Fallacious Proof
From Math Images
- The erroneous proof claiming that 1=2. Can you spot the error?
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Basic Description
There is an erroneous proof that many fall victim to. This proof concludes in the fallacy that one is equal to two.The problem with this proof is between the fourth and fifth lines.
The transition from (a + b)(a – b) = b(a – b)
to a + b = b involves a discrete division of both sides by (a – b). According to the proof itself, a = b and therefore, a – b = 0.
Since division by zero is impossible, this entire proof is invalid and false.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Basic algebra
The author of the above proof is not the first to fall victim to division by zero. In fact, Albert Ei [...]
The author of the above proof is not the first to fall victim to division by zero. In fact, Albert Einstein, who is considered by many to be the greatest physicist of all time, has made the exact same error. Einstein's error was by then viewed as a mathematical “sin.” Even worse, he made the mistake in the publication of his theory of relativity, a theory that proved to be a breakthrough in science. The result of his error was the overlooking of the fact that the universe was expanding.
Mahāvīrā, a ninth century mathematician of India, argued that A ÷ 0 = A[1 p. 72]. Division by zero, or nothing, was thought by some to be equivalent to no change.
It was widely accepted, as Bhāskara of India suggested, that A ÷ 0 = ∞[1 p. 135]. Finally, a mathematical proof was offered to prove this belief false:
Suppose A ≠ B. Since any number multiplied by zero is zero, A x 0 = B x 0. Assuming that a number could be divided by zero, (A x 0) ÷ 0 = (B x 0) ÷ 0. Canceling out the zeros would leave the equation A = B, an equality that blatantly contradicts the initial premise of the proof.
Finally, it was rightfully proposed that A ÷ 0 does not mean anything, because it means too much[1 p. 116]. The fallacious proof on top is essentially a derivative of the proof used to prove that zero is a number that is impossible to divide by. The fallacious proof does not shake the pillars of mathematics. If anything, the proof reinforces them.
In 1078, before the idea of dividing by zero became a taboo, an interesting proof was proposed by Anselm, the Archbishop of Canterbury[1 p. 163]. Anselm began the proof with the equation m ÷ 0 = ∞. Not realizing that the first equation is impossible, Anselm proceeded to multiply both sides by zero, giving m = ∞ x 0. This, Anselm claimed, was proof of the existence of God. Anselm viewed infinity as God and zero as the universe before He filled it with matter. After accepting these terms, the equation reads out as God creating the universe out of nothing [1 p. 163].
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Division by zeroTeaching Materials
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References
[1] Kaplan, Robert. The Nothing That Is: A Natural History of Zero. Oxford: Oxford UP, 2000. Print.
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