Slide 1
College AlgebraSierra NicholsThe Real Numbers
ObjectivesIdentify real numbersIdentify real number
propertiesWrite inequality and interval notationSolve absolute
valuesSketch graphs of absolute value inequalitiesDetermine
distance using absolute value inequalities
The Real NumbersThe set of real numbers is, no doubt, familiar
to you. These numbers are used in day-to-day life to measure
quantities such as distance, time, and area. In this section, we
examine some of the important properties of this set of
numbers.
The Real Number SystemThe best way to visualize the set of real
numbers is with a coordinate line.
Each point on the coordinate line corresponds to a real number.
Conversely, each real number corresponds to a point on the
coordinate line.
The Real Number System
We say that there exists a one-to-one correspondence between the
set of real numbers and the set of points on the coordinate
line.
The Real Number System
The point corresponding to 0 is the origin of the coordinate
line.
The Real Number System
Note: The coordinate line is also called the real number line,
or simply, the number line.
The Real Number SystemThree sets of real numbers are especially
important in mathematics.
The set of numbers,
, -3, -2, -1, 0, 1, 2, 3,
is the set of integers.
The Real Number SystemThe ratios of integers form the set of
rational numbers. Precisely, a rational number can be expressed as
a/b, in which a and b are integers (b0). For example,
-1/211/72.16=216/100
are all rational numbers.Any integer a is a rational number as
well, since a can be written as the ratio of two integers:a=a/1
The Real Number SystemThe decimal representation of a rational
number either repeats in some block of digits or terminates.
For example, the decimal representations of the rational
numbers:
2/7 = 0.285714285714 = 0.285714-17/6 = -2.8333 = -2.83-7/4 =
-1.75
The Real Number SystemSome numbers, such as 2, cannot be
expressed as the ratio of two integers. These numbers are called
irrational numbers. The set of irrational numbers includes, in
part, any numerical expression that contains a radical sign when
simplified. Another very famous irrational number is , the ratio of
the circumference to the diameter of a circle. The decimal
representation of an irrational number neither repeats nor
terminates.
The Real Number SystemThe sets of rational numbers and
irrational numbers together form the set of real numbers.
Example 1: Identifying Real NumbersDetermine which of the
following real numbers are integers, rational numbers, and
irrational numbers:
-8,5,2,-32/7
Plot the numbers on a coordinate line.
Example 1: Identifying Real NumbersThe first number in the list,
-8, is an integer, and it is there fore also a rational number.
The second number, 5, is an irrational number because 5 is not a
perfect square.
Example 1: Identifying Real NumbersThe third number, 2, is also
an irrational number. We can prove this claim by first assuming
that 2 is a rational number and then showing that this assumption
leads to a contradiction.
Suppose for a moment that 2 is the ratio of two integers a and
b:2 = a/bFrom this equation, it follows that = a/2bimplying that is
the ratio of two integers a and 2b. But we know that is an
irrational number, so 2 must be an irrational number.
Example 1: Identifying Real NumbersThe fourth number, -32/7, is
a ratio of two integers, so it is a rational number.
Example 1: Identifying Real NumbersBy calculator, we find that
5, 2, and -32/7 are approximately 2.2, 6.3, and -4.6 (to the
nearest 0.1), respectively.
Example 1: Identifying Real NumbersIn summary:
Integers: -8
Rational Numbers: -8, -32/7
Irrational Numbers: 5, 2
Properties of Real NumbersMany properties of real numbers are
familiar to you from previous algebra courses, but we restate a few
of the important ones here. Notice that these properties are
expressed using the operations of addition and multiplication.
Properties of Real NumbersFirst, we say that the set of real
numbers is closed for the operations of addition and
multiplication. By this, we mean that for any two real numbers a
and b, the sum a+b and the product ab are also real numbers.
