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Math 75 Notes
Math 75 Notes
Figure 2-2
5
2-1
Real Numbers
Did all civilizations use zero?
Have they all used negative numbers? We don’t always
use all the kinds of numbers available to us.
of the numbers explored in this text.
Here you will learn to classify some
As you read the following terms refer to figure
2-1 and figure 2-2.
Natural numbers:
Your three year old can count to two.
one for each hand.
He asks for two cookies,
Natural numbers or counting numbers are the first kind of numbers
we learn.
Whole numbers:
One day your child asks, “How many cookies may I have?” and she
learns about another number, zero.
whole numbers.
The natural numbers and zero together make the
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Integers:
When you got your first checkbook, you learned first hand about another
kind of number, $-25, -3 etc. Whole numbers and negative numbers that are the opposite
of natural numbers are integers.
Rational
numbers:
Noticehttp://www.wenkuxiazai.com/doc/df3c0a8f6529647d27285238.html on the number
line (figure 2-2) that if integers were the only numbers possible there would be many
gaps.
Rarely does a cash receipt or checkbook balance come to an integer amount.
3.5 and 2 ? are examples of rational numbers.
of two integers it is rational.
If a number can be written as the ratio
5 can be written as
51
and 0.333… can be written as
13
so 5 and
0.333… are rational. Notice the rational numbers include the integers which include
the whole numbers which include the counting numbers. The gaps in the real number
line are now partially filled in.
Irrational numbers:
integers.
These are numbers that cannot be written as a ratio of two
These are numbers that have an infinite non-repeating decimal expansion.
2, e, and ? (pi) are examples of irrational numbers.
Real numbers:
All together the rational and irrational numbers comprise the real
numbers. Hhttp://www.wenkuxiazai.com/doc/df3c0a8f6529647d27285238.htmlere is a
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thought: If these are called real numbers, what does that suggest to you?
Practice: Identify the following numbers as Natural - ?, Whole- W , Integer-Z ,
Rational - Q, Irrational-? and Real-?
a) 0.25
b) 5
8.252525…
0 -5
c) ? Negative 5 degrees
balance of above zero
$23
5000 feet stocks d)
above sea down 1/4
level
200 feet tax rebate e) 9
below sea of $200
level
6
-1/2
Most will have more than one classification.
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2-1-2 Absolute Value
a
3
is read “the absolute value of three.”
?3 is read “the absolute value of
negative three.”
3 asks how far away from zero is three.” ?3 asks how far away from zero is negative
three.”
The answer to an absolute value question is never a negative number. What is5?
http://www.wenkuxiazai.com/doc/df3c0a8f6529647d27285238.htmlFive is five units
form zero so 5=5
What is?5?
Negative five is five units form zero so ?5=5
Practice: Simplify the following.
a)
|398|=
b)
|-2/3|=
|-398|=
|5 2/3|=
|-3.14|=
|-2.35|=
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c)
|245-34|=
|-98|=
|-?|=
Hint: Work the subtraction first.
d)
|-7|=
|-8|=
e)
|12|=
|-2.6|
f)
|2 1
/3|=
h)
|-3 1/3|=
|?|=
|-e|
|3.682|=
g)
|4 1/2|=
|-6|=
|3|=
|-2|=
|-6.42|= i)
j)
|43 - 35|=
|624 - 34|=
|324 - 58|=
7
|789 - 234|=
|5000 - 2458|=
|3210 - 2987|=
|4.642|=
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2-1-3 Square Roots a
9?3
25?5
because 3 times 3 is 9
because 5 times 5 is 25
Check a root by multiplying the answer by itself.
When working a squarehttp://www.wenkuxiazai.com/doc/df3c0a8f6529647d27285238.html
root problem, ask:
“What times itself is the number inside the root symbol?”
6.25?2.5
Check this by multiplying 2.5x2.5 to get 6.25
Practice: Simplify the following.
Do these in your head: ask yourself what times itself is the number inside the root
symbol.
a) b) c) d)
4= 49= 25=
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49
36= = 81=
= = =
9= 144= 1=
4981
=
Hint: do the numerator and denominator separately.
Do these on a calculator.
e) f)
= 79=
= =
= 30=
= =
Round to the nearest hundredth.
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What would you guess the square root is, without using a calculator, for the
following?
