chapter 1.1 properties of-real-numbers
TRANSCRIPT
Properties of Real Numbers Properties of Real Numbers
1) real numbers2) rational numbers3) irrational numbers
Classify real numbers.
Use the properties of real numbers to evaluate expressions.
All of the numbers that you use in everyday life are real numbers.
Properties of Real Numbers Properties of Real Numbers
All of the numbers that you use in everyday life are real numbers.
Each real number corresponds to exactly one point on the number line, and
Properties of Real Numbers Properties of Real Numbers
All of the numbers that you use in everyday life are real numbers.
Each real number corresponds to exactly one point on the number line, and
x
Properties of Real Numbers Properties of Real Numbers
All of the numbers that you use in everyday life are real numbers.
Each real number corresponds to exactly one point on the number line, and
x
0 1 2 3 4 5-5 -4 -2 -1-3
Properties of Real Numbers Properties of Real Numbers
All of the numbers that you use in everyday life are real numbers.
Each real number corresponds to exactly one point on the number line, and
x
0 1 2 3 4 5-5 -4 -2 -1-3
every point on the number line represents one real number.
Properties of Real Numbers Properties of Real Numbers
All of the numbers that you use in everyday life are real numbers.
Each real number corresponds to exactly one point on the number line, and
x
0 1 2 3 4 5-5 -4 -2 -1-3
21
2 2
every point on the number line represents one real number.
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
zeroRational numbers can be expressed as a ratio , where a and b areintegers and b is not ____! b
a
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
zeroRational numbers can be expressed as a ratio , where a and b areintegers and b is not ____! b
a
The decimal form of a rational number is either a terminating or repeating decimal.
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
zeroRational numbers can be expressed as a ratio , where a and b areintegers and b is not ____! b
a
The decimal form of a rational number is either a terminating or repeating decimal.
Examples: ratio form decimal form
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
zeroRational numbers can be expressed as a ratio , where a and b areintegers and b is not ____! b
a
The decimal form of a rational number is either a terminating or repeating decimal.
Examples: ratio form decimal form
9 0.3
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
zeroRational numbers can be expressed as a ratio , where a and b areintegers and b is not ____! b
a
The decimal form of a rational number is either a terminating or repeating decimal.
Examples: ratio form decimal form
9 0.3
83
375.0
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
zeroRational numbers can be expressed as a ratio , where a and b areintegers and b is not ____! b
a
The decimal form of a rational number is either a terminating or repeating decimal.
Examples: ratio form decimal form
9 0.3
83
375.0
73
428571.0
or . . . 714285714285714285.0
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
A real number that is not rational is irrational.
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
A real number that is not rational is irrational.
The decimal form of an irrational number neither __________ nor ________.
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
A real number that is not rational is irrational.
The decimal form of an irrational number neither __________ nor ________.terminates repeats
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
A real number that is not rational is irrational.
The decimal form of an irrational number neither __________ nor ________.terminates repeats
Examples:
. . . 141592654.3
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
A real number that is not rational is irrational.
The decimal form of an irrational number neither __________ nor ________.terminates repeats
Examples:
. . . 141592654.3 More Digits of PI?
e . . . 718281828.2
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
A real number that is not rational is irrational.
The decimal form of an irrational number neither __________ nor ________.terminates repeats
Examples:
. . . 141592654.3 More Digits of PI?
e . . . 718281828.2
2 3 5 7 11 13
Do you notice a pattern within this group of numbers?
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
A real number that is not rational is irrational.
The decimal form of an irrational number neither __________ nor ________.terminates repeats
Examples:
. . . 141592654.3 More Digits of PI?
e . . . 718281828.2
2 3 5 7 11 13
Do you notice a pattern within this group of numbers?
Properties of Real Numbers Properties of Real Numbers
Real numbers can be classified a either _______ or ________.rational irrational
A real number that is not rational is irrational.
The decimal form of an irrational number neither __________ nor ________.terminates repeats
Examples:
. . . 141592654.3 More Digits of PI?
e . . . 718281828.2
2 3 5 7 11 13
Do you notice a pattern within this group of numbers?
They’re all PRIME numbers!
Properties of Real Numbers Properties of Real Numbers
- Numbers Real
Relationships among the real numbers - (sets and subsets).
Properties of Real Numbers Properties of Real Numbers
- Numbers Real
Q = rationals
Q I
I = irrationals
Relationships among the real numbers - (sets and subsets).
Properties of Real Numbers Properties of Real Numbers
- Numbers Real
Q = rationals
Q I
I = irrationals
Z
Z = integers
Relationships among the real numbers - (sets and subsets).
Properties of Real Numbers Properties of Real Numbers
- Numbers Real
Q = rationals
Q I
I = irrationals
Z
Z = integers
W
W = wholes
Relationships among the real numbers - (sets and subsets).
Properties of Real Numbers Properties of Real Numbers
The square root of any whole number is either whole or irrational.