Properties of Real NumbersSuppose that a, b, and c are real
numbersCommutative PropertiesCommutative Property of Additiona + b
= b + aReordering the terms of a sum does not change the value of
the sumCommutative Property of Multiplicationab = baReordering the
factors of a product does not change the value of the product
Properties of Real NumbersSuppose that a, b, and c are real
numbersAssociative PropertiesAssociative Property of Addition( a +
b ) + c = a + ( b + c )The order in which the terms of a sum or
added does not change the value of the sumAssociative Property of
Multiplication(ab)c = a(bc)The order in which the factors of a
product are multiplied does not change the value of the product
Properties of Real NumbersSuppose that a, b, and c are real
numbersIdentitiesAdditive Identity (Zero)a + 0 = aAdding 0 to a
real number does not change its valueMultiplicative Identity
(One)a1 = aMultiplying a real number by 1 does not change its
value
Properties of Real NumbersSuppose that a, b, and c are real
numbersInversesAdditive Inverse (Opposite)a + (-a) = 0The sum of a
real number and its opposite (its additive inverse) is 0 (the
additive identity)Multiplicative Inverse (Reciprocal)a(1/a) = 1,
a0The product of a nonzero real number and its reciprocal (its
multiplicative inverse) is 1 (the multiplicative identity)
Properties of Real NumbersSuppose that a, b, and c are real
numbersDistributive Property of Multiplication Over Additiona(b +
c) = ab + acMultiplying a number by a sum is equivalent to
multiplying each term of the sum by the number and adding the
resulting products
Properties of Real NumbersThe distributive property is the
workhorse of algebraic manipulation. This property takes many
forms. All of the following are examples of the distributive
property:4(x + y + z) = 4x + 4y + 4z(2a + b)x = 2ax + bx-(p + 3q +
2r) = -p -3q -2r(x + 2)(y + 3) = x(y + 3) + 2(y + 3) = xy +3x +2y
+6
Properties of Real NumbersThe process of using the distributive
property to change the product a(b + c) to the sum ab + ac is
called simplifying, or expanding, the product.
Conversely, changing the sum ab + ac to the product a(b + c) is
described as factoring the sum.
Properties of Real NumbersThe other two operations of real
numbers, subtraction and division, are defined in terms of addition
and multiplication, respectively.
Subtraction of Real Numbersa b = a + (-b)Subtracting a number is
equivalent to adding its oppositeDivision of Real Numbersa b =
a(1/b), b0Dividing by a nonzero number is equivalent to multiplying
by its reciprocal
Example 2: Identifying Real Number PropertiesIdentify the
property of real numbers that is exhibited in the statement
given.
2(4 + y) = 2(y + 4)
(2a)(ac)=2(aa)c
3(x +y 7) = 3x + 3y - 21
Example 2: Identifying Real Number PropertiesThe terms in the
sum 4 + y are reordered as y + 4. Thus, the statement demonstrates
the commutative property of addition.The change in grouping symbols
reorders the operations, so this statement demonstrates the
associative property of multiplicationA product 3(x + y 7) is
changed to a sum by distributing the multiplication by 3 over x + y
7. The statement reflects the distributive property.
Intervals and InequalitiesAlgebraic statements that include the
symbols , , or , are called inequalities. Inequalities are often
used to describe intervals of real numbers. For example, the set of
real numbers greater than or equal to 1 is described by x 1
Intervals and InequalitiesSimilarly, the set of real numbers
that are both greater than -4 and less than -1 is described by
-4 < x < 1
An inequality with two inequality symbols, such as -4 < x
< 1, is a continued, or compound, inequality.
Intervals and InequalitiesIntervals of real numbers can also be
described by interval notation. For example, the interval x 1 is
described by the interval notation[1, +)and the interval -4 < x
< 1 is described by the interval notation(-4, 1)
Intervals and InequalitiesA parenthesis in interval notation
indicates that the corresponding endpoint is not included in the
interval, and a bracket is used when the endpoint is included.
Intervals and InequalitiesThe symbol (infinity) in the notation
[1, +) does not represent a real number. Rather, it is used to show
that the interval extends indefinitely without bound in the
positive direction. The interval (-, 1] extends indefinitely
without bound in the negative direction.
