CONCEPTS
5.3 THE RATIONAL NUMBERS
Rational numbers.
Convert a repeating decimal to a fraction.
5.4 IRRATIONALS
Pythagorean theorem
square root
+, -, ×, ÷
rationalizing the denominator
5.3 THE RATIONAL NUMBERS are numbers that
can be expressed as the quotient of two integers.
ex. Express the rational number in decimal form.
a)
3
4
b)
3
11
How can you tell if a fraction will be a terminating
or a repeating decimal?
A fraction can be written as a terminating decimal iff
the only prime factors of the reduced denominator are
2s or 5s.
ex. a)
2
3
b)
7
20
c)
4
35
d)
7
35
ex. Convert to a fraction.
a) 0.00045
b) 6.82
c) 26.0037
d) 0.6
e) 21.5342
Denseness Property: A set is dense iff between any
two members of the set there exists a third member
of the set.
ex. Which sets are dense?
a) integers
b) rational numbers
ex. Find a rational number between
a)
2
5
and
3
5
b) 0.07 and 0.076
5.4 THE IRRATIONAL NUMBERS AND THE
REAL NUMBER SYSTEM
Pythagorean theorem: Given a right triangle with
sides a and b, and hypotenuse c,
ex. Given a right triangle with sides a and b, find
the hypotenuse.
a) a = 3
b=4
b) a = 1
b=1
Irrational numbers: a real number whose decimal
representation is nonterminating and nonrepeating.
Square root (principle square root),
√
n,
is the positive number that when squared equals n.
Product rule for radicals:
√
a·b=
ex. Simplify.
√
3 40 =
Addition and Subtraction
ex. Simplify.
√
√
a) 4 15 + 7 15 =
√
√
√
b) 3 20 − 7 45 + 8 =
Division
√
a
√ =
b
√
15
ex. √ =
5
Rationalizing the denominator (eliminate the radical
in the denominator.
√
3
a) √
5
=
√
2
b) √
6
=
ex. A doorway is 12 feet x 9 feet. Find the diameter
of the largest circular mirror that will fit through
the door.