Find the Fallacies

Here are some examples of false Mathematical proofs:
Number 1:

Number 2:

Number 3 (the famous one):
let:

Number 4:
(1-1) + (1-1) + … + (1-1) + (1-1) + … = 0
1 + (-1+1) + (-1+1) + … + (-1+1) + … = 0
1 + 0 + 0 +… + 0 + .. = 0
1 = 0

 

A Simple Exercise in Laplace Transform

Prove that:

Solution:
First,

It follows that:

Then,

Hence,

This 2009!

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The number 5040 – according to Plato

5 040 – Greek philosopher Plato gave this number as the population of an ideal city.

This number has several factors, which implies an efficient division of population for the sake of land distribution, taxes, war etc.
In fact,
5, 040 = 7! = (1)(2)(3)(4)(5)(6)(7)
5, 040 = (24)(32)(51)(71)
Thus, 5 040 has (4+1)(2+1)(1+1)(1+1) = 60 factors

A Simple Technique #1

Compute in (less than) 30 seconds without using calculator.
My technique:
Since 2007 is the arithmetic mean of the numbers 2004, 2006, 2008 and 2010, then: Let x = 2007
So, x-3 = 2004, x-1 = 2006, x+1 = 2008, x+3 = 2010
Hence, the expression is equal to
Simplifying,
= x2 – 5 = (2007)2 – [...]

Value of a in ax+b

When the polynomial x2007 + 1 is divided by x2 – 3x +2, the remainder can be expressed in the form ax+b. Find the value of a.
My Solution:
Let k be the quotient, then:
x2007 + 1 = k(x2 – 3x +2) + (ax+b)
Since x2 – 3x +2 = (x-2)(x-1),
x2007 + 1 = k(x-2)(x-1) + (ax+b)
To omit [...]

Number of Perfect Squares

Compute the number of squares between 49 and 94
My short solution: The technique is to express these two numbers into perfect squares so that it’s easy to compute the number of squares between these two numbers.
49 = (22)9 = (29)2 = 5122 ;
94 = (32)4 = (34)2 = 812
Therefore, the number of squares between 49 [...]

25th number

Five digit numbers containing all digits from 1-5 are arranged from highest to lowest. What is the 25th number?
(Examples of these numbers are 54 321; 23 451; 41 532)
My Solution:
First, it’s impractical to list these (5)(4)(3)(2)(1)=120 numbers from highest to lowest and then determine the 25th number in the list. So, we can instead do [...]

A Simple Proof # 1

Prove that:    , for any
Solution: Since     , then in order to produce a short
proof, the expression   should be express in the form  
Doing so,

Hence,

Angle of Projection

At what angle of projection is the range maximum? Prove it mathematically.

To solve for the required angle of projection ( ), we need first to find a general expression for the range.
To find range, set y = 0 first and use:
,

Implies that,

(non zero value)
Solving for the general expression for range,

Regardless of the [...]

Mathematical Beauty

“A physical theory must possess mathematical beauty.”
In the most basic explanation, mathematical beauty is related to simplicity and surprise.
- Paul Dirac, quantum physicist

NASA’s top 10 views of Earth

To celebrate the beauty of our planet on Earth Day, NASA released its 10 favorite photos taken by astronauts on the International Space Station.
Click here: NASA’s top 10 views of Earth

CNN: Dr. Ronald Mallett’s Time Travel Machine [Video]

Video by pauleycamerieexodus

‘Memristor’ [from PhysOrg.com]

Researchers Prove Existence of New Basic Element for Electronic Circuits — ‘Memristor’ from PhysOrg.com
HP today announced that researchers from HP Labs have proven the existence of what had previously been only theorized as the fourth fundamental circuit element in electrical engineering.
This scientific advancement could make it possible to develop computer systems that have memories that [...]

Log-Base 2

(AIME 1994) Find the positive integer n for which
[log2 1] + [log2 2] + [log2 3] +…+ [log2 n] = 1994
where [x] denotes the greatest integer less than or equal to x (for example [Π] = 3)
Solution:
Observe that:
[log2 1] = 0(20)
[log2 2] + [log2 3] = 1(21)
[log2 4] + [log2 5] + [log2 6] [...]

From Infinity to Divinity

There was a young fellow from Trinity (Cambridge University?)…..
…..Who took √∞ ….. But the number of digits….. Gave him the fidgets…..
He dropped Math and took up Divinity…
Hehehehe … ∞
This is the phrase at the bottom of title page of the book One, Two, Three…Infinity by George Gamow.
I enjoyed reading it, since it is about facts [...]

Some Proofs on Irrationality

Prove that √5 + √3 is irrational
(No.4 of Problem Set 10 in Numbers: Rational and Irrational by Ivan Niven)
Solution:
The argument made here is a proof by contradiction or reductio ad absurdum; That is, assuming that the proposition is false and then derive a contradiction from this assumption.
Note: My solutions are also parallel to Prof. Niven’s
First, [...]

Points inside rectangles

Let P and P’ denote points inside rectangles ABCD and A’B’C’D’, respectively. If PA = a+b, PB=a+c, PC = c+d, PD =b+d, P’A=ab, P’B’=ac, P’C’=cd, prove that P’D’=bd.
(Pi Mu Epsilon Journal – 4(Spring 1967)258, proposed by Stanley Rabinowitz)
————————————————————–
*Let sides AB and CD are the horizontal sides, and sides BC and AD are the vertical sides.
*Draw [...]

Value of S

(AHSME 1996) When n standard 6 sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same probability of obtaining a sum of S. What is the smallest possible value of S?

————————————————————–
Observe that for any natural number a:
For 2a dice (even):

prob. of obtaining a sum of [...]