1.4 the irrational numbers and the real number system
TRANSCRIPT
![Page 1: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/1.jpg)
1.4
The Irrational Numbers and the Real Number System
![Page 2: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/2.jpg)
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
![Page 3: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/3.jpg)
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...
![Page 4: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/4.jpg)
• are all irrational numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand.
Radicals
2, 17, 53
![Page 5: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/5.jpg)
Principal Square Root
• The principal (or positive) square root of a number n, written is the positive number that when multiplied by itself, gives n.
• For example,
16 = 4 since 44 =16
49 = 7 since 77 = 49
n
![Page 6: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/6.jpg)
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.
![Page 7: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/7.jpg)
Product Rule for Radicals
• Simplify:a)
b)
ab a b, a 0, b 0
40 410 4 10 2 10 2 10
125 255 25 5 5 5 5 5
40
125
![Page 8: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/8.jpg)
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.
![Page 9: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/9.jpg)
Example: Adding or Subtracting Irrational Numbers
• Simplify: • Simplify: 4 7 3 7
4 7 3 7
(4 3) 7
7 7
8 5 125
8 5 125
8 5 25 5
8 5 5 5
(8 5) 5
3 5
![Page 10: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/10.jpg)
Multiplication of Irrational Numbers
• Simplify:
6 54
6 54 654 324 18
![Page 11: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/11.jpg)
Quotient Rule for Radicals
a
b
a
b, a 0, b 0
![Page 12: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/12.jpg)
Example: Division
• Divide:
• Solution:
• Divide:
• Solution:
16
4
144
2
16
4
16
4 4 2
144
2
144
2 72
362 36 2
6 2
![Page 13: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/13.jpg)
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.
![Page 14: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/14.jpg)
Example: Rationalize
• Rationalize the denominator of
• Solution:
8
12.
8
12
8
12
2
3
2
3
2
3
3
3
6
3
![Page 15: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/15.jpg)
1.5
Real Numbers and their Properties
![Page 16: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/16.jpg)
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 .
![Page 17: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/17.jpg)
Relationships Among Sets
Irrational numbers
Rational numbers
Integers
Whole numbersNatural numbers
Real numbers
![Page 18: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/18.jpg)
Properties of the Real Number System
• ClosureIf 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.
![Page 19: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/19.jpg)
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.
![Page 20: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/20.jpg)
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.
![Page 21: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/21.jpg)
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.
![Page 22: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/22.jpg)
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.
![Page 23: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/23.jpg)
Distributive Property
• Distributive property of multiplication over additiona • (b + c) = a • b + a • cfor any real numbers a, b, and c.
• Example: 6 • (r + 12) = 6 • r + 6 • 12 = 6r + 72
![Page 24: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/24.jpg)
1.6
Rules of Exponents and Scientific Notation
![Page 25: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/25.jpg)
Exponents
• When a number is written with an exponent, there are two parts to the expression: baseexponent
• The exponent tells how many times the base should be multiplied together.
45 44444
![Page 26: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/26.jpg)
Product Rule
• Simplify: 34 • 39
34 • 39 = 34 + 9 = 313
• Simplify: 64 • 65
64 • 65 = 64 + 5 = 69
amg an amn
![Page 27: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/27.jpg)
Quotient Rule
• Simplify: • Simplify:
am
anam n, a 0
75
72
915
98
75
7275 2 73
915
98915 8 97
![Page 28: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/28.jpg)
Zero Exponent Rule
• Simplify: (3y)0
(3y)0 = 1
• Simplify: 3y0
3y0 = 3 (y0) = 3(1) = 3
a0 1, a 0
![Page 29: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/29.jpg)
Negative Exponent Rule
• Simplify: 64
a m
1
am, a 0
6 4
1
64
1
1296
![Page 30: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/30.jpg)
Power Rule
• Simplify: (32)3
(32)3 = 32•3 = 36
• Simplify: (23)5
(23)5 = 23•5 = 215
(am )n amgn
![Page 31: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/31.jpg)
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.
![Page 32: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/32.jpg)
• 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.
Scientific Notation continued
![Page 33: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/33.jpg)
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.)
![Page 34: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/34.jpg)
Example
• Write each number in scientific notation.a) 1,265,000,000.
1.265 109
b) 0.0000000004324.32 1010
![Page 35: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/35.jpg)
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.
![Page 36: 1.4 The Irrational Numbers and the Real Number System](https://reader036.vdocuments.us/reader036/viewer/2022081420/56649f385503460f94c54717/html5/thumbnails/36.jpg)
Example
• Write each number in decimal notation.a) 4.67 105
467,000
b) 1.45 10–7
0.000000145