ALGEBRA I
LESSON 2-7
Pg. 103-106
SQUARE ROOTS AND REAL NUMBERS
OBJECTIVE: 1. Determine square root values of basic numbers and approximation of numbers
2. Classify and order real numbers
Bell Ringer Special:
1) What is the opposite of addition? (Subtraction)
2) What is the opposite of multiplication? (Division)
3) What is the opposite of “squaring a number”? (Square Root)
Terms
1) Square Root: One of two equal factors of a number
2) Perfect Square: A number whose square root is a rational number (Examples: 4, 9, 16, 25, 36, etc)
3) Real Numbers: the set of all rational and irrational numbers grouped together (all except imaginary)
4) Radical Sign: √16 (Read the “square root of 16”)
Square Roots
If (2)(2) or 22 equals 4, then the square root of 4 is 2. However, (-2)(-2) is also 4! So by definition,
each square root can have two answers! Both the positive and negative roots are solutions to square root
problems. However, most of the time, just the positive or “Principal Square Root” is requested.
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Examples: 1) √64 = Since 64 = (8)(8) or 8 then = 8 (or –8)
2) √0.16 = 0.4 (or –0.4)
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3) √1 = √1/64 = 1/8 (or –1/8)
√64
CLASSIFICATION OF REAL NUMBERS
1) Natural Numbers: Positive counting numbers starting at “1” and continuing to the right forever
2) Whole Number: Natural numbers with “zero” added at the beginning
3) Integers: Both positive and negative counting numbers (also including zero)
4) Rational Numbers: All integers and the fractions that can be expressed w/terminating or repeating
5) Irrational Numbers: Fractions and decimals that “don’t end” predictably. (Not rational and includes
most square roots, except perfect squares)
6) Real Numbers: All rational and irrational numbers grouped together
So if x > 2 that means are real numbers greater than 2 and can be graphed on a line graph accordingly.
Approximation of Square Roots
We should be able to approximate the value of any square root number by using previously known
values of “perfect squares”
22 = 4
32 = 9
42 = 16
52 = 25 62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
112 = 121
122 = 144 ….. So to find the square root, go backwards!
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Example: Approximate the value of √42
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A) Identify the upper and lower perfect squares
√49
= 7
1) There are 13 units between 36 and 49
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2) There are 6 units between 36 and 42
√42
≈ 6.46
3) Divide lower by the larger (6/13) ≈ 0.46
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4) Add the increase to the lower perfect square root
√36
=6
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Step 4 expanded: 6 + 0.46 or 6.46 ≈ √42
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Try √84
≈ 9.16
A) 81 = 9 and 100 = 10
1) There are 19 units between this pair
2) There are 3 units between 81 and 84
3) Divide 3/19 ≈ 0.16
4) 9 + 0.16 = 9.16 (Calculator value is 9.16515139…)
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Rearrange real numbers in order from least to greatest
Note: Rewrite all in common terms (decimal is usually good and works with a calculator)
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#64. √0.42, 0.63, √4/3 [0.63, 0.64807, .667]
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#66. –1.46, 0.2, √2, -1/6 [-1.46, -0.166, 0.2, 1.41421] so…{-1.46, -1/6, 0.2, √2}
START PREPARING FOR THE TEST!