1. Home
  2. Intro to logic
  3. Introduction
  4. 90 = 100: a proof

90 = 100: a proof

<< Chapter < Page
  Intro to logic   Page 1 / 1
Chapter >> Page >
A (purported) geometric proof that 90=100.

Construct a four-sided figure ABED as follows:

  • |ABE| = 90
  • |DEB| = 100
  • |AB| = |ED|
Using that as a starting point, we now tinker a bit to show that 90=100:
  • Draw the perpendicular bisectors to BE and AD; call the point where they meetC.
Actually, we must prove that those two perpendicular bisectors really do meet at all ( i.e. , that the point C even exists). In this case, it turns out to be pretty clearit's not hard to argue that lines AD and BE aren't parallel,and therefore their perpendicular bisectors aren't parallel, and so they must intersect (in Euclidean geometry).Still, be alert for people making glib assertions in proofs.

A construction to help prove that 90=100

Looking at this figure, some warning flags should be going up: How do we know C lies below BD? Might it lie above BD? Or exactly on BD?It turns out that the argument below is the same in all of these cases, though you'll certainly want to verify this to yourself later.

1 |AB| = |ED| By construction.
2 |BC| = |EC| C is on the perpendicular bisector of BE (thus△BEC is isosceles).
3 CBEBEC Base angles of isosceles triangle BEC are congruent.
4 |CBE| = |BEC| Congruent angles have equal measures; line 3.
5 |AC| = |DC| C is on the perpendicular bisector of AD (thus△ADC is isosceles).
6 △ABC△DEC (!!) Triangles with three congruent sides are congruent (Euclid's Side-Side-Side congruence theorem); lines 1,2,5.
7 (From here, it's just routine steps to conclude 90=100:)
8 ABCDEC Corresponding parts of congruent triangles are congruent; line 6.
9 |ABC| = |DEC| Congruent angles have equal measures; line 8.
10 |ABC| = |ABE| + |CBE| By construction.
11 |DEC| = |DEB| + |BEC| By construction.
12 |DEC| = |DEB| + |CBE| Substituting equals with equals; lines 11 and 4.
13 |ABC| = |DEB| + |CBE| Substituting equals with equals; lines 12 and 9.
14 |ABE| + |CBE| = |DEB| + |CBE| Substituting equals with equals; lines 13 and 10.
15 |ABE| = |DEB| Subtracting equals from equals remains equal.
16 90 = |DEB| By construction, and substituting equals with equals; line 15.
17 90 = 100 By construction, and substituting equals with equals; line 16.

A useful corollary: 0=1.

1 90 = 100 Previous theorem.
2 0 = 10 Subtracting equals (90) from equals remains equal.
3 0 = 1 Dividing equals by non-zero equals (10) remains equal.

If you feel this result is incorrect,then the challenge for you is to find the first line which is false.

The flaw is extremely hard to find. We won't actually give the solution, but here's a hint onhow to go about attacking the puzzle:

Note that finding the bug in the proof is the same skill as debugging a program.A good approach is to try various degenerate inputs. In this case, there are a couple ofinputsto the constructionthe length of CD is arbitrary; no matter how long or short the proof should apply equally well.Similarly, the angle 100seems arbitrary; fiddling with inputs like these (making them very small or very large)might give you some clues as to where the bug is. A very careful drawing will clear things up.

Got questions? Get instant answers now!

You may have noticed that the proof given here has some very minuscule steps e.g. Congruent angles have equal measure.Usually such simple steps can be omitted, since they are obvious to any reader. We include them for a few reasons:

  • As a careful thinker, you should recognize that such small steps really are part of the complete reasoning,even if they're not worth mentioning continually.
  • If a computer is checking a proof, it needs to actually include those steps.
  • Programmers do need to be concerned with distinctions about (abstract) typesthe difference between angles and their measures, in this case.
  • Sometimes a line's justification is glibly given asby construction, when that may not even be correct !-) .
In this course, we'll spend a few weeks working with proofs which do include all the small, pedantic steps, to instill a mental framework for what a rigorous proof is.But after that, you can relax your proofs to leave out such low-levelsteps, once you appreciate that they are being omitted.

Questions & Answers

prostaglandin and fever
Maha Reply
Discuss the differences between taste and flavor, including how other sensory inputs contribute to our  perception of flavor.
John Reply
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
which drugs can we use for ulcers
Ummi Reply
omeprazole
Kamara
what
Renee
what is this
Renee
is a drug
Kamara
of anti-ulcer
Kamara
Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
Nency Reply
Not really sure
Eli
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
what is anatomy
Oyindarmola Reply
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Oyerinde Reply
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Mohammed Reply
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
Wulku Reply
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
Ali Reply
function of digestive
Ali
the diagram of the lungs
Adaeze Reply
what is the normal body temperature
Diya Reply
37 degrees selcius
Xolo
37°c
Stephanie
please why 37 degree selcius normal temperature
Mark
36.5
Simon
37°c
Iyogho
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
37A c
Wulku
what is anaemia
Diya Reply
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
Diya Reply
how does Lysin attack pathogens
Diya
acid
Mary
I information on anatomy position and digestive system and there enzyme
Elisha Reply
anatomy of the female external genitalia
Muhammad Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Intro to logic. OpenStax CNX. Jan 29, 2008 Download for free at http://cnx.org/content/col10154/1.20
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Intro to logic' conversation and receive update notifications?

Ask