June 2016

# 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

"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.

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.

• Union Public Service Commission - www.upsc.gov.in
• IIT-Kharagpur - www.iitkgp.ac.in
• Indian Statistical Institute - www.isical.ac.in
• Indian Institute of Technology Madras - www.iitm.ac.in
• Indian Institute of Management, Ahmedabad - www.iimahd.ernet.in
• Indian Institute of Mass Commission - www.iimc.nic.in
• IIT Bombay - www.iitb.ac.in