Intervals and InequalitiesInequality Notation and Interval
NotationDescription and GraphInequality NotationInterval
NotationOpen Intervala < x < b(a, b)Closed Intervala x b[a,
b]Half-Open Intervala x < b
a < x b[a, b)
(a, b]Open Infinite Intervalx < a
x > a(-, a)
(a, +)Closed Infinite Intervalx a
x a(-, a]
[a, -)
Example 3: Using Inequality and Interval NotationSketch the
graph of the set described by the inequality notation, and describe
it using interval notation.
-3 x < 2
x 5
Example 3: Using Inequality and Interval NotationThe compound
inequality -3 x < 2 describes a half-open interval with
endpoints -3 and 2.
The interval notation for this interval is [-3, 2).
The closed infinite interval described by x 5
has interval notation (-, 5]
Absolute ValueThe absolute value of a real number a, written
|a|, is the distance from the origin to the point corresponding to
a.For example, |-2| = 2|2| = 2since -2 and 2 are each 2 units from
the origin.
The absolute value of a number must be nonnegative because it is
the measure of a distance.
Absolute ValueIn practical terms, the absolute value of a
positive number a is a itself, the absolute value of 0 is 0; and
the absolute value of a negative number a is its opposite a.
Example 4: Using Absolute ValuesWrite each expression without
absolute value symbols
|11 4|
|x + 6|, given that x > -5
Example 4: Using Absolute ValuesThe problem hinges on whether 11
4 is a positive or negative quantity. The value of 11 is
approximately 3.3, so 11 4 is negative. Thus, by the definition of
absolute value, we get| 11 4 | = -(11 4 ) = 4 - 11
The value of x is greater than -5, so x + 6 is greater than 1,
and is therefore positive. If follows that |x + 6| = x + 6
Absolute ValueAbsolute Value Equations and
InequalitiesDescriptionGraphSet of Points|u| < aAll real numbers
u that are less than a units from the origin-a < u < a
(-a, a)|u| = aAll real numbers u that are exactly a units from
the origin -a au = a|u| > aAll real numbers u that are more than
a units from the originu < -a or u > a
(-, -a) or (a, +)
Inequalities involving absolute values arise frequently in
science and in fields ofmathematics such as calculus. This table
shows a summary of how absolute valuesare used to describe
intervals of real numbers.
Example 5: Sketching the Graphs of Absolute Value
InequalitiesSketch each set of numbers described on a coordinate
line.
|x| < 5
|t| = 4
|p| 2
Example 5: Sketching the Graphs of Absolute Value
InequalitiesThe inequality |x| < 5 is of the form |u| < a,
where u represents x and a represents 5. By the table, we rewrite
this absolute value inequality as the compound inequality:-5 < x
< 5
Example 5: Sketching the Graphs of Absolute Value
InequalitiesFrom the table, the equation |t| = 4 is equivalent to t
= 4. Thus, |t| =4 describes the points -4 and 4
Example 5: Sketching the Graphs of Absolute Value
InequalitiesThe solution to |p| 2 is the set of real numbers that
are at least 2 units from the origin:p -2orp 2
Absolute ValueThe distance between two real numbers a and b is
the absolute value of their difference.That is, d(a, b) = |a b|
Absolute ValueThe reason we use absolute value in this
definition is that the distance between two points is a nonnegative
quantity. For example,d(8, 3) = |8-3| = 5d(2, -7) = |2-(-7)| =
9d(6, 6) = |6-6| = 0
Example 6: Distance and Absolute Value InequalitiesSketch each
set of numbers described on a coordinate line.
|x 2| < 0.5
|x + 4| 2
Example 6: Distance and Absolute Value InequalitiesThe
expression |x 2| represents d(x, 2), the distance between x and 2,
so the real numbers x that are described by the inequality |x 2|
< 0.5 are those within 0.5 units of 2. This interval is
described by 1.5 < x< 2.5
Example 6: Distance and Absolute Value InequalitiesBecause |x +
4| = |x (-4)|, this quantity represents d(x, -4), the distance
between x and -4. Those numbers x that are 2 or more units from -4
make up the set of real numbers described by x -6 or x -2
HomeworkThe Real Numbers Practice SheetThe Real Numbers
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