Download 1.7 Square Roots and Real Numbers

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Chapter 1. Real Numbers Operations
Review Answers
1.
2.
3.
4.
5.
1
(a) 101
2
(b) 8
(c) − 21
19
(d) 71
2
(e) − 2xy
z3
(a) 10
9
(b) 14
(c) 35
66
(d) 1
7x
(e) − 10
2y
(f) x
(g) 95
(h) 2
(i) − 44
x
At 48 square feet per pint I get less coverage.
Time = 16
9 hour(1 hr46 min40 sec)
mass = 20
21 kg
45
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1.7. Square Roots and Real Numbers
1.7 Square Roots and Real Numbers
Learning Objectives
•
•
•
•
•
Find square roots.
Approximate square roots.
Identify irrational numbers.
Classify real numbers.
Graph and order real numbers.
Find Square Roots
The square root of a number is a number which, when multiplied by itself gives the original number. In algebraic
terms, the square root of x is a number, b, such that b2 = x.
Note: There are two possibilities for a numerical value for b. The positive number that satisfies the equation b2 = x
is called the principal square root. Since (−b) · (−b) = +b2 = x, −b is also a valid solution.
√
√
The square root of a number, x, is written as x or sometimes as 2 x . For example, 22 = 4, so the square root of 4,
√
4 = ±2 .
Some numbers, like 4, have integer square roots. Numbers with integer square roots are called perfect squares. The
first five perfect squares (1, 4, 9, 16, 25) are shown below.
You can determine whether a number is a perfect square by looking at its prime factors. If every number in the factor
tree appears an even number of times, the number is a perfect square. Further, to find the square root of that number,
simply take one of each pair of factors and multiply them together.
Example 1
Find the principal square root of each of these perfect squares.
a) 121
b) 225
c) 324
d) 576
a) 121 = 11 × 11
Solution
√
b) 225 = (5 × 5) × (3 × 3)
46
121 = 11
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Chapter 1. Real Numbers Operations
Solution
√
225 = 5 × 3 = 15
c) 324 = (2 × 2) × (3 × 3) × (3 × 3)
Solution
√
324 = 2 × 3 × 3 = 18
d) 576 = (2 × 2) × (2 × 2) × (2 × 2) × (3 × 3)
Solution
√
576 = 2 × 2 × 2 × 3 = 24
Approximate Square Roots
When we have perfect squares, we can write an exact numerical solution for the principal square root. When we have
one or more unpaired primes in the factor tree of a number, however,
√ we√do not get
√ integer values for the square root
and we have seen that we leave a radical in the answer. Terms like 2, 3 and 7 (square roots of prime numbers)
cannot be written as rational numbers. That is to say, they cannot be expressed as the ratio of two integers. We call
them irrational numbers. In decimal form, they have an unending, seemingly random, string of numbers after the
decimal point.
√
√
To find approximate values for square roots, we use the
or x button on a calculator. When the number we are
finding the square root of is a perfect square, or the square of a rational number, we will get an exact answer. When
the number is a non-perfect square, the decimals will appear random and we will have an irrational number as our
answer. We call this an approximate answer. Even though we may have an answer to eight or nine decimal places,
it still represents an approximation of the real answer which has an infinite number of non-repeating decimals.
Example 4
Use a calculator to find the following square roots. Round your answer to three decimal places.
√
a) 99
√
b) 5
√
c) 0.5
√
d) 1.75
a) The calculator returns 9.949874371.
Solution
√
99 ≈ 9.950
b) The calculator returns 2.236067977.
Solution
√
5 ≈ 2.236
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1.7. Square Roots and Real Numbers
c) The calculator returns 0.7071067812 .
Solution
√
0.5 ≈ 0.707
d) The calculator returns 1.322875656.
Solution
√
1.75 ≈ 1.323
Identify Irrational Numbers
Any square root that cannot
√ be simplified to a form without a square root is irrational,
√ but not all square√roots are
irrational. For example, √ 49 = 7. This is a terminating decimal, which makes 49 is rational, but 50 does
50≈ 7.071067812. The fact that it is a non-terminating non-repeating decimal makes
not
√ simplify perfectly.
50irrational.
Example 5
Identify which of the following are rational numbers and which are irrational numbers.
a) 23.7
b) 2.8956
c) π
√
d) 6
¯
e) 3.27
7
. This is clearly a rational number.
a) 23.7 = 23 10
8956
b) 2.8956 = 2 10000
. Again, this is a rational number.
c) π = 3.141592654 . . . The decimals appear random, and from the definition of π we know they do not repeat. This
is an irrational number.
√
√
√
√
d) 6 = 2.44949489743 . . . Again
the
decimals
appear
to
be
random.
We
also
know
that
6
=
2
×
3. Square
√
roots of primes are irrational. 6 is an irrational number.
¯ = 3.2727272727 . . . Although these decimals are recurring they are certainly not unpredictable. This is a
e) 3.27
¯ = 36 )
rational number (in actual fact, 3.27
11
Classify Real Numbers
We can now see how real numbers fall into one of several categories.
