4 = 5; where's the catch?
My friend (the one who had tried to prove that 1 = 2 was at it again; this time he was bent upon convincing me that 4 = 5!
"Hey, I can prove that 4 = 5," he said. I was reading a very interesting story book, and I wanted him to go away. But you have to be civil; you can't tell your friends to go away, especially those friends who lend you story books. The story book I was reading had, in fact, been borrowed from him.
I suppressed a sigh, and trying to appear curious I asked him to show me how four could equal five.
The friend always carried a notebook and pen with him; even on the playground! He whipped out the notebook and wrote:
-20 = -20
"Is that right?" he asked.
"Of course," I said. "Even a kid knows that 1 = 1, 5 = 5, 10000 = 10000, -20 = -20."
"But I can write -20 as 16 - 36; again, -20 is also 25 - 45."
"Sure," I said. "We all know that 16 - 36 = -20, and 25 - 45 = -20."
"So, instead of writing -20 = -20, I can write 16 - 36 = 25 - 45," he said looking at me.
I saw nothing wrong in that.
But 16 = 42, 36 = 4 x 9, 25 = 52 and 45 = 5 x 9, he went on.
I knew that much.
Therefore, 42 - 4 x 9 = 52 - 5 x 9.
That was alright with me. I had no objections.
"Allow me to add 81/4 to both sides; since I am adding 81/4 to both sides the equality holds," he said.
I gave him my permission.
So, now we have 42 - 4 x 9 + 81/4 = 52 - 5 x 9 + 81/4.
I nodded my head sagely in agreement.
So, (4 - 9/2)2 = (5 - 9/2)2.
That stumped me for a while; but, then, I remembered that a2 - 2ab + b2 = (a - b)2. So, LHS = (4 - 9/2)2 = 42 - 2x4x9/2 + (9/2)2 = 42 - 4 x 9 + 81/4. Similarly, the RHS was 52 - 5 x 9 + 81/4. Everything was fine.
Since (4 - 9/2)2 = (5 - 9/2)2, this means on taking the square roots (4 - 9/2) = (5 - 9/2), he said looking me straight into the eyes.
I had to agree.
We can cancel out -9/2 from both sides; so what we have is 4 = 5. There, I have proved that 4 = 5; anything wrong in the proof? he asked me grinning from ear to ear.
I scratched my head; he had pulled a fast one on me once again. Will you help me detect the catch? Please help.