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11.1 Square Roots and Irrational
Numbers
Perfect Squares
• Area of a square
• Multiplying a number by itself
List of Perfect Squares
• 1,4,9,16,25,36,49,64,81,
100,121,144,169,196,225…
Review:
Square root is to find the side
lengths of a square given its area
. Square Root
An operation that yields a number which, when
multiplied by itself, produces the given number
16  4 , because 4 • 4  16.
2
If a  0 and a  b, then a is the square root of b.
Square Roots
Squares
02 0
00
2
1 1
1 1
42
22 4
32  9
93
2
4  16
16  4
Every positive number has two roots.
Positive Square Root
(Principal Square Root)
16  4
Consider the following:
x 2  100
x  10 or 10
Two solutions
Negative Square Root
 16  4
x  100
x  10
One solution
Simplify each root below and identify as a
positive root or a negative root.
1)
49  7
positive
root
5)  36  6 negative
root
2)  25  5 negative 6)
root
36  6 positive
root
3) 144  12 positive 7)  64  8 negative
root
root
4) 16  4
positive
root
8)
100
Not a
Real Root
Negative numbers do not have real roots.
Rational vs Irrational
Rational –Anything that can
be written as a fraction:
• Perfect Square
• Whole Number
• Negative Number
• Decimal that ends
• Decimal that repeats
• Fraction
• Improper fraction
• Mixed number
Irrational: Can not be
written as a fraction
• Square root that does not
have a perfect square
• A decimal that never ends
and never repeats
Math Humor
Perfect Squares Rational Roots
0 0
0
1 1
1
4 2
4
93
9
16
16  4
25  5
25
36  6
36
49  7
49
64  8
64
81 9
81
Irrational Roots
2  1.414...
3  1.732...
5  2.236...
6  2.449...
7  2.646...
8  2.828...
10  3.162...
11  3.317...
12  3.464...
13  3.606...
Identify each root as rational or irrational.
1) 10 irrational
2)
25 rational
6)
62 irrational
7) 81 rational
3) 15 irrational
8)  16 rational
4)  49 rational
9)
5)
50 irrational
99
irrational
10) 121 rational
Estimating Irrational Square
Roots
Step 1: Find the two perfect squares the
irrational root lies between. (hint use the list
of perfect squares: 1, 4, 9, 16, 25, 36, 49,
64, 81, 100..etc)
Determine what two consecutive integers each
irrational root lies between.
1) 7
4 7 9
lies between 2 and 3
2
2)
3
20 16  20  25
4
lies between 4 and 5
5
3) 85 81  85  100 lies between 9 and 10
9
4)
45
10
36  45  49
6
7
lies between 6 and 7
Estimating Irrational Square
Roots
• Step 2: Subtract the irrational root from the
perfect squares to determine which number
it is closest to.
Estimate each irrational root to the nearest integer
without using a calculator.
1) 8  3
4 9
2) 17  4
16  25
3)
24  5
16  25
5)
53  7
49  64
6)
65  8
64  81
7)
76  9
64  81
35  6
8) 140  12
25  36
121  144
4)
1)
Finding the Side of a Square
2)
A  49 un s
2
s
2
As
2
49  s
 49 or 49  s
A  40 un
What
2
As
would the
2
40  s
perimeter
be?? 40 or 40  s
7s
Distance can’t
be negative
2
40  s
1.) 28 un
2.)25.2 un
6.3  s
Homework
• Page 562 (2-38) even