1/8 Vs. 1/4 ( 1/8 LARGER than 1/4 )

1/8 Vs. 1/4 (1/8 LARGER than 1/4)

Can anyone can tell me which is greater among 1/4 and 1/8 ? Normally we will answer this question with 1/4 . Lets make a difference in this . I say 1/8 is larger than 1/4 . Let me Prove 1/8 is larger than 1/4 .

We Know that ,

3 > 2   
3 log(1/2) > 2 log(1/2)    // Multiply log (1/2) on both sides ..( log means log base 10 )
log[(1/2)³] > log[(1/2)²]  // Property of logrithms
(1/2)³ > (1/2)²   // Take anti-log on both sides
1/8 > 1/4

Here is my solution to prove something not possible . Can you find the fallacy I have used in this Proof ? 

Ans :  The Answer is that the multiplication of log(1/2) in the second step is the fallacy . In reality , log(1/2) is a negative value .

1 Equals 0 (FALLACY)

1=0

(n+1)2 = n2+2n+1
                                //Expansion

(n+1)2-(2n+1) = n2

                                //Subtract from both sides

(n+1)2-(2n+1)-n(2n+1) = n2-n(2n+1)
                                //Add to both sides

(n+1)2-(n+1)(2n+1) = n2-n(2n+1)
                                //Factor

(n+1)2-(n+1)(2n+1)+(2n+1)2/4 = n2-n(2n+1)+(2n+1)2/4
                                //Add to both sides

[(n+1)-(2n+1)/2]2 = [n-(2n+1)/2]2
                                //Factor

(n+1)-(2n+1)/2 = n-(2n+1)/2
                                //Take square roots of both sides

n+1 = n
                                //Subtract from both sides

1 = 0

Impossible!

The operations listed above utilize basic arithmetics to arrive at the false conclusion. Starting by simply expanding a squared equation, we can subtract 2n+1 from both sides to isolate n2. Subtracting n(2n+1) from both sides now allows the left side to be factored. Adding (2n+1)2/4 to both sides once again will enable both sides of the equation to be factored down to squared forms. By taking the square roots and then subtracting the n-(2n+1)/2, the proof is complete and 1=0.

If two numbers are equal, their squares are also equal. However, the reverse form of such a statement does not hold. In short, u = v does not imply square root of u equals square root of v due to the fact that the result of a square root is not unique. Without this fact, the above proof becomes actually legitimate.

Is Zero A Prime Number Or A Composite Number ?

Zero - Prime Or Composite

                   Most of us will be wondering whether the number (awesome & unique number) is either prime or composite . The general explanation you might have received is that ZERO is not a natural number and so it is neither prime nor composite . Well , of course , the answer is correct . Zero is Neither Prime Nor Composite but the way of explaining is not correct and is not logically valid .

                   We all know the definition of the PRIME Number . The Prime Number is a number which has exactly two factors 1 and itself . But Zero has infinitely many factors i.e., zero is divisible by all numbers except zero and so zero can never be a Prime Number .

                    Now lets jump on to the definition of a COMPOSITE Number . The Composite Number is a number which can be represented by the product of any two positive integers , neither of which can be itself . Since Zero can never be represented as a product of two non-zero positive integers , it can also never be a Composite Number . 

                     As ZERO can never be a Prime number and also a Composite number , ZERO is NEITHER PRIME NOR COMPOSITE .

Different Methodology In Multiplication

Different Methodology In Multiplication


           This is a new methodology in Multiplication used to make the kids understand the multiplication more clearly and perfectly . This can help the kids do the multiplication correctly using this method .