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3.2: Signed Numbers

  • Page ID
    49355
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    Overview

    • Positive and Negative Numbers
    • Opposites

    Positive and Negative Numbers

    When we studied the number line in Section 2.3 we noted that

    Each point on the number line corresponds to a real number, and each real number is located at a unique point on the number line.

    A number line with arrows on each end, labeled from negative six to six in increments of one. There are two closed circles at negative two and four, respectively.

    Positive and Negative Numbers
    Each real number has a sign inherently associated with it. A real number is said to be a positive number if it is located to the right of 0 on the number line. It is a negative number if it is located to the left of 0 on the number line.

    THE NOTATION OF SIGNED NUMBERS
    A number is denoted as positive if it is directly preceded by a "+" sign or no sign at all.
    A number is denoted as negative if it is directly preceded by a "−" sign.

    The "+" and "−" signs now have two meanings:

    + can denote the operation of addition or a positive number.
    − can denote the operation of subtraction or a negative number.

    Read the "−" Sign as "Negative"
    To avoid any confusion between "sign" and "operation," it is preferable to read the sign of a number as "positive" or "negative."

    Sample Set A

    Example \(\PageIndex{1}\)

    −8 should be read as "negative eight" rather than "minus eight."

    Example \(\PageIndex{2}\)

    \(4+(−2)\) should be read as "four plus negative two" rather than "four plus minus two."

    Example \(\PageIndex{3}\)

    \(−6+(−3)\) should be read as "negative six plus negative three" rather than "minus six plus minus three."

    Example \(\PageIndex{4}\)

    \(−15−(−6)\) should be read as "negative fifteen minus negative six" rather than "minus fifteen minus minus six."

    Example \(\PageIndex{5}\)

    \(−5+7\) should be read as "negative five plus seven" rather than "minus five plus seven."

    Example \(\PageIndex{6}\)

    \(0−2\) should be read as "zero minus two."

    Practice Set A

    Write each expression in words.

    Practice Problem \(\PageIndex{1}\)

    \(4 + 10\)

    Answer

    four plus ten

    Practice Problem \(\PageIndex{2}\)

    \(7 + (-4)\)

    Answer

    seven plus negative four

    Practice Problem \(\PageIndex{3}\)

    \(-9 + 2\)

    Answer

    negative nine plus two

    Practice Problem \(\PageIndex{4}\)

    \(-16 - (+8)\)

    Answer

    negative sixteen minus positive eight

    Practice Problem \(\PageIndex{5}\)

    \(-1 -(-9)\)

    Answer

    negative one minus negative nine

    Practice Problem \(\PageIndex{6}\)

    \(0 + (-7)\)

    Answer

    zero plus negative seven

    Opposites

    Opposites
    On the number line, each real number 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. Notice that the letter \(a\) is a variable. Thus, "\(a\)" need not be positive, and "\(−a\)" need not be negative.

    If \(a\) is a real number, \(−a\) is opposite \(a\) on the number line and \(a\) is opposite \(−a\) on the number line.

    Two number lines with arrows on each end. The first number line has three labels, zero at the center, negative a to the left of zero and a to the right of zero. Negative a and a are equidistant from zero. The second line has three labels, zero at the center, a to the left of zero and negative a to the right of zero. The points a and negative a are equidistant from zero.

    \(−(−a)\) is opposite \(−a\) on the number line. This implies that \(−(−a)=a\).

    This property of opposites suggests the double-negative property for real numbers.

    The Double-Negative Property

    If a is a real number, then
    \(−(−a)=a\)

    Sample Set B

    Example \(\PageIndex{7}\)

    If \(a=3\), then \(−a=−3\) and \(−(−a)=−(−3)=3\).

    A number line with arrows on each end, labeled from negative three to three in increments of three. Negative three is labeled as negative a, and three is labeled as a. There is an additional label for three as the opposite of negative a.

    Example \(\PageIndex{8}\)

    If \(a=−4\), then \(−a=−(−4)=4\) and \(−(−a)=a=−4\).

    A number line with arrows on each end, labeled from negative four to four in increments of three. Negative four is labeled as a, and four is labeled as negative a. There is an additional label for negative four as the opposite of negative a.

    Practice Set B

    Find the opposite of each real number.

    Practice Problem \(\PageIndex{7}\)

    \(8\)

    Answer

    \(-8\)

    Practice Problem \(\PageIndex{8}\)

    \(17\)

    Answer

    \(-17\)

    Practice Problem \(\PageIndex{9}\)

    \(-6\)

    Answer

    \(6\)

    Practice Problem \(\PageIndex{10}\)

    \(-15\)

    Answer

    \(15\)

    Practice Problem \(\PageIndex{11}\)

    \(-(-1)\)

    Answer

    \(-1\), since\(-(-1) = 1\)

    Practice Problem \(\PageIndex{12}\)

    \(-[-(-7)]\)

    Answer

    \(7\)

    Practice Problem \(\PageIndex{13}\)

    Suppose that a is a positive number. What type of number is \(−a\) ?

    Answer

    If \(a\) is positive, \(-a\) is negative.

    Practice Problem \(\PageIndex{14}\)

    Suppose that \(a\) is a negative number. What type of number is \(−a\) ?

