Mathematics

Zero Divided by zero – Yes



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I fear that people may have been over-complicating this problem. The argument seems to be twofold, the first part being is division by zero possible in computers which rely on real numbers to make their calculations? The short answer to this is no (at least not yet, but never say never, especially with the incredible advances in computing technology that are a sign of the age that we live in). Division by zero yields the result of 'infinity', which is an conceptual number of unimaginable magnitude, and not a real number, therefore the computer is unable to imagine it. And since it can't imagine it, it can't calculate it. The second part could be described as can you divide by zero in an abstract maths way, treating zero and infinity as symbols, i.e. concepts with specific properties attributed to them (i.e. 1x0 = 0, and 1/∞ = 0). In this case, I believe you can.

Obviously in the real number system division by any real number by zero yields an indeterminate value, normally assigned the conceptual value 'infinity'. The value of infinity is an number beyond imagining, true, but this does not mean that purely because infinity is the result of an equation, that said equation is impossible. For example dividing the number 1 by 1 = 1. 1/0.1 = 10, 1/0.01 = 100, and so on and so forth. It is no giant leap to imagine then that the smaller the number you divide one by, the larger the outcome will be. As more and more zeros are placed after the decimal point in the above examples, the denominator approaches zero, then the result approaches infinity. From a purely abstract conceptual point of view, this makes sense, even if it is true that any equation involving division by zero (and therefore infinity) is incalculable. It might not be calculable, but it is possible, and the subject of this debate merely asked if it was possible.

In summary, division by zero is certainly possible, if not always calculable. Computers work in real numbers, not abstracts, therefore it is not possible with respect to computers (i.e. computational/applied maths), but from a pure maths point of view, it is very possible.

Now consider another paradox: If 1/0 = ∞ then 1/∞ = 0 both of which make sense, now consider 0/0 = 1 as you would expect from any number divided by itself, therefore 0x∞ = 1 (where intuition would lead us to believe that any number multiplied by zero equals zero). So does 0/0 equal 1 or 0?

I believe 0/0 = 1 because such is the supreme unimaginable magnitude of infinity, that it could actually break the nx0=0 rule. Obviously it wouldn't work in computer calculations, but it certainly merits consideration as a simple, elegant solution to this conceptual conundrum.

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