FREE ESSAY ON LOOKS CAN BE DECEIVING

LOOKS CAN BE DECEIVING

Looks Can Be Deceiving
Paradoxes are sometimes composed of contradictory ideas presented together, ultimately
leading to an unworkable situation. Paradoxes, however, are not simply ambiguous
questions. Paradoxes are the essence of the inherent complexity of systems (Internet 1).
Each paradox must be analyzed and clearly understood before it can be explained. Since
mathematics is, in a sense, a universal language, certain paradoxes and contradictions
have arisen that have troubled mathematicians, dating from ancient times to the present.
Some are false paradoxes; that is, they do not present actual contradictions, and are
merely slick logic tricks. Others have shaken the very foundations of mathematics -
requiring brilliant, creative mathematical thinking to resolve. Others remain unresolved
to this day, but are assumed to be solvable. One recurring theme concerning paradoxes is
that each of them can be solved to some degree of satisfaction, but are never completely
conclusive. In other words, new answers will likely replace older ones, in an attempt to
solidify the answer and clarify the problem.
A paradox can be defined as an unacceptable conclusion derived by apparently acceptable
reasoning from apparently acceptable premises. This essay provides an introduction to a
range of paradoxes and their possible solutions. In addition, a questionnaire was
composed in order to demonstrate the extent of knowledge that the general population has
pertaining to paradoxes.
Paradoxes are useful things, despite their mind-boggling appearance. Generally, however,
most paradoxes can be "solved" by searching for specific properties that they may
contain. Therefore, if you try to describe a situation and you end up with a paradox
(contradictory outcome), it usually means that the theory is wrong, or the theory or the
definitions break down along the way. Also, it is possible that the situation cannot
possibly occur, or the question may simply be meaningless for some other reason. Any of
these possibilities are relevant, and if you exhaust all the possible interpretations,
one of them should prove to be incorrect (Internet 1).
The following type of paradox is called Simpson's Paradox. This paradox involves an
apparent contradiction, because when the data are presented one way, one particular
conclusion is inferred. However, when the same data are presented in another form, the
opposite conclusion results.
Paradox 1:
Acceptance Percentages for College A and College
Chart 1
Section A Section B 
Accepted Rejected Total Percent Accepted Accepted Rejected Total Percent Passing
Women 400 250 650 61% 50 300 350 14%
Men 50 25 75 67% 125 300 425 29%
Total 450 275 725 175 600 775 
As is evident in Chart 1, when the data are presented in two separate tables, it looks as
if men are accepted more often than women, because in each case (College A and College
B), men are accepted at a higher ratio than women. However, when the same data are
combined into one table (Chart 2), a contradicting result is implied.
Acceptance Percentage Totals for the University
Chart 2
Accepted Rejected Total Percent Accepted
Women 450 550 1000 45%
Men 175 325 500 35%
Total 625 875 1500 
This table shows women actually having a higher overall acceptance rate than men. This is
an example of Simpson's Paradox because it involves misleading data. Obviously, the
presentation of the data is very important, and can lead to incorrect assumptions if the
data are not used properly (Internet 2).
Paradox 2: 
An Arrow in Flight
One can imagine an arrow in flight, toward a target. For the arrow to reach the target,
the arrow must first travel half of the overall distance from the starting point to the
target. Next, the arrow must travel half of the remaining distance. For example, if the
starting distance was 10m, the arrow first travels 5m, then 2.5m.
If one extends this concept further, one can imagine the resulting distances getting
smaller and smaller. 
Will the arrow ever reach the target? (Internet 3) 
The answer is, of course, yes the arrow will reach the target. Our common sense tells us
so. But, mathematically, this fact can be proven because the sum of an infinite series
can be a finite number. The question contains a premise, which implies that the infinite
series will result in an infinite number. Thus, 1/2 + 1/4 + 1/8 + ... = 1 and the arrow
hits the target (Internet 3).
Paradox 3:
Two Equals One?
Assume that
a = b. (1)
Multiplying both sides by a,
a? = ab. (2)
Subtracting b? from both sides,
a? - b? = ab - b? . (3)
Factoring both sides,
(a + b)(a - b) = b(a - b). (4)
Dividing both sides by (a - b),
a + b = b. (5)
If now we let a = 1 = b, we conclude, from step (5), that 2 = 1. Or we can subtract b
from both sides and conclude that a, which can be taken as any number, must be equal to
zero. Or we can substitute b for a and conclude that any number is double itself. Our
result can thus be interpreted in a number of ways, all equally ridiculous.
The paradox arises from a disguised breach of the arithmetical prohibition on division by
zero, occurring at Step (5). Namely, since a = b, dividing both sides by (a - b) is
dividing by zero, which renders the equation meaningless. As Northrop goes on to show,
the same trick can be used to prove, for example, that any two unequal numbers are equal,
or that all positive whole numbers are equal (Internet 4).
Paradox 4:
Squares and Rectangles
The area of the square, shown above, is 8 x 8 = 64 units?. The square is cut in the four
parts A, B, C, and D, which are rearranged into the rectangle shown below. This
rectangle, however, has an area of 13 x 5 = 65 units?. 
