Mathematical Fallacy: A Classic Proof That 1 Equals 0

Mathematical proofs are one of the most distinguishing beauty of mathematics. But the taken for granted beauty “on the other side” are the fallacies. These are construction of mathematical proofs that conceals some errors and inconsistencies and therefore presents an untrue statement to be true. One example of such is a classic proof that 1 equals 0. I’m sure, you will see more of this if you search the internet.

Below is the proof. Find the step/s that makes the proof wrong.

Theorem: 1 equals 0

Proof:

$latex a = b$

Multiplying both sides by $latex a$ gives

$latex a^2 = ab$

Subtract $latex b^2$ from both sides results to

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Proofs and Jokes Have a Lot in Common

Proofs and jokes have a lot in common.

One similarity is their structure. Just as the conclusion of a proof must be justified by the premises, the punch line of a joke must be supported by a series of statements. The following joke provides an example.

  1. During a flight from Warsaw to New York, members of the flight crew became ill. When the pilot and copilot passed out, flight attendants began asking if anyone on board could fly the plane.
  2. An elderly Polish gentleman said he had flown supply planes in the army many years ago.
  3. He was escorted to the cockpit, but upon looking at the controls, he realized that this plane was far more complicated than the ones he had flown years ago.
  4. He told the flight attendant that he would not be able to fly the plane.
  5. “I’m sorry,” he said. “I’m just a simple Pole in a complex plane.”

Notice that each statement 1-4 alludes to a different part of the punch line in statement 5. Without any of them, the joke fails. Similarly, a proof with an unsupported conclusion will be invalid. Continue reading

A Mathematical Proof that Two Equals One

Introduction

Since it’s Christmas let us break a leg for a while from learning and thinking about serious mathematical proofs. In this post, I am going to prove something fun — something that is counterintuitive. If you love challenge, think about the proof and justify why it is right or wrong.

In this post, I will prove that two equals one.

Theorem: 2 = 1

Proof

Let $latex x = y$.

Multiply both sides by $latex x$: $latex x^2 = xy$

Add $latex x^2$ to both sides: $latex 2x^2 = x^2 + xy$

Subtract $latex 2xy$ from both sides: $latex 2x^2 – 2xy = x^2 – xy$.

Factor out $latex x^2 – xy$ : $latex 2(x^2 – xy) = 1(x^2-xy)$

Divide both sides by $latex x^2 – xy$: $latex 2 = 1$. $latex \blacksquare$