?means approximately equal to. Try one at a time and then see how close
you were with your guess.
Example:8 would be between 4 and 9.
The answer is between 2 and 3. 8 is
clhttp://www.wenkuxiazai.com/doc/df3c0a8f6529647d27285238.htmloser to 9 so the
guess should be closer to 3.
Maybe 2.9?
How close is this to the calculator answer
of 2.83?
If you are in a store and need an approximation to a square root that doesn’t need
to be to exact, you can guess close. An educated guess is called an estimate.
g) h) i) j)
6? 82? 24?
?
48? ? 39?
?
?
?
? ? ?
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2? ?
?
61?
? ?
?
k)
?
?
?
?
8
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Review Topic 1: Place Value
The number in the chart above is read “fifty-one and three hundred forty-five
thousandths.” Read the number before the decimal.
Read the number following the decimal.
Say “and.” for the decimal.
Add the word for the place value of the last
digit.
0.04 is read 4 hundredths.
://www.wenkuxiazai.com/doc/df3c0a8f6529647d27285238.htmlar3.00013 is read "three
and thirteen hundred thousandths."
"Fourteen and five hundred twelve millionths" must end on the millionths place,
14.000512
Practice: Write the following in words. Use the chart above to write in the number
to help you find the word that ends number.
12.00017
1.0041
0.0000074
c)
1.0101
a) 2.30007
123.321
0.034
21.000034
b)
0.12345
In money, $23.45, the 2 is worth $20 and the 4 is worth 4 tenths of a dollar.
In 0.02345 the 3 represents 3/1000 or three thousandths. The 5 represents 5/100000
or five hundred thousandths.
Practice: State the value of the specified digit in the following number 23.4516879
4
6
5
2
7
9
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9
10
We have many ways to say the same amount. The form depends on how you are using the
number.
e)
12
42/56= f)
Review
5 25/35=
Topic
Fractionalhttp://www.wenkuxiazai.com/doc/df3c0a8f6529647d27285238.html
Conversions
1 7/28=
6
2
90/135= 4 12/16=
/8=
8 8/64=
14
/21= 1 9/54=
2:
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24
1 10/15= /90=
45
8 /60=
7281
21
g)
3 24/72= 5 2/8= /28=
100
h)
5 5/10= 1 56/144= /1000=
1080432
i) 180/240= /1215= /720=
23012000
j) 200/300= /340= /24000=
Practice: Raise the following to the indicated denominator.
1
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k)
2/3 to 12ths /2 to 12ths ? to 12ths
23
l)
3/5 to 35ths /3 to 24ths /8 to 24ths
/56= /81=
567
/1008= /5600=
150
/270= /480=
4800320
? to 12ths
55
2
/5 to 10ths
/12 to 24ths /6
? to 24ths
to 18ths
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5
m)
5/6 to 12ths
n) 5/12 to 24
ths
o) 3 2/5 to 10ths
p) 7 1/10 thttp://www.wenkuxiazai.com/doc/df3c0a8f6529647d27285238.htmlo 100ths
57
/6
to 24ths /12 to 36ths /8 to 56ths
573
/6 to 30ths /6 to 48ths
/12 to 48ths ? to 48ths
32
? to 48ths
35
3
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/8 to 144ths /8 to 16ths /8 to 24ths
5 2/3 to 144ths 1 8/15 to 45ths /25 to 100ths /7 to 105ths
11
Fraction to Decimal
Remember the fraction bar means divide. To change 5
34
to a decimal, save the whole number “5” and
0.75
3
work only with the ?.
Divide 3 by 4. 43.00 Then put the five back in position. 5=5.75.
4
Practice: Change the following to decimal numbers. Round to the thousandths place.
5
a) 2/3
b)
?
4 1/8
?
/4
3 3/8
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2
c) 1/3
/5
2 3/10
32
d) 5/12
/16
17e) /4
5 5/8
439f) /3
/5
/7
/8
/8
://www.wenkuxiazai.com/doc/df3c0a8f6529647d27285238.htmlpar364g) /25
/7
/5
Decimal to Fraction
Read the number.
Then write the number using the place value and reduce.
Example: 3.45 Save the 3. Read .45 as forty-five hundredths. Write The result is 3
9/20.
