chapter 1.1 properties of-real-numbers
TRANSCRIPT
![Page 1: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/1.jpg)
![Page 2: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/2.jpg)
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.
![Page 3: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/3.jpg)
All of the numbers that you use in everyday life are real numbers.
Properties of Real Numbers Properties of Real Numbers
![Page 4: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/4.jpg)
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
![Page 5: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/5.jpg)
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
![Page 6: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/6.jpg)
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
![Page 7: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/7.jpg)
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
![Page 8: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/8.jpg)
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
![Page 9: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/9.jpg)
Real numbers can be classified a either _______ or ________.
Properties of Real Numbers Properties of Real Numbers
![Page 10: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/10.jpg)
Real numbers can be classified a either _______ or ________.rational irrational
Properties of Real Numbers Properties of Real Numbers
![Page 11: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/11.jpg)
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
![Page 12: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/12.jpg)
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
![Page 13: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/13.jpg)
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
![Page 14: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/14.jpg)
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
![Page 15: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/15.jpg)
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
![Page 16: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/16.jpg)
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
![Page 17: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/17.jpg)
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
![Page 18: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/18.jpg)
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
![Page 19: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/19.jpg)
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
![Page 20: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/20.jpg)
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
![Page 21: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/21.jpg)
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
![Page 22: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/22.jpg)
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
![Page 23: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/23.jpg)
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
![Page 24: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/24.jpg)
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
![Page 25: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/25.jpg)
- Numbers Real
Relationships among the real numbers - (sets and subsets).
Properties of Real Numbers Properties of Real Numbers
![Page 26: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/26.jpg)
- Numbers Real
Q = rationals
Q I
I = irrationals
Relationships among the real numbers - (sets and subsets).
Properties of Real Numbers Properties of Real Numbers
![Page 27: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/27.jpg)
- 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
![Page 28: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/28.jpg)
- 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
![Page 29: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/29.jpg)
The square root of any whole number is either whole or irrational.
Properties of Real Numbers Properties of Real Numbers
![Page 30: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/30.jpg)
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
![Page 31: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/31.jpg)
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
![Page 32: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/32.jpg)
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
![Page 33: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/33.jpg)
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
![Page 34: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/34.jpg)
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
![Page 35: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/35.jpg)
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
![Page 36: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/36.jpg)
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
![Page 37: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/37.jpg)
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
![Page 38: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/38.jpg)
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
![Page 39: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/39.jpg)
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
![Page 40: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/40.jpg)
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
![Page 41: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/41.jpg)
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
![Page 42: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/42.jpg)
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
![Page 43: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/43.jpg)
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?
![Page 44: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/44.jpg)
)(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
![Page 45: Chapter 1.1 properties of-real-numbers](https://reader031.vdocuments.us/reader031/viewer/2022012405/555f110ed8b42a5c388b51c7/html5/thumbnails/45.jpg)
Consider these Java Applets to better understand theDistributive Property
Algebra Tiles 1
Algebra Tiles 2