10.2: Signed Numbers
- Page ID
- 48893
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- be able to distinguish between positive and negative real numbers
- be able to read signed numbers
- understand the origin and use of the double-negative product property
Positive and Negative Numbers
Each real number other than zero has a sign associated with it. A real number is said to be a positive number if it is to the right of 0 on the number line and negative if it is to the left of 0 on the number line.
+ and − Notation
A number is denoted as positive if it is directly preceded by a plus sign or no sign at all.
A number is denoted as negative if it is directly preceded by a minus sign.
Reading Signed Numbers
The plus and minus signs now have two meanings:
The plus sign can denote the operation of addition or a positive number.
The minus sign can denote the operation of subtraction or a negative number.
To avoid any confusion between "sign" and "operation," it is preferable to read the sign of a number as "positive" or "negative." When "+" is used as an operation sign, it is read as "plus." When "-" is used as an operation sign, it is read as "minus."
Read each expression so as to avoid confusion between "operation" and "sign."
-8 should be read as "negative eight" rather than "minus eight."
\(4 + (-2)\) should be read as "four plus negative two" rather than "four plus minus two."
\(-6 + (-3)\) should be read as "negative six plus negative three" rather than "minus six plus minus three."
\(-15 - (-6)\) should be read as "negative fifteen minus negative six" rather than "minus fifteen minus minus six."
\(-5 + 7\) should be read as "negative five plus seven" rather than "minus five plus seven."
\(0 - 2\) should be read as "zero minus two."
Practice Set A
Write each expression in words.
\(6 + 1\)
- Answer
-
six plus one
Practice Set A
\(2 + (-8)\)
- Answer
-
two plus negative eight
Practice Set A
\(-7 + 5\)
- Answer
-
negative seven plus five
Practice Set A
\(-10 - (+3)\)
- Answer
-
negative ten minus three
Practice Set A
\(-1 - (-8)\)
- Answer
-
negative one minus negative eight
Practice Set A
\(0 + (-11)\)
- Answer
-
zero plus negative eleven
Opposites
Opposites
On the number line, each real number, other than zero, has an image on the opposite side of 0. For this reason, we say that each real number has an opposite. Opposites are the same distance from zero but have opposite signs.
The opposite of a real number is denoted by placing a negative sign directly in front of the number. Thus, if \(a\) is any real number, then \(-a\) is its opposite.
The letter "\(a\)" is a variable. Thus, "\(a\)" need not be positive, and "\(-a\)" need not be negative.
If \9a\) is any real number, \(-a\) is opposite \(a\) on the number line.
The Double-Negative Property
The number \(a\) is opposite \(-a\) on the number line. Therefore, \(-(-a)\) is opposite \(-a\) on the number line. This means that \(-(-a) = a\)
From this property of opposites, we can suggest the double-negative property for real numbers.
Double-Negative Property: \(-(-a) = a\)
If \(a\) is real number, then
\(-(-a) = a\)
Find the opposite of each number.
If \(a = 2\), then \(-a = -2\), Also, \(-(-a) = -(-2) = 2\).
If \(a = -4\), then \(-a = -(-4) = 4\), Also, \(-(-a) = a = -4\).
Practice Set B
Find the opposite of each number.
8
- Answer
-
-8
Practice Set B
17
- Answer
-
-17
Practice Set B
-6
- Answer
-
6
Practice Set B
-15
- Answer
-
15
Practice Set B
-(-1)
- Answer
-
-1
Practice Set B
−[−(−7)]
- Answer
-
7
Practice Set B
Suppose \(a\) is a positive number. Is \(-a\) positive or negative?
- Answer
-
\(-a\) is negative
Practice Set B
Suppose \(a\) is a negative number. Is \(-a\) positive or negative?
- Answer
-
\(-a\) is positive
Practice Set B
Suppose we do not know the sign of the number \(k\). Is \(-k\) positive, negative, or do we not know?
- Answer
-
-17
We must say that we do not know.
Exercises
Exercise \(\PageIndex{1}\)
A number is denoted as positive if it is directly preceded by .
- Answer
-
+ (or no sign)
Exercise \(\PageIndex{2}\)
A number is denoted as negative if it is directly preceded by .
How should the number in the following 6 problems be read? (Write in words.)
Exercise \(\PageIndex{3}\)
-7
- Answer
-
negative seven
Exercise \(\PageIndex{4}\)
-5
Exercise \(\PageIndex{5}\)
15
- Answer
-
fifteen
Exercise \(\PageIndex{6}\)
11
Exercise \(\PageIndex{7}\)
-(-1)
- Answer
-
negative negative one, or opposite negative one
Exercise \(\PageIndex{8}\)
-(-5)
For the following 6 problems, write each expression in words.
Exercise \(\PageIndex{9}\)
5 + 3
- Answer
-
five plus three
Exercise \(\PageIndex{10}\)
3 + 8
Exercise \(\PageIndex{11}\)
15 + (-3)
- Answer
-
fifteen plus negative three
Exercise \(\PageIndex{12}\)
1 + (-9)
Exercise \(\PageIndex{13}\)
-7 - (-2)
- Answer
-
negative seven minus negative two
Exercise \(\PageIndex{14}\)
0 - (-12)
For the following 6 problems, rewrite each number in simpler form.
Exercise \(\PageIndex{15}\)
-(-2)
- Answer
-
2
Exercise \(\PageIndex{16}\)
-(-16)
Exercise \(\PageIndex{17}\)
-[-(-8)]
- Answer
-
-8
Exercise \(\PageIndex{18}\)
-[-(-20)]
Exercise \(\PageIndex{19}\)
7 - (-3)
- Answer
-
7 + 3 = 10
Exercise \(\PageIndex{20}\)
6 - (-4)
Exercises for Review
Exercise \(\PageIndex{21}\)
Find the quotient; \(8 \div 27\).
- Answer
-
\(0.\overline{296}\)
Exercise \(\PageIndex{22}\)
Solve the proportion: \(\dfrac{5}{9} = \dfrac{60}{x}\)
Exercise \(\PageIndex{23}\)
Use the method of rounding to estimate the sum: \(5829 + 8767\)
- Answer
-
\(6,000 + 9,000 = 15,000\) \((5,829 + 8,767 = 14,596)\) or \(5,800 + 8,800 + 14,600\)
Exercise \(\PageIndex{24}\)
Use a unit fraction to convert 4 yd to feet.
Exercise \(\PageIndex{25}\)
Convert 25 cm to hm.
- Answer
-
0.0025 hm