3: Integers
- Page ID
- 21679
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)At over 29,000 feet, Mount Everest stands as the tallest peak on land. Located along the border of Nepal and China, Mount Everest is also known for its extreme climate. Near the summit, temperatures never rise above freezing. Every year, climbers from around the world brave the extreme conditions in an effort to scale the tremendous height. Only some are successful. Describing the drastic change in elevation the climbers experience and the change in temperatures requires using numbers that extend both above and below zero. In this chapter, we will describe these kinds of numbers and operations using them.
- 3.1: Introduction to Integers (Part 1)
- The opposite of a number is the number that is the same distance from zero on the number line, but on the opposite side of zero. Just as the same word in English can have different meanings, the same symbol in algebra can have different meanings. So, in opposite notation, -a means the opposite of the number a. Integers are counting numbers, their opposites, and zero. The absolute value of a number is its distance from 0 on the number line.
- 3.2: Introduction to Integers (Part 2)
- To solve real-world problems, we first need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we translate math notation into a sentence to answer the question.
- 3.3: Add Integers (Part 1)
- In order to understand how addition and subtraction of negative numbers works, we will use two color counters. We let a blue counter represent a positive number and a red counter will represent a negative number. When the numbers' signs are the same, the counters would be all the same color, so add them together. When the numbers' signs are different, some counters would make neutral pairs, so subtract them to see how many are left.
- 3.4: Add Integers (Part 2)
- To solve real-world problems, we first need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we translate math notation into a sentence to answer the question.
- 3.5: Subtract Integers (Part 1)
- We continue to use the blue counters to represent positive numbers and the red counters to represent negative numbers. When there are enough counters of a certain color to take away, subtract. But, when there are not enough of the counters to take away, add neutral pairs and then subtract. The Subtraction Property states that that subtraction of signed numbers can be done by adding the opposite.
- 3.6: Subtract Integers (Part 2)
- To solve real-world problems, we first need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we translate math notation into a sentence to answer the question.
- 3.7: Multiply and Divide Integers (Part 1)
- Multiplying or dividing two numbers with the same sign, positive or negative, will produce a positive number. While multiplying or dividing two numbers with different signs will produce a negative number. Also, multiplying or dividing a number by -1 gives its opposite.
- 3.8: Multiply and Divide Integers (Part 2)
- To solve real-world problems, we first need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we translate math notation into a sentence to answer the question.
- 3.9: Solve Equations Using Integers; The Division Property of Equality (Part 1)
- To determine if a number is a solution to an equation, first substitute the number for the variable in the equation. Then simplify the expressions on both sides of the equation and determine whether the resulting equation is true or not. If it is true, the number is a solution. But if it is not true, the number is not a solution. The subtraction, addition, and division properties of equality can be used to find solutions to equations.
- 3.10: Solve Equations Using Integers; The Division Property of Equality (Part 2)
- To solve real-world problems, we first need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we translate math notation into a sentence to answer the question.
Figure 3.1 - The peak of Mount Everest. (credit: Gunther Hagleitner, Flickr)