Sunday, 12 April 2015

Tuesday, 7 April 2015

Monday, 1 December 2014

SOME FALLACIOUS MATH PROOFS



 Math has a lot of proofs. Some of these proofs prove statements that we all know cannot be true. When a fallacious statement is proven in math, then there has been a violation in the process.
We should note well, some of these violations so that when they appear in a proof, we can easily point them out as violations:
Ø  Dividing a number by zero: Dividing a number by zero is undefined and hence is a violation.
Ø  Any number multiplied by zero yields zero ( spencer 1998)
Ø  If a2 /b2 = c2 /d2  then a/b = ± c/d .This means that either the positive is true or the negative is true and so one should not always be quick to pick the positive as the answer.
Let us now proceed as more violations will be noticed in the process.
PROVE THAT 1 + 1 = 0
Step 1: a = 1
  Step 2:  b = 1
Step 3: a = b
Step 4: a2 = b2
Step 5: a2 -  b2 =o this becomes (a – b)(a + b) = 0
Step 6: (a – b)(a + b) /(a – b) = 0/(a – b)
Step 7: 1(a + b) = 0
Step 8: a + b = 0
Step 9: 1 + 1 = 0 (this is because of step 1 and 2) and this end the proof that 1 + 1 = 0
Why is this a false  proof?
In step 6, (a – b)(a + b) was divided by (a – b). (a – b) is 0 because a = b. The division by (a – b) in step 6 is the same as dividing by zero which is undefined. In the same step 6, (a – b)(a + b) is 0 because (a – b) is 0. Any number multiplied by zero yields zero (Spencer, 1998).






PROVE THAT 2 = 1
Step 1: a2 = ab( let a = b)
Step 2: a2 – b2 = ab – b2 ( that is subtract b2 from both sides)
Step 3: (a – b)(a + b)/(a – b) = b(a – b)/(a – b) ( that is dividing both sides by (a – b)
Note: ab – b2 = b(a – b) (that is, b factorized out)
Step 4: (a + b) = b
Step 5: But a = b
Step 6: b + b = b
Step 7: 2b = b
Step 8: 2b/b = b/b
Step 9: 2 = 1
Which step makes this proof a fallacy?
We should note that a = b.
 In step  2, a2 – b2 = 0 , ab – b2 = 0 and so 0 = 0 which means there is nothing to prove.
In step three ( a – b) = 0. (a – b)(a + b) = 0,
(a – b)(a + b)/(a – b) = 0/0 which is undefined.
In the same step, b(a –b)/(a – b) becomes b(0)/0 which is also undefined. Upon all these “violations”, the proof still continued and that makes it a fallacious proof.

 TO PROVE THAT 1 = 0
1.       Let x = 1
2.       Multiply both sides by x
          x2 = x
3.       Subtract 1 from both sides
x2 -1 = x – 1
4.       Divide both sides by x – 1
(x2 – 1 )/(x – 1) = 1
5.       Simplify
(x – 1)(x + 1)/(x – 1) = 1 note that x2 – 1 = (x – 1)(x + 1)
 x + 1 = 1
6.       Subtract 1 from both sides
X = 0   
Substitute the value of x from step 1
1 = 0
The fallacy here is subtle in step 2,
Multiplying both sides by x introduces an extraneous solution to the equation of x = 0. Then in step 4, there is division by x – 1 which is an illegal operation because x – 1 = 0 and you cannot divide by zero.
SEE THE FOLLOWING:
1.                y = 100, z = 0
               x = (y + z)/2
              2x = y + z
              2x(y –z) = (y + z)(y –z)
              2xy – 2x2 = y2 – z2
              Z2 = 2xz = y2 – 2xy
             z2 – 2xz + x2 = y2 – 2xy + x2
             (z – x)2 = (y – x)2
              Z – x = y – x
              Z = y
             0 = 100
            Shaun; (December, 2008)
Then 1 + 1 = 3 becomes x + x = 3
2x = 3
2x/2 = 3/2
X = 1.5
Substituting into the expression we get 3 which imply that 1 + 1 = 3 as required.
Are the above two fallacious proofs? If they are, point out the wrong steps and explain why they are are wrong. You ideas will be appreciated.

