0.1: Integers
- Page ID
- 45021
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The ability to work comfortably with negative numbers is essential for success in algebra. Hence, we discuss adding, subtracting, multiplying and dividing of integers in this section.
Integers are zero, all the positive whole numbers and their opposites (negatives).
The first set of rules for working with negative numbers was written out by the Indian mathematician Brahmagupa.
Adding Integers
When adding integers we have two cases to consider.
Case 1: Adding integers with the same signs, i.e., the addends, the numbers being added, are both positive or both negative. If the signs are the same, we add the numbers and keep the sign.
Add: \(3 + 6\)
Solution
\[\begin{array}{rl} 3 + 6 & \text{Addends are both positive} \rightarrow \text {Add } 3 + 6 \rightarrow \text{ Keep the positive} \\ 9 & \text{Sum}\end{array}\nonumber\]
Add: \(-5 + (-3)\)
Solution
\[\begin{array}{rl} - 5 + (- 3) & \text{Addends are both negative} \rightarrow \text{Add } 5 + 3 \rightarrow \text{Keep the negative} \\ - 8 & \text{Sum} \end{array}\nonumber\]
Add: \(-7 + (-5)\)
Solution
\[\begin{array}{rl} - 7 + (- 5) & \text{Addends are both negative} \rightarrow \text{Add } 7 + 5 \rightarrow \text{Keep the negative} \\ - 12 & \text{Sum} \end{array}\nonumber\]
Case 2. The signs are different, where one number is positive and one number is negative. We subtract the absolute values of the numbers and then keep the sign from the larger number. This means if the larger number is positive, the answer is positive, or if the larger number is negative, the answer is negative.
When we say “keep the sign of the larger number,” we mean to take the absolute value of each addend, and then determine the larger number, e.g., \(-10 + 7\): \[|-10| = 10 \text{ and } |7|=7\nonumber\] Hence, the larger number is 10 and so we would keep the negative sign in our result.
Add: \(-7+2\)
Solution
\[\begin{array}{rl}-7+2&\text{Addends are opposite signs }\rightarrow\text{ Subtract }7-2\rightarrow\text{ Keep the sign of the larger} \\ &\text{number, negative} \\ -5&\text{Sum}\end{array}\nonumber\]
Add: \(-4+6\)
Solution
\[\begin{array}{rl}-4+6&\text{Addends are opposite signs }\rightarrow\text{ Subtract }6-4\rightarrow\text{ Keep the sign of the larger } \\ &\text{number, positive} \\ 2&\text{Sum}\end{array}\nonumber\]
Add: \(4 + (-3)\)
Solution
\[\begin{array}{rl}4+(-3)&\text{Addends are opposite signs }\rightarrow\text{ Subtract }4-3\rightarrow\text{ Keep the sign of the } \\&\text{larger number, positive} \\ 1&\text{Sum}\end{array}\nonumber\]
Add: \(7 + (-10)\)
Solution
\[\begin{array}{rl}7+(-10)&\text{Addends are opposite signs }\rightarrow\text{ Subtract }10-7\rightarrow\text{ Keep the sign of the }\\ &\text{larger number, negative} \\ -3&\text{Sum}\end{array}\nonumber\]
Subtracting Integers
For subtracting with negative integers, we will rewrite the expression as addition by changing the subtraction sign to an addition sign and rewriting the number after the subtraction sign as its opposite. Then simplify using the methods of adding integers.
This method is often referred to as “adding the opposite.”
Subtract: \(8-3\)
Solution
\[\begin{array}{rl}8-3&\text{Change the sign to addition and rewrite }3\text{ as its opposite} \\ 8+(-3)&\text{Addends are opposite signs }\to\text{ Subtract }8-3\to\text{ Keep the sign of the} \\ &\text{larger number, positive} \\ 5&\text{Difference}\end{array}\nonumber\]
Subtract: \(-4 - 6\)
Solution
\[\begin{array}{rl}-4-6&\text{Change the sign to addition and rewrite }6\text{ as its opposite} \\ -4+(-6)&\text{Addends are same signs }\to\text{ Add }4+6\to\text{ Keep the sign, negative} \\ -10&\text{Difference}\end{array}\nonumber\]
Subtract: \(9 - (-4)\)
Solution
\[\begin{array}{rl}9-(-4)&\text{Change the sign to addition and rewrite }4\text{ as its opposite} \\ 9+(4)&\text{Addends are same signs }\to\text{ Add }9+4\to\text{ Keep the sign, positive} \\ 13&\text{Difference}\end{array}\nonumber\]
Subtract: \(- 6 - (- 2)\)
Solution
\[\begin{array}{rl}-6-(-2)&\text{Change the sign to addition and rewrite }-2\text{ as its opposite} \\ -6+(2)&\text{Addends are opposite signs }\to\text{ Subtract }6-2\to\text{ Keep the sign of the} \\&\text{larger number, negative} \\ -4&\text{Difference}\end{array}\nonumber\]
Multiplying and Dividing Integers
To multiply two integers, we multiply as usual and follow the following properties:
- If the two numbers have signs that are the same, both integers are positive or both are negative, then the product is positive.
