Division by zero: Is it really impossible?

No

3 of 43

by Jimmy Nightingale

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In 2006, Dr James Anderson, a computer science professor at the University of Reading (UK) boldly announced that he had solved a very important problem. It was a problem that has perplexed academics and anyone of a vaguely scientific or mathematical ilk for centuries, well ever since about 800AD. The big announcement, carried by the BBC was that Dr Anderson had devised a proof that it was possible to divide by zero.

His proof is not possible to write here because of the mathematical symbols involved, but it was based on the idea of using 0/0 and what he called a "transreal" number. Rather than being a proof, Anderson's workings assume that it is possible from the start, so it is a logical fallacy. What it did do thought was create an uproar in academic circles. Given that there hasn't been a major discovery in the field of mathematics for a very long time, perhaps they felt a little aggrieved at all Dr Anderson's attention. As a number of critics pointed out, you can use Dr Anderson's logic to prove anything:

0 x 1 = 0 and 0 x 2 = 0; therefore

0 x 1 = 0 x 2, so dividing both sides by zero gives:

1 = 2 (and you can substitute 1 and 2 with anything).

It assumes that the very thing it is trying to prove is valid at the start and is clearly an abuse of logic. It is along the same plane as the old Monty Python logic:

Statement 1 = I like to eat kippers for breakfast,
Statement 2 = Kippers live in water, and
Statement 3 = Water comes from rain.

And coming up with the brilliant conclusion:

If I don't eat kippers for breakfast, it will not rain.

Anyway, to move away from Dr Anderson and to some discussion that supports the proposition that it is possible to divide by zero.

In the 12th century, an interesting Indian mathematician called Bhaskara and astronomer came up with a brilliant treatise called the Lilavati that dealt with all manner of arithmetic concepts, one of which was the properties of zero and things that you could do with it. I say interesting because he is also known as Bhaskara II and Bhaskara Acharya (which means Bhaskara the Teacher) and he was one of a long line of mathematicians and astronomers. One can only imagine what their dinner table conversation would have been like.

Bhaskari was into astrology, fair enough given his interest in mathematics and astronomy, and one of the enduring legends concerning him has to do with his daughter, Lilavati (his work on arithemtic is named after her and said to be a gift to her). Bhaskari read her horoscope as she was planning to wed and somehow worked out that her husband to be would die soon after their marriage unless the wedding took place at a particular time. To help work out when that time would be, he took a cup and made a tiny hole in it before placing it in a water filled vessel. He aligned the volume of the cup and the hole in such a way that the cup would sink during the hour when she needed to be wed. Like Eve and the Forbidden Fruit, she was told to go nowhere near the thing and couldn't help herself. Sure enough, her curiousity led to disaster. Her nose ring popped out and fell into the cup, disturbing it so that she ended up marrying at the wrong time. Her husband died shortly after the wedding. Pity he didn't invent a clock, it would have saved a lot of grief.

Anyway, his big pronouncement was that any finite number divided by zero yielded infinity. He stated it in slightly more grandiose terms, namely "In this quantity which has zero as its divisor there is no change even when many [quantities] have entered into it or come out [of it], just as at the time of destruction or creation when throngs of creatures enter into and come out of the infinite and unchanging [Vishnu]". Regardless of the flowery language, it was a brilliant deduction that prevails to the current day.

To think of this in more understandable terms, if you take any number, for argument's sake we'll use 148. If you deduct 2 from it, it becomes 146 and you can do this 74 times until it becomes zero. If you deduct 1, you can do twice as many times, ie 148 times. What this means is that as you reduce the number you are deducting (in this case going from 2 to 1), you can do the exercise more often (74 to 148). Now, subtract zero off your 148. It stays as 148. You can keep doing this forever and it will always be 148. As Bhaskari suggested, you can do this an infinite number of times, though it would have to wait another 500 years until John Wallis came up with the commonly accepted symbol for infinity: .

On this basis, you can clearly divide a number by zero. Anything divided by zero = . This is not regarded as a real number, then again, this debate does not specify limits with regard to real numbers.

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