Wednesday, 26 November 2014

REVIEW OF THE BOOK “EASY WAYS FOR PERFORMING ARITHMETIC COMPUTATIONS”



EASY WAYS FOR PERFORMING ARITHMETIC COMPUTATIONS” is a book written by Bagiliko John, an undergraduate student of the University for Development Studies (UDS), offering BSc. Pure Mathematics. He is currently in Level 200. He is currently researching on why most people do not like mathematics. He believes that through his effort, the depending perception of most people that mathematics or arithmetic is difficult will be eradicated.
In the book, he presents easy ways, more or less simple tricks that can be used in performing some arithmetic computations. Why his choice of this topic? He aims at proving to people that arithmetic computations can be done in a very easy and interesting way, contrary to what they think about it. What is his definition for arithmetic? He defines arithmetic as the basic tool to mathematics. He gives a scenario in the introduction; when a baby is born, a point is reached where it learns to walk. It does this by first crawling. This crawling he says can be likened to arithmetic. The walking itself is mathematics.
The book is presented in a way that will arouse the interest of the reader to learn arithmetic. For example, 11 × 35 can be computed by placing the 3 and 5 as 3….5. The 3 and 5 are again added and placed in between 3….5. This becomes 385. This is very amazing and confirms that arithmetic computation can be very interesting. In the book, there are many of these which will help the reader to know how to compute similar ones. The book also presents to the reader various simple tricks to be used in computing many arithmetic questions.
Someone might have concluded that then the book is for basic school children. No, the book is for everybody, whether a worker or a student of any level. Arithmetic is actually needed in our daily activities. Sometimes we go to buy many things in a shop at a go. With strong knowledge of arithmetic computations, we will be able to compute the total amount to be paid within some few seconds. This book explains to the reader how to do that without thinking there is magic involved.
What will the reader be able to do after reading this book? The author presents the book such that after the reader has gone through it, he or she should be able to perform the following:
               I.            Multiply a double digit number by eleven.
             II.            Multiply a number by 10, 100, 1000 and so on.
           III.            Multiply a number by 5
           IV.            Multiply two numbers ranging from 10 to 19
             V.            Find the square of any number ending with a 5.
           VI.            Divide a number by 10, 100, 1000 and so on.
         VII.            Divide a number by 5
       VIII.            Multiply numbers ranging from 1 to 10 by 9 using an amazing fact.
These are all things the reader will be able to do without the use of a calculator. The reader is also expected to practice these things well to be able give out answers to any of those computations within a few seconds. The book is actually a must read since we all need arithmetic in our daily activities.
The book appears to concentrate on multiplication and division for now. More will come from the author as time goes on. He will revise this current book by adding more to the content.

WHY DO SO MANY PEOPLE HATE MATH?



I hate being the first to tell you that mathematics has never been liked by many people since the beginning of time. I hope I am not the first to tell you this? Why have many people out of the many subjects thought in schools just decided to give hatred to math? I used to hate math at the basic school. Now math is the subject I like best. I think that is the reason why I am offering BSc. Pure Mathematics at the University. I have always been trying hard to find an answer to this question and have ended up having more than one answer. I am convinced I am not the first person to take up this headache upon myself. 

Math is poorly taught in schools   
Math is poorly taught in schools of all levels. Some teachers read out what they have also read to students. Sometimes without any explanation, probably they do not also understand. If that is the case, then I will say that is hypocrisy on their part to pretend to know what they not know. Gordon Barker on March 7, 2013, said, most math teachers should be put against a wall and shot. He argued that math is taught badly at all levels (including first Year University) with no connection to real world. Not a quote though. This really kills the desire of most students to pursue math in higher levels and also increasing their hatred for it.  

Passing of antipathy on to kids by parents and role models
It is sad to know that the perception of most people that math is difficult is as a result of antipathy passed on to them by parents or row models. A woman told the daughter who failed a math paper in the open air, not to worry at all since math is difficult and that she herself wasn’t good at it when she was in school. This woman indirectly has killed the daughters’ zeal that she might have had to learn math. So it is evidently clear that no matter the effort of the teacher, this girl cannot make it in math. She will always be consoled by the fact that her mother was not also good at it so what’s the big deal. There was a case, back at the primary school, where my math teacher was solving a math problem and made a mistake along the line and when it was pointed out to him, he simply said “adwen no nnye mma maths.” This is Asante Twi, meaning the brain is not good for math. We laughed and were consoled that after all, our math teacher’s brain is not also good for doing math, how much more we children. 

