The Real Numbers

C O L L E G E A L G E B R A
S I E R R A N I C H O L S
The Real Numbers

Objectives
 Identify real numbers
 Identify real number properties
 Write inequality and interval notation
 Solve absolute values
 Sketch graphs of absolute value inequalities
 Determine distance using absolute value inequalities

The Real Numbers
The 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 System
The 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.

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 point corresponding to 0 is the origin of the
coordinate line.

Note: The coordinate line is also called the real
number line, or simply, the number line.

Three sets of real numbers are especially important in
mathematics.
1. The set of numbers,
…, -3, -2, -1, 0, 1, 2, 3, …
is the set of integers.

2. The 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 (b≠0). For example,
-1/2 11/7 2.16=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 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

Some 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 π, the ratio
of the circumference to the diameter of a circle. The
decimal representation of an irrational number
neither repeats nor terminates.

The sets of rational numbers and irrational numbers
together form the set of real numbers.

Example 1: Identifying Real Numbers
Determine which of the following real numbers are
integers, rational numbers, and irrational numbers:
-8, √5, 2π, -32/7
Plot the numbers on a coordinate line.

The 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.

The third number, 2π, is also an irrational number. We can
prove this claim by first assuming that 2π is a rational number
and then showing that this assumption leads to a contradiction.
Suppose for a moment that 2π is the ratio of two integers a and
b:
2π = a/b
From this equation, it follows that
π = a/2b
implying that π is the ratio of two integers a and 2b. But we
know that π is an irrational number, so 2π must be an irrational
number.

The fourth number, -32/7, is a ratio of two integers, so
it is a rational number.

By calculator, we find that √5, 2π, and -32/7 are
approximately 2.2, 6.3, and -4.6 (to the nearest 0.1),
respectively.

In summary:
Integers: -8
Rational Numbers: -8, -32/7
Irrational Numbers: √5, 2π

Properties of Real Numbers
Many 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.

First, 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.

Suppose that a, b, and c are real numbers
Commutative Properties
Commutative Property of
Addition
a + b = b + a Reordering the terms of a
sum does not change the
value of the sum
Commutative Property of
Multiplication
ab = ba Reordering the factors of
a product does not change
the value of the product

Associative Properties
Associative 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 sum
Associative 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

Identities
Additive Identity (Zero) a + 0 = a Adding 0 to a real
number does not change
its value
Multiplicative Identity
(One)
a·1 = a Multiplying a real number
by 1 does not change its
value

Inverses
Additive Inverse
(Opposite)
a + (-a) = 0 The sum of a real number
and its opposite (its
additive inverse) is 0 (the
additive identity)
Multiplicative Inverse
(Reciprocal)
a·(1/a) = 1, a≠0 The product of a nonzero
real number and its
reciprocal (its
multiplicative inverse) is 1
(the multiplicative
identity)

Distributive Property of Multiplication Over Addition
a(b + c) = ab + ac Multiplying a number by a sum is
equivalent to multiplying each term of
the sum by the number and adding the
resulting products

The 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

The 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.

The other two operations of real
numbers, subtraction and division, are
defined in terms of addition and
multiplication, respectively.
Subtraction of Real
Numbers
a – b = a + (-b) Subtracting a number is
equivalent to adding its
opposite
Division of Real Numbers a ÷ b = a·(1/b), b≠0 Dividing by a nonzero
number is equivalent to
multiplying by its
reciprocal

Example 2: Identifying Real Number Properties
Identify the property of real numbers that is exhibited
in the statement given.
a) 2(4 + y) = 2(y + 4)
b) (2a)(ac)=2(aa)c
c) 3(x +y – 7) = 3x + 3y - 21

Example 2: Identifying Real Number Properties
a) The terms in the sum 4 + y are reordered as y + 4.
Thus, the statement demonstrates the commutative
property of addition.
b) The change in grouping symbols reorders the
operations, so this statement demonstrates the
associative property of multiplication
c) A 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 Inequalities
Algebraic 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

Similarly, 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 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)

A 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.

The 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.

Inequality Notation and Interval Notation
Description and Graph Inequality Notation Interval Notation
Open Interval
a < x < b (a, b)
Closed Interval
a ≤ x ≤ b [a, b]
Half-Open Interval
a ≤ x < b
a < x ≤ b
[a, b)
(a, b]
Open Infinite Interval
x < a
x > a
(-∞, a)
(a, +∞)
Closed Infinite Interval
x ≤ a
x ≥ a
(-∞, a]
[a, -∞)

Example 3: Using Inequality and Interval
Notation
Sketch the graph of the set described by the inequality
notation, and describe it using interval notation.
a) -3 ≤ x < 2
b) x ≤ 5

Example 3: Using Inequality and Interval
Notation
a) The compound inequality -3 ≤ x < 2 describes a
half-open interval with endpoints -3 and 2.
The interval notation for this interval is [-3, 2).
b) The closed infinite interval described by x ≤ 5
has interval notation (-∞, 5]

Absolute Value
The 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| = 2
since -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 Value
In 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 Values
Write each expression without absolute value symbols
a) |√11 – 4|
b) |x + 6|, given that x > -5

Example 4: Using Absolute Values
a) The 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
a) 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 Value
Absolute Value Equations and Inequalities
Description Graph Set of Points
|u| < a All real numbers u
that are less than a
units from the
origin
-a < u < a
(-a, a)
|u| = a All real numbers u
that are exactly a
units from the
origin
-a a
u = ±a
|u| > a All real numbers u
that are more than
a units from the
origin
u < -a or u > a
(-∞, -a) or (a, +∞)
Inequalities involving absolute values arise frequently in science and in fields of
mathematics such as calculus. This table shows a summary of how absolute values
are used to describe intervals of real numbers.

Example 5: Sketching the Graphs of Absolute
Value Inequalities
Sketch each set of numbers described on a coordinate
line.
a) |x| < 5
b) |t| = 4
c) |p| ≥ 2

Value Inequalities
a) The 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

Value Inequalities
b) From the table, the equation |t| = 4 is equivalent to
t = ±4. Thus, |t| =4 describes the points -4 and 4

Value Inequalities
c) The solution to |p| ≥ 2 is the set of real numbers
that are at least 2 units from the origin:
p ≤ -2 or p ≥ 2

Absolute Value
The distance between two real numbers
a and b is the absolute value of their
difference.
That is,
d(a, b) = |a – b|

Absolute Value
The 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| = 5
d(2, -7) = |2-(-7)| = 9
d(6, 6) = |6-6| = 0

Example 6: Distance and Absolute Value
Inequalities
Sketch each set of numbers described on a coordinate
line.
a) |x – 2| < 0.5
b) |x + 4| ≥ 2

Inequalities
a) The 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

Inequalities
b) Because |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

Homework
 The Real Numbers Practice Sheet
 The Real Numbers Discussion Forum

The Real Numbers

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