Practice: Change the following to fraction numbers. a)
0.4
2.01
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b) 3.05
c) 0.3
0.25
4.1
d) 0.92
2.025
e) 1.0012
3.375
f) 0.875
0.625
g) 2.005
1.0002
45100
. Reduce to
920
.
3.75 7.8 8.005 0.375 25.25 0.3 5.02
12
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Percent -- Cent means 100.
Per is divide or out of.
PerCent - how many out of
100.
Percent to Decimal or Fraction
To change a percent to either a decimal
or fraction, divide by 100.
If a decimal is required, use decimal division.
If a fraction is
http://www.wenkuxiazai.com/doc/df3c0a8f6529647d27285238.htmlrequired use fraction
division.
Practice: Change the following to decimal numbers. a) 20%
15%
5%
c) 0.5%
0.2%
d) 120%
350%
e) 0.02%
0.05%
f)
0.008%
2.5%
Practice: Change the following to fraction numbers. g) 20%
15%
b) 7%
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h) 7%
5%
i) 0.5%
0.2%
j) 120%
350%
l)
k) 0.02%
0.008%
25%
8%
0.75%
100%
0.125%
34%
3%
0.75%
100%
2.5%
0.3%
3%
?%
1/20%
415%
1.25%
0.4%
25%
8%
0.3%
415%
1 ? %
0.4%
34%
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0.125%
?%
13
Decimal or Fraction to Percent
To change either a decimal or fraction to a percent, multiply by 100.
If a decimal is given, use decimal multiplication.
If a fraction is given, use
fraction multiplication.
Practice:
Change
the
followihttp://www.wenkuxiazai.com/doc/df3c0a8f6529647d27285238.htmlng to percent
numbers. a) 0.41
c) 3.05
d) 0.3
e) 0.92
54
0.25
4.1
0.21
0.51
7.8
0.35
8.005
0.72
2.025
0.375
0.234
0.25 b)
0.4
2.01
3.75
0.08
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f) 2/3
4 1/8
/4 /5
6
g)
?
?
3 3/8 /7
23
h) 1/3
/5
2 3/10 /25
329
i) 5/12
173j) /4
/16
/7 /8
5 5/8
/8 /5
3
The shaded area is2.
It is also 13/5.
5
Think about how you would solve the Think about how you would solve the following
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problem: I have 135 pencils in boxes following problem: I have 5 crates of apples
that hold 12 each. How many full boxes and how and one crate with 7 apples left.
Each full crate many in the part box? holds 30 apples. How many apples in all?
Improper to Mixed Mixed to Improper 135/12 is an improper fraction.
Multiply the
whohttp://www.wenkuxiazai.com/doc/df3c0a8f6529647d27285238.htmlle number by the
denominator, The numerator is larger or equal to the then add the numerator.
is the new denominator. numerator.
The denominator remains the same. Divide to
change it to a mixed number.
11R375?30?7157
315??
12135 yields 11?11 303030
124
The fraction bar means divide.
14
Practice: Change the following to mixed numbers. Reduce as necessary.
a)
51/3 4/1 52/3
b) 14/8 28/5 29/
This
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8 c) 72/12 41/8 56/12
d) 9
/2 12/8 634/12 e) 45/2 45/8 63/
12 f) 78/2 1002/81 634/12
g) 91/ 3 188/5 934/21
Practice: Change the following to improper fractions. h) 1 ?
2 2/3
8 3/4 i) 10 2/3 4 2/5 3 1/8
j) 3 2/5 7 1/8 1 1/8
k) 3 1/5 65 2/7 6 7/
8 l) 33 1/3 66 2/3 16 2/3
m) 6 1 /8 4 3/8 10 5/8
14
/3 53
/5 17://www.wenkuxiazai.com/doc/df3c0a8f6529647d27285238.html
/15 523
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/5 523
/15 553
/24 93
/4 5 4/5 2 3/8 1 3/8 7 1/4 37 1/2 2 7/15 14
/5 32
/3 3
/2 27
/3 207
/3
27
/13 87
/4
1 1/5 21 1/3 100 2/3 8 4/5 5 4/9 1 4/7
15
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16
Think about a 50% off sale.
as easy.
The arithmetic is easy.
Just divide by 2. 25% is just
Divide by 4. For 75%, divide by 4 then multiply by 3.
10% is found by
dividing by 10 which for decimals in your head is just moving the decimal to the left
one space.
17
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