2-8: Square Roots and Real Numbers
OBJECTIVE:
You must be able to find a square root, classify numbers, and graph
solution of inequalities on number lines.
•square root - one of two equal factors of a number
A number that will multiply by itself to get another given number.
•perfect square - a rational number whose square root is a rational
number
For an example of these two terms, 9 * 9 = 81.
•9 is the square root of 81 since 9 times itself yields 81.
•81 is a perfect square since it is a rational number and its square root, 9,
is a rational number.
•Your calculators should have a square root key. It looks something like
this:
x
© William James Calhoun, 2001
2-8: Square Roots and Real Numbers
2.8.1 DEFINITION OF SQUARE ROOT
If x2 = y, then x is a square root of y.
•radical sign - the symbol for square root
There are three modes of square roots:
indicates the principal square root of 81.
81 = 9
81 indicates the negative square root of 81.
− 81 = −9 − 81 indicates both square roots of 81.
± 81 = ±9 ± 81 is read “plus or minus the square root of 81.”
81 without a calculator, you ask yourself, “What times
To find a square± root
itself will get me this number?”
What times itself gives you 16?
Answer: 4, so 16 = 4
© William James Calhoun, 2001
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2-8: Square Roots and Real Numbers
EXAMPLE 1: Find each square root.
A. 25
B. − 144
This represents the principal
square root of 25.
Since 52 = 25, you know the
answer is:
5
C. ± 0.16
This represents the negative
square root of 144.
Since 122 = 144, you know
the answer is:
-12
This represents both the
positive and negative square
roots of 144.
0.42 = 0.16, so:
-12
Remember: The easy way to answer these problems is use your
calculator to get the principal square root of the number, the put the sign
from the problem on your answer.
In fact, unless it is an easily-remembered perfect square (like 4, 16, 25,
144, etc.) you will need to use a calculator.
© William James Calhoun, 2001
2-8: Square Roots and Real Numbers
EXAMPLE 2: Use a calculator to evaluate each expression if x = 2401,
a = 147, and b = 78.
A. x
B. ± a + b
Replace x with 2401. = 2401
Replace a with 147
= ± 147 + 78
and b with 78.
Use calculator:
= 49
Combine like terms. = ± 225
Use calculator:
= ± 15
Now from square roots with nice rational answers
to
square roots and other numbers which can not be written as fractions.
© William James Calhoun, 2001
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2-8: Square Roots and Real Numbers
Question: What is the value of
2?
Your calculator should give you 1.412136…
Notice this decimal does not appear to terminate or repeat.
The decimal continues indefinitely without repeating.
This brings up some new options for our Number Sets.
Remember the chart and Venn diagrams from earlier:
Sets
Natural numbers
Whole numbers
Integers
Examples
1, 2, 3, 4, 5, …
0, 1, 2, 3, 4, …
…, -2, -1, 0, 1, 2, …
Whole Numbers
Integers
Natural
Numbers
Venn Diagram
Well, there is more, as evidenced on the next slide.
© William James Calhoun, 2001
2-8: Square Roots and Real Numbers
Sets
Natural numbers
Whole numbers
Integers
Rational Numbers
Examples
Symbol
1, 2, 3, 4, 5, …
N
0, 1, 2, 3, 4, …
W
…, -2, -1, 0, 1, 2, …
Z
any number that can be written as a fraction
Q
includes repeating and terminating decimals
Irrational Numbers numbers that cannot be written as fractions
I
non-repeating and non-terminating
Real Numbers
the set of all Rational and Irrational Numbers R
Integers
Whole Numbers
Irrationals
Natural Numbers
Real Numbers
Real Numbers
Rationals
© William James Calhoun, 2001
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2-8: Square Roots and Real Numbers
EXAMPLE 3: Name the set or sets of numbers to which each real
number belongs.
14
A. 0.833333333…
B. − 16
C. 2
D. 120
This is a repeating
decimal, so it is rational.
It is not an integer, whole
number, or natural
number.
The only answer is then:
Rational
This simplifies to:
-4
which can be written as a
fraction, so it is rational.
Also, -4 is one of the
integers.
The answer is:
Rational
Integer
This simplifies to:
7
which can be written as a
fraction, so it is rational.
Also, 7 is an integer, a
whole number, and a
natural number.
The answer is:
Plus this into a
calculator.
The result is:
10.95445115…
which is non-repeating
and non-terminating, so
there can be only one
answer:
Rational
Integer
Whole
Natural
Irrational
© William James Calhoun, 2001
2-8: Square Roots and Real Numbers
EXAMPLE 3: The area of a square is 235 square inches. Find its
perimeter to the nearest hundredth.
First find the length of each side.
Area of a square = (side)2.
So, side = square root of Area.
235in2
s= A
One side is found by plugging in for A.
Remember the perimeter of a square
has the formula:
P = 4s
We will simplify this with the answer:
s= A
s = 325
s = 18.02775638
P = 4s
P = 4(18.02775638)
P = 72.111026551
The perimeter is about 72.11 inches.
Now we switch gears and do some graphing of inequalities.
© William James Calhoun, 2001
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2-8: Square Roots and Real Numbers
Rules for graphing inequalities on a number line:
1) Use the initial rules for graphing points on a number line from
Section 2.1.
2) For ≠, > and <, we use an open circle to signify the point is not
included.
3) For =, > and <, we use a closed circle to signify the point is included.
4) Greater than has an arrow to the right. Less than has an arrow to the
left. Not equal goes in both directions.
EXAMPLE 5: Graph each solution set.
A. y > -7
B. p ≠ 3/4
-7
> so full circle
greater than so to right
3/
4
not equal so open circle
not equal so both directions
© William James Calhoun, 2001
2-8: Square Roots and Real Numbers
HOMEWORK
Page 123
#21 - 59 odd
© William James Calhoun, 2001
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