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MTH 208_ Wks 1_5 DQs

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1. Draw the real line and label from about –5 to 5 in increments of 1. There are an infinite
amount of integers on the real line. Based on your sketch you’ll also agree with me that there are
twice as many total integers as there are positive integers, right? But how many positive integers
are there? That’s it… an infinite amount. So there are infinite amounts of integers both on the
entire real line and on each half of the real line. Are you scratching your head yet?
Now consider the real numbers. In fact, lets narrow our focus way down to just the
part of the real line between 0 and 1. Circle that part of the real line. How many real
numbers are there in between 0 and 1? What can you conclude from all this?
Based on the question asking for real numbers and not whole numbers, one could
obviously state that there would be an infinite amount of numbers between 0 and 1. The
division of a number into smaller parts is, in theory, an infinite process. The real world
contains the actuality to do so. However the human mind will only go so far, thereby
making this process seem impossible. It is natural for most humans to look at this
question and say there are X amount of numbers between 0 and 1.
I'm not sure if you would consider this relative to the DQ, but infinite numbers were
made concrete in the mathematical and scientific world by Georg Cantor with his theory
that a group of numbers is infinite when a part of that group is larger than the whole?
For example = we count 1,2,3,4,5.... but if you were to count by even numbers, those
numbers would still match on a 1 for 1 basis....
1 - 2 - 3 - 4 - 5 - 6 - n
2 - 4 - 6 - 8 - 10-12- n2
2. Given the following numbers: 8.5, π, 0, -1, -6.75, 32/7, (-2)
2
, -4/3
Place the numbers above in their relative positions on a real number line from -10 to
10.
Also, consider the following words: real, rational, irrational, integer, counting (or
natural). For each of the numbers above, write all of the above words that apply to
that number
(Hint: Check out the graphic on page 5 of the text)
-6.75,-4/3,-1,0,π,(-2)
2
,32/7,8.5
-6.75 is a rational number

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-4/3 is a rational number
-1, 0, and (-2)
2
, which is actually 4, are counting, rational, integer, and whole
π is an irrational number
32/7 is a rational number
8.5 is a rational number
3. Evaluate the following expressions:
a. -13(5 + -7)^2– 6
+
5
3
10
b. 1 – (1 – (1 – (1 – 2))) Hint: start from the inside and work out
c. ((4
2
+ 3)
2
– 5) – (1/2 + 2/3)/6
A)
-13 (5+-7)^2-6
10 +5^3
-13 (-2)^2-6
10 +5^3
-13 * 4 -6
10 +5^3
-52 - 6
10 +5^3
-58
10 + 5^3
-5.8 + 125
ANSWER = 119.2
B) 1 – (1 – (1 – (1 – 2)))
1-(1-(1-(-1)))
1-(1-(1+1))
1-(1-2)
1-(-1)
1+1
ANSWER = 2
C) ((4
2
+ 3)
2
– 5) – (1/2 + 2/3)/6
((16+3)^2-5)-(3/6 + 4/6)/6

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