Absence of constant practice
Math and constant practice are synonymous. Due to laziness or other reasons for people to constantly practice math, they end up forgetting the basic things in math that are must-know to the problem to be solved. I think I now know why I did not like math at the basic school. In the Senior High School, I started learning math regularly and I made sure I was always ahead of my math teacher. Within some few months in the Senior High School, my interest in math started increasing and I eventually became one of the best students in math in my class. Now I think working math questions is my hobby. So most people do not practice constantly and so forget the basic concepts in math and end up hating it because they lack the basic concepts to solve any math problem. An example, to compute 64 ÷ 8, one must know the times table for 8. That is, one must know that, 8 × 8 is 64. 

Poor preparation on the part of some teachers before going to teach a lesson
It sometimes baffles my mind why some teachers enter the class to teach without doing any good preparation or having a well prepared lesson notes by them. This constant practice in math applies to everyone, whether a teacher or a student of any level. Some teachers go to the classroom having done barely any preparation. They end up in the class not being able to solve a particular math problem. When the students become fully convinced that their teacher could not solve a math problem, they also relent in their effort to learn math since even their teacher could not also do so.

Lack of good fundamental preparation
Luck of good fundamental preparation is factor to most people hating math. Most people do not like mathematics because they are not good at it. They are not good at it because they might not have the basic preparation. As stated earlier, people begin to hate math if they do not have knowledge on the basic concepts of math. Children are not thought basic concepts in math as the way they are thought poems. If basic math concepts were made known to basic school children as poems in the basic schools, the issue of someone growing to hate math would reduce. For instance, I do not know the number of days in the months of the year off head. I can only tell you the number of days in a particular month of the year by reciting a poem in my head. There are some things I do remember now only by singing a song or by reciting a poem. Math has not been taught that way. I wish I could come out with a song or a poem containing most of the basic concepts in math which would be taught in basic schools to children, so that as they go physically, they also grow mentally with math concepts. I am sure someone is good at writing songs or writing poems and is reading this article of mine. This person I hope will be touched and will come out with a very nice song or poem that will make children like math. 

Is math a language? 
 This I came across, in a comment by someone to an article. This is from John Giano, March 8, 2013, “I think you hit the nail on the head. Math is a language. It is a set of symbols and concepts that relate and work together. Why everyone hate’s math is probably because the language is not spoken from birth and its rarely spoken in a dialect that people can understand. For most people, it will be like trying to learn Japanese; reading it as a native English speaker, and having the instructor pronouncing with a Russian accent.” A Nigerian comedian also joked about this issue of math in one of his comedies using a scenario; a teacher giving an expression like 2x +3 = 4 and asking the students to find x when x is already in the expression, not a quote though. His argument is valid. The math language used is not understood by many and is barely made to be understood. 
  
No sense of application of math
Math as thought in most cases is not pointed out what the application of that topic in real life situations is. What math seeks to do is prepare the individual and providing him with the basic concepts to be applied in the next difficult topics. Math for example, prepares an individual in pre-calculus for calculus. Maths is thought as if we were going to be students forever without having to apply the concepts in normal life activities. I remember I once asked my math teacher what the importance of construction was. I was given a bogus answer. He only told me, “You will know its importance as you go higher.” Where is this higher everyone talk about? 

Who likes to told openly that he’s wrong
If you are the type who hates to be told the simple truth that you are wrong, then, you will think math is not your field. Math has a unique answer to a particular problem and you cannot have two different answers to the same math problem. All these answers are expected to be proven, some in a somewhat simple way, some too, very abstract. In any given math problem, one is expected to follow some specific procedures in coming up with that answer, if not, wrong!! Math is unlike those other subjects where you may be asked a question and you are expected to share your ideas. Your ideas no matter what, cannot be totally marked as wrong.
This math is issue is a problem of the whole universe. Your comments to this article will be very necessary for coming out with a solution to it. There is this video; “Khan academic video” that I will like people to watch in order to build their desire for math. I also recommend the article titled, “Six Easy Ways of Performing Arithmetic Computations” and also a book titled “EASY WAYS FOR PERFORMING ARITHMETIC COMPUTATIONS BY: BAGILIKO JOHN”. They seek to prove to readers how computations in arithmetic or math can be made so easily and interesting.