3.2: Signed Numbers
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.
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
−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.
\(4 + 10\)
- Answer
-
four plus ten
\(7 + (-4)\)
- Answer
-
seven plus negative four
\(-9 + 2\)
- Answer
-
negative nine plus two
\(-16 - (+8)\)
- Answer
-
negative sixteen minus positive eight
\(-1 -(-9)\)
- Answer
-
negative one minus negative nine
\(0 + (-7)\)
- 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.
\(−(−a)\) is opposite \(−a\) on the number line. This implies that \(−(−a)=a\).
This property of opposites suggests the double-negative property for real numbers.
If a is a real number, then
\(−(−a)=a\)
Sample Set B
If \(a=3\), then \(−a=−3\) and \(−(−a)=−(−3)=3\).
If \(a=−4\), then \(−a=−(−4)=4\) and \(−(−a)=a=−4\).
Practice Set B
Find the opposite of each real number.
\(8\)
- Answer
-
\(-8\)
\(17\)
- Answer
-
\(-17\)
\(-6\)
- Answer
-
\(6\)
\(-15\)
- Answer
-
\(15\)
\(-(-1)\)
- Answer
-
\(-1\), since\(-(-1) = 1\)
\(-[-(-7)]\)
- Answer
-
\(7\)
Suppose that a is a positive number. What type of number is \(−a\) ?
- Answer
-
If \(a\) is positive, \(-a\) is negative.
Suppose that \(a\) is a negative number. What type of number is \(−a\) ?
- Answer
-
If \(a\) is negative, \(-a\) is positive.
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
A number is denoted as positive if it is directly preceded by ____________________ .
- Answer
-
a plus sign or no sign at all
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.)
\(-5\)
- Answer
-
a negative five
\(-3\)
\(12\)
- Answer
-
twelve
\(10\)
\(-(-4)\)
- Answer
-
negative negative four
\(-(-1)\)
For the following problems, write the expressions in words.
\(5 + 7\)
- Answer
-
five plus seven
\(2 + 6\)
\(11 + (-2)\)
- Answer
-
eleven plus negative two
\(1 + (-5)\)
\(6 - (-8)\)
- Answer
-
six minus negative eight
\(0 - (-15)\)
Rewrite the following problems in a simpler form.
\(−(−8)\)
- Answer
-
\(−(−8)=8\)
\(−(−5)\)
\(−(−2)\)
- Answer
-
\(2\)
\(−(−9)\)
\(−(−1)\)
- Answer
-
\(1\)
\(−(−4)\)
\(−[−(−3)])\)
- Answer
-
\(-3\)
\(−[−(−10)]\)
\(−[−(−6)]\)
- Answer
-
\(-6\)
\(−[−(−15)]\)
\(−\{−[−(−26)]\}\)
- Answer
-
\(26\)
\(−\{−[−(−11)]\}\)
\(−\{−[−(−31)]\}\)
- Answer
-
\(31\)
\(−\{−[−(−14)]\}\)
\(−[−(12)]\)
- Answer
-
\(12\)
\(−[−(2)]\)
\(−[−(17)]\)
- Answer
-
\(17\)
\(−[−(42)]\)
\(5−(−2)\)
- Answer
-
\(5−(−2)=5+2=7\)
\(6−(−14)\)
\(10−(−6)\)
- Answer
-
\(16\)
\(18−(−12)\)
\(31−(−1)\)
- Answer
-
\(32\)
\(54−(−18)\)
\(6−(−3)−(−4)\)
- Answer
-
\(13\)
\(2−(−1)−(−8)\)
\(15−(−6)−(−5)\)
- Answer
-
\(26\)
\(24−(−8)−(−13)\)
Exercises for Review
There is only one real number for which \((5a)^2=5a^2\). What is the number?
- Answer
-
\(0\)
Simplify \((3xy)(2x^2y^3)(4x^2y^4)\).
Simplify \(x^{n + 3} \cdot x^5\)
- Answer
-
\(x^{n + 8}\)
Simplify \((a^3b^2c^4)^4\).
Simplify \((\dfrac{4a^2b}{3xy^3})^2\)
- Answer
-
\(\dfrac{16a^4b^2}{9x^2y^6}\)