Download Section 5.4 -- The Irrational Numbers

Thinking
Mathematically
Number Theory and the Real Number
System
5.4 The Irrational Numbers
The Irrational Numbers
The set of irrational numbers is the set of
numbers whose decimal representations are
neither terminating nor repeating.
An irrational number cannot be written as the
ratio of two integers.
One of the most well known irrational numbers is
the ratio between the circumference and diameter
of a circle known as “pi” and written π.
3.14.159
The Golden Ratio
The golden ratio has the value
5 1
2
which is approximately 1.618 033 988
Fibonnaci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34
Square Roots
The principal “square root” of a positive number, n,
is the positive number that when multiplied times
itself produces n. This number is written n . The
square root of zero is zero.
Some square roots are rational numbers. For
example since 6 x 6 = 36, 36 = 6.
But, the square root operation provides us with many
examples of irrational numbers. E.g. 2
Example: Square Roots
Exercise Set 5.4 #3, #11
25 = ?
173 = ?
Simplifying Square Roots
a
b
a
b

a
b
2
2
Simplify by looking for perfect square factors
Exercise Set 5.4 #19
Simplify 80
The Product Rule for Square Roots
If a and b represent nonnegative numbers,
then √(ab) = a• b and a• b = (ab).
Exercise Set 5.4 #29
3 x 6 = ?
The Quotient Rule for Square Roots
If a and b represent nonnegative real numbers
and b ≠ 0, then
a a
a a
 and

b b
b b
The quotient of two square roots is the square root
of the quotient.
Example: Quotients of Square Roots
Exercise Set 5.4 #35
90
?
2
Adding and Subtracting Square Roots
ac + bc = (a + b)c
Important:
ac - bc = (a - b)c
a
b a b
Exercise Set 5.4 #49
50 - 18 = ?
Rationalizing the Denominator
The process of rewriting a radical expression to
remove the square root from the denominator
without changing the value of the expression is
called rationalizing the denominator.
Exercise Set 5.4 #59
Rationalize the denominator of
21
7
Thinking
Mathematically
Number Theory and the Real Number
System
5.4 The Irrational Numbers