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1.4
The Irrational Numbers and the Real
Number System
Pythagorean Theorem
• Pythagoras, a Greek mathematician, is
credited with proving that in any right
triangle, the square of the length of one side
(a2) added to the square of the length of the
other side (b2) equals the square of the length
of the hypotenuse (c2) .
• a2 + b2 = c2
Irrational Numbers
• An irrational number is a real number whose
decimal representation is a nonterminating,
nonrepeating decimal number.
• Examples of irrational numbers:
5.12639573...
6.1011011101111...
0.525225222...
Radicals
•
2, 17,
53 are all irrational numbers.
The symbol
is called the radical sign.
The number or expression inside the radical
sign is called the radicand.
Principal Square Root
• The principal (or positive) square root of a
number n, written n is the positive number
that when multiplied by itself, gives n.
• For example,
16 = 4 since 4  4 = 16
49 = 7 since 7  7 = 49
Perfect Square
• Any number that is the square of a natural
number is said to be a perfect square.
• The numbers 1, 4, 9, 16, 25, 36, and 49 are the
first few perfect squares.
Product Rule for Radicals
a  b  a  b,
a  0, b  0
• Simplify:
a)
40
40  4 10  4  10  2  10  2 10
b)
125
125  25  5  25  5  5  5  5 5
Addition and Subtraction of Irrational
Numbers
• To add or subtract two or more square roots
with the same radicand, add or subtract their
coefficients.
• The answer is the sum or difference of the
coefficients multiplied by the common radical.
Example: Adding or Subtracting
Irrational Numbers
• Simplify: 4 7  3 7
• Simplify: 8 5  125
4 7 3 7
8 5  125
 (4  3) 7
 8 5  25  5
7 7
8 5 5 5
 (8  5) 5
3 5
Multiplication of Irrational Numbers
• Simplify:
6  54
6  54  6  54  324  18
Quotient Rule for Radicals
a
b

a
,
b
a  0, b  0
Example: Division
• Divide:
16
• Divide:
4
• Solution:
16
16

 42
4
4
144
2
• Solution:
144
144

 72
2
2
 36  2  36  2
6 2
Rationalizing the Denominator
• A denominator is rationalized when it contains
no radical expressions.
• To rationalize the denominator, multiply BOTH
the numerator and the denominator by a
number that will result in the radicand in the
denominator becoming a perfect square.
Then simplify the result.
Example: Rationalize
• Rationalize the denominator of
• Solution:
8
8


12
12

2
3
6

3

2
3
2
3

3
3
8
12
.
1.5
Real Numbers and their Properties
Real Numbers
• The set of real numbers is formed by the
union of the rational and irrational numbers.

.

• The symbol for the set of real numbers is
Relationships Among Sets
Real numbers
Rational numbers
Integers
Whole numbers
Natural numbers
Irrational
numbers
Properties of the Real Number
System
• Closure
If an operation is performed on any two elements
of a set and the result is an element of the set, we
say that the set is closed under that given
operation.
Commutative Property
• Addition
a+b=b+a
for any real numbers a
and b.
• Multiplication
a • b = b •a
for any real numbers a
and b.
Example
• 8 + 12 = 12 + 8 is a true statement.
• 5  9 = 9  5 is a true statement.
• Note: The commutative property does not
hold true for subtraction or division.
Associative Property
• Addition
(a + b) + c = a + (b + c),
for any real numbers
a, b, and c.
• Multiplication
(a • b) • c = a • (b • c),
for any real numbers
a, b, and c.
Example
• (3 + 5) + 6 = 3 + (5 + 6) is true.
• (4  6)  2 = 4  (6  2) is true.
• Note: The associative property does not hold
true for subtraction or division.
Distributive Property
• Distributive property of multiplication over
addition
a • (b + c) = a • b + a • c
for any real numbers a, b, and c.
• Example: 6 • (r + 12) = 6 • r + 6 • 12
= 6r + 72
1.6
Rules of Exponents and Scientific
Notation
Exponents
• When a number is written with an exponent,
there are two parts to the expression: base
• The exponent tells how many times the base
should be multiplied together.
exponent
4  44444
5
Product Rule
a ga  a
m
• Simplify: 34 • 39
34 • 39 = 34 + 9 = 313
• Simplify: 64 • 65
64 • 65 = 64 + 5 = 69
n
mn
Quotient Rule
m
a
mn

a
, a0
n
a
5
7
• Simplify:
2
7
75
52
3

7

7
2
7
• Simplify:
915
8
9
915
158
7

9

9
8
9
Zero Exponent Rule
a0  1, a  0
• Simplify: (3y)0
(3y)0 = 1
• Simplify: 3y0
3y0 = 3 (y0)
= 3(1) = 3
Negative Exponent Rule
a
m
1
 m, a0
a
• Simplify: 64
6
4
1
1
 4 
1296
6
Power Rule
(a )  a
m n
mgn
• Simplify: (32)3
(32)3 = 32•3 = 36
• Simplify: (23)5
(23)5 = 23•5 = 215
Scientific Notation
• Many scientific problems deal with very large
or very small numbers.
• 93,000,000,000,000 is a very large number.
• 0.000000000482 is a very small number.
Scientific Notation continued
• Scientific notation is a shorthand method used
to write these numbers.
• 9.3  1013 and 4.82  10–10 are two examples
of numbers in scientific notation.
To Write a Number in Scientific Notation
1. Move the decimal point in the original number to
the right or left until you obtain a number greater
than or equal to 1 and less than 10.
2. Count the number of places you have moved the
decimal point to obtain the number in step 1.
If the decimal point was moved to the left, the
count is to be considered positive. If the decimal
point was moved to the right, the count is to be
considered negative.
3. Multiply the number obtained in step 1 by 10
raised to the count found in step 2. (The count
found in step 2 is the exponent on the base 10.)
Example
• Write each number in scientific notation.
a) 1,265,000,000.
1.265  109
b) 0.000000000432
4.32  1010
To Change a Number in Scientific
Notation to Decimal Notation
1. Observe the exponent on the 10.
2. a) If the exponent is positive, move the decimal
point in the number to the right the same
number of places as the exponent. Adding
zeros to the number might be necessary.
b) If the exponent is negative, move the decimal
point in the number to the left the same
number of places as the exponent. Adding
zeros might be necessary.
Example
• Write each number in decimal notation.
a) 4.67  105
467,000
b) 1.45  10–7
0.000000145