CHAPTER 11
Rational and
Irrational
Numbers
Rational Numbers
11-1 Properties
of Rational
Numbers
Rational Numbers
• A real number that
can be expressed as
the quotient of two
integers.
Examples
• 7 = 7/1
• 5 2/3 = 17/3
• .43 = 43/100
• -1 4/5 = -9/5
Write as a quotient of
integers
•3
• 48%
• .60
• - 2 3/5
Which rational
number is greater
8/3 or 17/7
Rules
• a/c > b/d if and only if
ad > bc.
• a/c < b/d if and only if
ad < bc
Examples
• 4/7 ? 3/8
• 7/9 ? 4/5
• 8/15 ? 3/4
Density Property
• Between every pair of
different rational
numbers there is
another rational
number
Implication
• The density property
implies that it is
possible to find an
unlimited or endless
number of rational
numbers between two
given rational numbers.
Formula
If a < b, then to find the
number halfway from
a to b use:
a + ½(b – a)
Example
• Find a rational
number between -5/8
and -1/3.
Rational Numbers
11-2 Decimal
Forms of
Rational
Numbers
Forms of Rational
Numbers
• Any common fraction
can be written as a
decimal by dividing
the numerator by the
denominator.
Decimal Forms
• Terminating
• Nonterminating
Examples
Express each fraction
as a terminating or
repeating decimal
5/6
7/11
3 2/7
Rule
• For every integer n and
every positive integer d,
the decimal form of the
rational number n/d
either terminates or
eventually repeats in a
block of fewer than d
digits.
Rule
• To express a
terminating decimal
as a common fraction,
express the decimal
as a common fraction
with a power of 10 as
the denominator.
Express as a fraction
• .38
• .425
Solutions
• .38 = 38/100 or 19/50
• .425 = 425/1000= 17/40
Express a Repeating
Decimal as a fraction
• .542
• let N = 0.542
• Multiply both sides of
the equation by a
power of 10
Continued
• Subtract the original
equation from the
new equation
• Solve
Rational Numbers
11-3 Rational
Square Roots
Rule
2
a
If = b, then a is a
square root of b.
Terminology
• Radical sign is 
• Radicand is the
number beneath the
radical sign
Product Property of
Square Roots
For any nonnegative
real numbers a and b:
ab = (a) (b)
Quotient Property of
Square Roots
For any nonnegative
real number a and any
positive real number b:
a/b = (a) /(b)
Examples
• 36
• 100
• - 81/1600
• 0.04
Irrational Numbers
11-4 Irrational
Square Roots
Irrational Numbers
• Real number that
cannot be expressed
in the form a/b where
a and b are integers.
Property of
Completeness
• Every decimal
number represents a
real number, and
every real number can
be represented as a
decimal.
Rational or Irrational
• 17
• 49
• 1.21
• 5 + 2 2
Simplify
• 63
• 128
• 50
• 6108
Simplify
• 63 = 9 7 = 37
• 128 = 64 2 = 82
• 50 = 25 5 = 55
• 6108= 636 3=36 3
Rational Numbers
11-5 Square
Roots of Variable
Expressions
Simplify
• 196y2
• 36x8
2
• m -6m + 9
3
• 18a
Solutions
• 196y2 = ± 18y
• 36x8 = ± 6x4
2
• m -6m + 9 = ±(m -3)
3
• 18a = ± 3a 2a
Solve by factoring
• Get the equation
equal to zero
• Factor
• Set each factor equal
to zero and solve
Examples
•
= 64
• 45r2 – 500 = 0
2
• 81y – 16= 0
2
9x
Irrational Numbers
11-6 The
Pythagorean
Theorem
The Pythagorean
Theorem
In any right triangle, the
square of the length of
the hypotenuse equals
the sum of the squares
of the lengths of the
2
2
2
legs. a + b = c
Example
c
a
b
Example
c
8
15
Solution
2
a
2
b
2
c
+ =
2
2
2
8 + 15 = c
64 + 225 =c2
289 =c2
17 = c
Example
The length of one side of
a right triangle is 28 cm.
The length of the
hypotenuse is 53 cm.
Find the length of the
unknown side.
Solution
2
a
2
b
2
c
+ =
2
2
2
a + 28 = 53
a2 + 784 =2809
a2 =2025
a = 45
Converse of the
Pythagorean Theorem
If the sum of the squares of the
lengths of the two shorter
sides of a triangle is equal to
the square of the length of the
longest, then the triangle is a
right triangle. The right side
is opposite the longest side.
Radical Expressions
11-7 Multiplying,
Dividing, and
Simplifying
Radicals
Rationalization
The process of
eliminating a radical
from the denominator.
Simplest Form
• No integral radicand has
a perfect-square factor
other than 1
• No fractions are under a
radical sign, and
• No radicals are in a
denominator
Simplify
• 3/5
• 7/ 8
• 3 3/7
• 9 3/ 24
Solution
• 3/5 = 3 5 /5
• 7/ 8= 14/4
• 3 3/7= 22
• 9 3/ 24 = 9 2/4
Radical Expressions
11-8 Adding and
Subtracting
Radicals
Simplifying Sums or
Differences
• Express each radical in
simplest form.
• Use the distributive
property to add or
subtract radicals with like
radicands.
Examples
• 47 + 57
• 36 - 213
• 73 - 46 + 248
Solution
• 97
• 86 - 213
• 153 -46
Radical Expressions
11-9 Multiplication
of Binomials
Containing
Radicals
Terminology
• Binomials – variable
expressions containing
two terms.
• Conjugates – binomials
that differ only in the sign
of one term.
Rationalization of
Binomials
• Use conjugates to
rationalize
denominators that
contain radicals.
Simplify
• (6 + 11)(6 - 11)
2
• (3 + 5)
• (23 - 57) 2
• 3/(5 - 27)
Solution
• 25
• 14 + 65
• 187 – 2021
• -5 - 2 7
Radical Expressions
11-10 Simple
Radical Equations
Terminology
• Radical equation – an
equation that has a
variable in the radicand.
Examples
• d = 1000
•x = 3
•x = ± 3
Solutions
• 140 = 2(9.8)d
• (5x +1) + 2 = 6
• (11x2 – 63) -2x = 0
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