Properties of Real Numbers Properties of Real Numbers
The square root of any whole number is either whole or irrational.
For example, is a whole number, but , since it lies between 5 and 6, must be irrational.
36 30
Properties of Real Numbers Properties of Real Numbers
The square root of any whole number is either whole or irrational.
x
0 1 32 4 5 6 7 98 10
For example, is a whole number, but , since it lies between 5 and 6, must be irrational.
36 30
36
. . . 477225575.5
25
30
Properties of Real Numbers Properties of Real Numbers
The square root of any whole number is either whole or irrational.
x
0 1 32 4 5 6 7 98 10
For example, is a whole number, but , since it lies between 5 and 6, must be irrational.
36 30
36
. . . 477225575.5
25
30
Common Misconception:
Do not assume that a number is irrational just because it is expressed using the square root symbol. Find its value first!
Properties of Real Numbers Properties of Real Numbers
The square root of any whole number is either whole or irrational.
x
0 1 32 4 5 6 7 98 10
For example, is a whole number, but , since it lies between 5 and 6, must be irrational.
36 30
36
. . . 477225575.5
25
30
Common Misconception:
Do not assume that a number is irrational just because it is expressed using the square root symbol. Find its value first!
Study Tip:
KNOW and recognize (at least) these numbers,
169644936251694 14412110081
Properties of Real Numbers Properties of Real Numbers
The real number system is an example of a mathematical structure called a field.
Some of the properties of a field are summarized in the table below:
Properties of Real Numbers Properties of Real Numbers
The real number system is an example of a mathematical structure called a field.
Some of the properties of a field are summarized in the table below:
Real Number Properties
For any real numbers a, b, and c.
Property Addition Multiplication
Associative
Identity
Inverse
Distributive
Properties of Real Numbers Properties of Real Numbers
Commutative
The real number system is an example of a mathematical structure called a field.
Some of the properties of a field are summarized in the table below:
Real Number Properties
For any real numbers a, b, and c.
Property Addition Multiplication
Commutative
Associative
Identity
Inverse
Distributive
ba ab ba ab
Properties of Real Numbers Properties of Real Numbers
The real number system is an example of a mathematical structure called a field.
Some of the properties of a field are summarized in the table below:
Real Number Properties
For any real numbers a, b, and c.
Property Addition Multiplication
Commutative
Associative
Identity
Inverse
Distributive
ba ab ba ab
cba cba cba cba
Properties of Real Numbers Properties of Real Numbers
The real number system is an example of a mathematical structure called a field.
Some of the properties of a field are summarized in the table below:
Real Number Properties
For any real numbers a, b, and c.
Property Addition Multiplication
Commutative
Associative
Identity
Inverse
Distributive
ba ab ba ab
cba cba cba cba
0a a a0 1a a a1
Properties of Real Numbers Properties of Real Numbers
The real number system is an example of a mathematical structure called a field.
Some of the properties of a field are summarized in the table below:
Real Number Properties
For any real numbers a, b, and c.
Property Addition Multiplication
Commutative
Associative
Identity
Inverse
Distributive
ba ab ba ab
cba cba cba cba
0a a a0 1a a a1
aa 0 aa then0, a If
a
a1
1 aa1
Properties of Real Numbers Properties of Real Numbers
The real number system is an example of a mathematical structure called a field.
Some of the properties of a field are summarized in the table below:
Real Number Properties
For any real numbers a, b, and c.
Property Addition Multiplication
Commutative
Associative
Identity
Inverse
Distributive
ba ab ba ab
cba cba cba cba
0a a a0 1a a a1
aa 0 aa then0, a If
a
a1
1 aa1
)( cba and acab acb )( caba
Properties of Real Numbers Properties of Real Numbers
Reciprocals
• The Reciprocal of a is providing a does NOT equal 0.
• Definition of Subtraction:– Adding the opposite:
• Definition of Division:– Multiplying by the reciprocal:
a
1
)( baba
ba
b
a 1
Operations with Real Numbers
A. The sum of -5 and -13 is
B.The difference of -17 and -8 is
C. The product of -3 and -6 is
D. The quotient of 28 and -7 is
18
)8(17
9817
1863
47
28
Real Life Wind Farms• One barrel of oil can generate 545 kilowatt-hours of
electricity. In 1990, the 17,000 windmills in California could generate up to 1.5 million kilowatt-hours per hour. At peak capacity, how many barrels of oil could be saved each hour? Operating at 75% of peak efficiency, how many barrels of oil could be saved in a year?
)(545
)(
barrelhourperhoursKilowatt
windhourperhoursKilowattOilofBarrels
545
000,500,1OilofBarrels yearperbarrels752,2
At 75% peak capacity for a year
year
days
day
hours
hour
barrels 36524275275.0
year
barrels61018
Consider these Java Applets to better understand theDistributive Property
Algebra Tiles 1
Algebra Tiles 2