets}\\\scriptsize\colorbox{yellow}{\text{Deal with Square Roots first}}\end{array}}{{\underbrace{\bco{(2\times 3-\colorbox{yellow}{$2^2$})}^3}_{\begin{array}{c}\scriptsize\text{Brackets}\\\scriptsize\colorbox{yellow}{\text{Deal with Exponents first}}\end{array}}}}

=5(2)+402−(2−1)⏟Brackets(2×3−4)3⏟BracketsDeal with Multiplication next=5(2)+\dfrac{40}{2}-\underbrace{\bco{(2-1)}}_\text{Brackets}{{\underbrace{\bco{(\colorbox{yellow}{$2\times 3$}-4)}^3}_{\begin{array}{c}\scriptsize\text{Brackets}\\\scriptsize\colorbox{yellow}{\text{Deal with Multiplication next}}\end{array}}}}

=5(2)+402−(2−1)⏟Brackets(6−4)⏟Braclet3=5(2)+\dfrac{40}{2}-\underbrace{\bco{(2-1)}}_\text{Brackets}{{{\underbrace{\bco{({6}-4)}}_\text{Braclet}}}^3}

=5(2)+402−(1)(2)3=5(2)+\dfrac{40}{2}-\bco{(1)}\bco{(2)}^3

=5(2)+402−(1)(2)3⏟Exponent=5(2)+\dfrac{40}{2}-(1)\underbrace{\bct{(2)^3}}_\text{Exponent}

=5(2)+402−(1)(8)=5(2)+\dfrac{40}{2}-(1)\bm(\colorTwo{8})

=5(2)⏟Multiplication+402⏟Division−(1)(8)⏟Multiplication=\underbrace{\bcth{5(2)}}_\text{Multiplication}+\underbrace{\bcth{\dfrac{40}{2}}}_\text{Division}-\underbrace{\bcth{(1)(8)}}_\text{Multiplication}

=10+20−8=\bcth{10}+\bcth{20}-\bcth{8}

=10+20−8⏟Do the Addition & Subtraction from left to right=\underbrace{\bcfi{10+20-8}}_\text{Do the Addition \& Subtraction from left to right}

=22=\boxed{22}

Practice: Order of Operations

Without using a calculator, evaluate 23+7(14−6)2^3+7\left(14-6\right).

Practice: Order of Operations

Without using a calculator, evaluate 5×44−3(4+2×3)225\times44-\dfrac{3\left(4+2\times 3\right)^2}{2}.

Practice: Order of Operations

Simplify 32+2(14−22×3)413−253^2+\dfrac{2(14-2^2\times 3)^4}{13-\sqrt{25}}.