    Answer

    If \(a\) is negative, \(-a\) is positive.

    Practice Problem \(\PageIndex{15}\)

    Suppose we do not know the sign of the number \(m\). Can we say that \(−m\) is positive, negative, or that we do not know ?

    Answer

    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

    a plus sign or no sign at all

    Exercise \(\PageIndex{2}\)

    A number is denoted as negative if it is directly preceded by ____________________ .

    For the following problems, how should the real numbers be read ? (Write in words.)

    Exercise \(\PageIndex{1}\)

    \(-5\)

    Answer

    a negative five

    Exercise \(\PageIndex{2}\)

    \(-3\)

    Exercise \(\PageIndex{3}\)

    \(12\)

    Answer

    twelve

    Exercise \(\PageIndex{4}\)

    \(10\)

    Exercise \(\PageIndex{5}\)

    \(-(-4)\)

    Answer

    negative negative four

    Exercise \(\PageIndex{6}\)

    \(-(-1)\)

    For the following problems, write the expressions in words.

    Exercise \(\PageIndex{7}\)

    \(5 + 7\)

    Answer

    five plus seven

    Exercise \(\PageIndex{8}\)

    \(2 + 6\)

    Exercise \(\PageIndex{9}\)

    \(11 + (-2)\)

    Answer

    eleven plus negative two

    Exercise \(\PageIndex{10}\)

    \(1 + (-5)\)

    Exercise \(\PageIndex{11}\)

    \(6 - (-8)\)

    Answer

    six minus negative eight

    Exercise \(\PageIndex{12}\)

    \(0 - (-15)\)

    Rewrite the following problems in a simpler form.

    Exercise \(\PageIndex{13}\)

    \(−(−8)\)

    Answer

    \(−(−8)=8\)

    Exercise \(\PageIndex{14}\)

    \(−(−5)\)

    Exercise \(\PageIndex{15}\)

    \(−(−2)\)

    Answer

    \(2\)

    Exercise \(\PageIndex{16}\)

    \(−(−9)\)

    Exercise \(\PageIndex{17}\)

    \(−(−1)\)

    Answer

    \(1\)

    Exercise \(\PageIndex{18}\)

    \(−(−4)\)

    Exercise \(\PageIndex{19}\)

    \(−[−(−3)])\)

    Answer

    \(-3\)

    Exercise \(\PageIndex{20}\)

    \(−[−(−10)]\)

    Exercise \(\PageIndex{21}\)

    \(−[−(−6)]\)

    Answer

    \(-6\)

    Exercise \(\PageIndex{22}\)

    \(−[−(−15)]\)

    Exercise \(\PageIndex{23}\)

    \(−\{−[−(−26)]\}\)

    Answer

    \(26\)

    Exercise \(\PageIndex{24}\)

    \(−\{−[−(−11)]\}\)

    Exercise \(\PageIndex{25}\)

    \(−\{−[−(−31)]\}\)

    Answer

    \(31\)

    Exercise \(\PageIndex{26}\)

    \(−\{−[−(−14)]\}\)

    Exercise \(\PageIndex{27}\)

    \(−[−(12)]\)

    Answer

    \(12\)

    Exercise \(\PageIndex{28}\)

    \(−[−(2)]\)

    Exercise \(\PageIndex{29}\)

    \(−[−(17)]\)

    Answer

    \(17\)

    Exercise \(\PageIndex{30}\)

    \(−[−(42)]\)

    Exercise \(\PageIndex{31}\)

    \(5−(−2)\)

    Answer

    \(5−(−2)=5+2=7\)

    Exercise \(\PageIndex{32}\)

    \(6−(−14)\)

    Exercise \(\PageIndex{33}\)

    \(10−(−6)\)

    Answer

    \(16\)

    Exercise \(\PageIndex{34}\)

    \(18−(−12)\)

    Exercise \(\PageIndex{35}\)

    \(31−(−1)\)

    Answer

    \(32\)

    Exercise \(\PageIndex{36}\)

    \(54−(−18)\)

    Exercise \(\PageIndex{37\)

    \(6−(−3)−(−4)\)

    Answer

    \(13\)

    Exercise \(\PageIndex{38}\)

    \(2−(−1)−(−8)\)

    Exercise \(\PageIndex{39}\)

    \(15−(−6)−(−5)\)

    Answer

    \(26\)

    Exercise \(\PageIndex{40}\)

    \(24−(−8)−(−13)\)

    Exercises for Review

    Exercise \(\PageIndex{41}\)

    There is only one real number for which \((5a)^2=5a^2\). What is the number?

    Answer

    \(0\)

    Exercise \(\PageIndex{42}\)

    Simplify \((3xy)(2x^2y^3)(4x^2y^4)\).

    Exercise \(\PageIndex{43}\)

    Simplify \(x^{n + 3} \cdot x^5\)

    Answer

    \(x^{n + 8}\)

    Exercise \(\PageIndex{44}\)

    Simplify \((a^3b^2c^4)^4\).

    Exercise \(\PageIndex{45}\)

    Simplify \((\dfrac{4a^2b}{3xy^3})^2\)

    Answer

    \(\dfrac{16a^4b^2}{9x^2y^6}\)


    This page titled 3.2: Signed Numbers is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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