This can lead to the potential of making 65 units? of gold out of only 64 units?. How can
you justify this transformation in area and the creation of matter? 
The picture of the rectangle is deceptive! The line XY shown in the picture of the
rectangle (see above) is not a line at all. The parts XU and VY have a gradient of 2 / 5
= 0.4, and the parts XV and UY have a gradient of 3 / 8 = 0.375. So, in fact, XUYV is a
parallelogram with an area of 1, not a line! 
Paradox 5:
Where Is The Missing Dollar?
Three people check into a hotel. They pay $30 to the manager and go to their room. The
manager remembers that the room rate is $25 and gives $5 to the bellboy to return. On the
way to the room, the bellboy reasons that $5 would be difficult to share among three
people so he pockets $2 and gives $1 to each person.
Now each person paid $10 and got back $1. So they each paid $9, totaling $27. The bellboy
has $2, bringing the total up to $29. Where is the missing $1?
The correct response to this question is that since all three people paid $9 each, we are
looking at a total of $27. The manager has $25 for the room while the bellboy has $2 for
himself. The bellboy's $2 should be added to the manager's $25 or subtracted from the
tenant's $27, not added to the tenant's $27.
The existence of a paradox is proof that either, at least one of the propositions are
false, or the logic used to arrive at the paradox is false, at which point you do not
really have a paradox. As stated previously, there really is no such thing as a paradox,
for its own existence proves that the assumptions it is based on are wrong. (Internet 5)
Searching For Answer's
A survey was composed in order to demonstrate the extent of comprehension that the
general public has in terms of paradoxes. Ten individuals, whom of which ranged from the
ages 16-42, answered the questionnaires. The survey consisted of five paradoxes that were
randomly chosen, each individual was given an opportunity to choose from one of three
responses (yes, no, or uncertain) for each paradox. The survey showed that 32% responded
yes, 16% responded no and 46% responded uncertain to the ten questions that were asked.
These results justify that the individuals, who answered yes to most of the questions,
were tricked by false propositions. These individuals ignored common sense and allowed
themselves to be deceived. Moreover, the majority of individuals who answered no to most
of the questions were aware that the paradoxes were somewhat misleading. However, they
were unable to explain any further. Also, the questions that were answered with an
uncertain apparently left the individuals pondering. 
Survey Results In Chart Form
Number of answers which fall in each category
Individual Yes No Uncertain
Person #1 4 0 1
Person #2 3 1 1
Person #3 3 1 1
Person #4 2 2 1
Person #5 1 2 2
Person #6 2 0 3
Person #7 3 0 2
Person #8 0 1 4
Person #9 0 0 5
Person #10 1 1 3
Total 19/50 8/50 23/50
Percentages 38% 16% 46%
Survey Results Represented On A Pie Graph
Conclusion
Paradoxes
Survey
Question 1:
Acceptance Percentages for College A and College B
College A College B 
Accepted Rejected Total Percent Accepted Accepted Rejected Total Percent Passing
Women 400 250 650 61% 50 300 350 14%
Men 50 25 75 67% 125 300 425 29%
Total 450 275 725 175 600 775 
Do the women have reason to claim sexual discrimination against the university?
a) Yes. Explain:
b) No. Explain:
c) Uncertain
Question 2:
An Arrow In Flight
One can imagine an arrow in flight, toward a target. For the arrow to reach the target,
the arrow must first travel half of the overall distance from the starting point to the
target. Next, the arrow must travel half of the remaining distance. For example, if the
starting distance was 10m, the arrow first travels 5m, then 2.5m.
If one extends this concept further, one can imagine the resulting distances getting
smaller and smaller. 
Will the arrow ever reach the target? (2)
a) Yes. Explain:
b) No. Explain:
c) Uncertain
Question 3:
Does Two Equal One?
Assume that
a = b. (1)
Multiplying both sides by a,
a? = ab. (2)
Subtracting b? from both sides,
a? - b? = ab - b?. (3)
Factoring both sides,
(a + b)(a - b) = b(a - b). (4)
Dividing both sides by (a - b),
a + b = b. (5)
Let a=1=b,
2=1.
Do you agree?
a) Yes. Explain:
b) No. Explain:
c) Uncertain.
Question 4:
Squares and Rectangles
The area of the square, shown above, is 8 x 8 = 64 units?. The square is cut in the four
parts A, B, C, and D, which are rearranged into the rectangle shown below. This
rectangle, however, has an area of 13 x 5 = 65 units?. 
This can lead to the potential of making 65 units? of gold out of only 64 units?. Is this
a valid transformation in area? 
a) Yes. Explain:
b) No. Explain:
c) Uncertain
Question 5
The Missing Dollar
Three people check into a hotel. They pay $30 to the manager and go to their room. The
manager remembers that the room rate is $25 and gives $5 to the bellboy to return. On the
way to the room, the bellboy reasons that $5 would be difficult to share among three
people so he pockets $2 and gives $1 to each person.
Now each person paid $10 and got back $1. So they each paid $9, totaling $27. The bellboy
has $2, bringing the total up to $29. Is a dollar missing?
a) Yes. Explain:
b) No. Explain:
c) Uncertain.
Work Cited
(Internet 1) http://www.colchsfe.ac.uk/