If a real number can be expressed as a rational number, it falls into one of two categories. If the denominator of its
simplest form is one, then it is an integer. If not, it is a fraction (this term is used here to also include decimals, as
27
3.27 = 3 100
. Rational numbers will always be terminating decimals or non-terminating repeating decimals.
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Chapter 1. Real Numbers Operations
If the number cannot be expressed as the ratio of two integers (i.e. as a fraction), it is irrational. Irrational numbers
will always be non-terminating, non-repeating decimals.
Example 6
Classify the following real numbers.
a) 0
b) −1
c)
d)
e)
π
3
√
2
3
√
36
9
a) Solution
Zero is an integer.
b) Solution
−1 is an integer.
c) Although
π
3
is written as a fraction, the numerator (π) is irrational.
Solution
π
3
is an irrational number.
√
d) 3 2 cannot be simplified to remove the square root.
Solution
√
2
3 is an irrational number.
√
√
36
e) 9 can be simplified to 336 =
√
36 is a rational number.
9
6
9
= 23 Solution
Graph and Order Real Numbers
We have already talked about plotting integers on the number line. It gives a visual representation of which number
is bigger, smaller, etc. It would therefore be helpful to plot non-integer rational numbers (fractions) on the number
line also. There are two ways to graph rational numbers on the number line. You can convert them to a mixed
number (graphing is one of the few instances in algebra when mixed numbers are preferred to improper fractions),
or you can convert them to decimal form.
Example 7
Plot the following rational numbers on the number line.
a)
2
3
b) − 73
c)
17
3
d)
57
16
If we divide the intervals on the number line into the number on the denominator, we can look at the fraction’s
numerator to determine how many of these sub-intervals we need to include.
49
1.7. Square Roots and Real Numbers
a)
2
3
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falls between 0 and 1. We divide the interval into three units, and include two sub-intervals.
b) − 37 falls between 0 and −1. We divide the interval into seven units, and move left from zero by three sub-intervals.
2
c) 17
5 as a mixed number is 2 5 and falls between 3 and 4. We divide the interval into five units, and move over two
sub-intervals.
d)
57
16
9
as a mixed number is 3 16
and falls between 3 and 4. We need to make sixteen sub-divisions.
Example 8
Plot the following numbers, in the correct order, on a number line.
a) π
b)
22
7
c) 3.14
√
d) 10
We will use a calculator to find decimal expansions for each of these, and use a number line divided into 1000
sub-divisions. When we have two extremely close numbers, we will ensure that we place them in the correct order
by looking at the expansion to the 3rd decimal place and writing as a fraction of 1000.
142
a) π = 3.14159 . . . ≈ 3 1000
b)
22
7
143
= 3.14288 . . . ≈ 3 1000
140
c) 3.14 ≈ 3 1000
√
162
d) 10 = 3.16227 . . . ≈ 3 1000
Solution
Lesson Summary
• The square root of a number is a number which gives the original number
√ when multiplied by itself. In
2
algebraic terms, the square root of x is a number, b, such that b = x, or b = x
50
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Chapter 1. Real Numbers Operations
• There are two possibilities for a numerical value for b. A positive value called the principal square root and
a negative value (the opposite of the positive value).
• A perfect square is a number with an integer square root.
• Here are some mathematical properties of square roots.
√
√
√
a × b = ab
√
√
a÷ b =
√
√
√
A a × B b = AB ab
√
√
A
A a÷B b =
B
r
a
b
r
a
b
• Square roots of prime numbers are irrational numbers. They cannot be written as rational numbers (the ratio
of two integers). In decimal form, they have an unending, seemingly random, string of numbers after the
decimal point.
• Computing a square root on a calculator will produce an approximate solution since there are a finite number
of digits after the decimal point.
Review Questions
1. Find the following square roots exactly without using a calculator, giving your answer in the simplest form.
√
(a) √25
(b) √24
(c) √20
(d) √200
(e) r2000
1
(f)
r4
9
(g)
√4
(h) √0.16
(i) √0.1
(j) 0.01
2. Use a calculator to find the following square roots. Round to two decimal places.
√
(a) √13
(b) √99
(c) √123
(d) √2
(e) √2000
(f) √0.25
(g) √1.35
(h) √0.37
(i) √0.7
(j) 0.01
3. Classify the following numbers as an integer, a rational number or an irrational number.
√
(a) 0.25
51
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1.7. Square Roots and Real Numbers
√
(b) √1.35
(c) √20
(d) √25
(e) 100
4. Place the following numbers in numerical order, from lowest to highest.
√
6
2
61
50
√
1.5
16
13
5. Use the marked points on the number line and identify each proper fraction.
Review Answers
1.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
2.
3.
(j)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(a)
(b)
(c)
(d)
(e)
5√
2 √6
2 √5
10 √2
20 5
1
2
3
2
0.4
√
√1 or 1010
10
0.1
3.61
9.95
11.09
1.41
44.72
0.5
1.16
0.61
0.84
0.1
rational
irrational
irrational
integer
integer
4.
61
50
5.
52
(a)
(b)
(c)
(d)
2
3
9
5
9
4
34
9
√
6
2
16
13
√
1.6