- If the two numbers have opposite signs, one number is positive and the other is negative, then the product is negative.
For dividing with integers, we follow the same properties as multiplication.
Multiply: \((4)(-6)\)
Solution
\[\begin{array}{rl} (4) (- 6) & \text{Integers have opposite signs} \rightarrow \text{Product is negative}\\ - 24 & \text{Product} \end{array}\nonumber\]
Divide: \(\dfrac{- 36}{- 9}\)
Solution
\[\begin{array}{rl} \dfrac{- 36}{- 9} & \text{Integers are same sign} \rightarrow \text{Quotient is positive}\\ 4 & \text{Quotient} \end{array}\nonumber\]
Multiply: \(-2(-6)\)
Solution
\[\begin{array}{rl} -2(- 6) & & \text{Integers are same sign} \rightarrow \text{Product is positive}\\ 12 & & \text{Product} \end{array}\nonumber\]
Divide: \(\dfrac{15}{-3}\)
Solution
\[\begin{array}{rl} \dfrac{15}{-3} & \text{Integers have opposite sign} \rightarrow \text{Quotient is negative}\\ -5 & \text{Quotient} \end{array}\nonumber\]
- Be sure to see the difference between problems like \(- 3 - 8\) and \(- 3 (- 8)\).
- Notice \(-3(-8)\) is a multiplication problem because there is nothing between the \(-3\) and the parenthesis. If there is no operation written in between the parts, then we assume that means we are multiplying.
- The \(- 3 - 8\) is a subtraction problem because the subtraction sign separates the \(-3\) from the next number.
- Be sure to distinguish between the patterns for adding and subtracting integers and for multiplying and dividing integers. These operations can look very similar.
- For example, if the signs match on addition, then we keep the negative, e.g., \(- 3 + (- 7) = - 10\), but if the signs match on multiplication, then the answer is positive, e.g., \((- 3) (- 7) = 21\).
Integers Homework
Evaluate each expression.
\(1-3\)
\((−6) − (−8)\)
\((−3) − 3\)
\(3 − (−5)\)
\((−7) − (−5)\)
\(3-(-1)\)
\(6-3\)
\( (−5) + 3\)
\(2-3\)
\((−8) − (−5)\)
\((−2) + (−5)\)
\(5 − (−6)\)
\((−6) + 3\)
\(4 − 7\)
\((−7) + 7\)
\(4 − (−1)\)
\((−6) + 8\)
\((−8) − (−3)\)
\(7-7\)
\((−4) + (−1)\)
\((−1) + (−6)\)
\((−8) + (−1)\)
\((−1) − 8\)
\(5 − 7\)
\((−5) + 7\)
\(1 + (−1)\)
\(8 − (−1)\)
\((−3) + (−1)\)
\(7 − 3\)
\((−3) + (−5)\)
Find each product.
\((4)(−1)\)
\((10)(−8)\)
\((−4)(−2)\)
\((−7)(8)\)
\((9)(−4)\)
\((−5)(2)\)
\((−5)(4)\)
\((7)(−5)\)
\((−7)(−2)\)
\((−6)(−1)\)
\((6)(−1)\)
\((−9)(−7)\)
\((−2)(−2)\)
\((−3)(−9)\)
Find each quotient.
\(\frac{30}{-10}\)
\(\frac{-12}{-4}\)
\(\frac{30}{6}\)
\(\frac{27}{3}\)
\(\frac{80}{-8}\)
\(\frac{50}{5}\)
\(\frac{48}{8}\)
\(\frac{54}{-6}\)
\(\frac{-49}{-7}\)
\(\frac{-2}{-1}\)
\(\frac{20}{10}\)
\(\frac{-35}{-5}\)
\(\frac{-8}{-2}\)
\(\frac{-16}{2}\)
\(\frac{60}{-10}\)