Infinity, Zero, and Yesh Maayin
(This dvar Torah is so fundamental, please feel free to copy and distribute.)
Infinity and zero are opposites. Zero, more down to earth, is somewhat more manageable than infinity, but both are transcendental incomprehensible opposite forces, the eternal versus the void.
Mathematically, infinity and zero have unusual qualities. Unlike standard numbers, zero times or divided by any number is still zero, and infinity times or divided by any number is still infinity.
Multiplication and division are opposites. The smaller/larger the number you multiply by, the smaller/larger the result. But the smaller/larger the number you divide by, the larger/smaller the result. What happens when you divide by zero? The closer the denominator gets to zero, the further from zero the quotient becomes. The result of any number divided by zero seems to be infinity. Conventional mathematics says that division by zero is undefined. Infinity, likewise, cannot be defined, so division by zero being undefined and equaling infinity may actually be the same thing. Yes, there are those who differentiate between the two, but if division by zero does not equal infinity, it still has to be something very close to it. There seems to be more of a consensus, though, that any number divided by infinity equals zero. Some say this, too, is undefined, but this even more so coincides with infinity being undefined. As 6/3=2 and 6/2=3, it is logical that x/infinity=0 and x/0=infinity exist as a pair. We may indeed say all this is undefined, but that is because we can’t understand it.
Division by zero opens the flood gates. With nothing to confine the number, it spreads forth infinitely. Conversely, dividing by infinity so confines the number that it becomes zero. Multiplication and division are opposites, as mentioned. Dividing by zero is like multiplying by infinity; the result is infinity. And dividing by infinity is like multiplying by zero; the result is zero. There is one exception to the 0*x=0 and infinity*x=infinity rule. There is just one force strong enough to break the zero rule—infinity. There is just one force strong enough to break the infinity rule—zero. Since any number divided by infinity equals zero (z/y=x), and any number divided by zero equals infinity (z/x=y), it follows that infinity times zero is any number (x*y=z). There are similar exceptions to the x/0=infinity and x/infinity=0 rule—zero divided by zero and infinity divided by infinity. Since any number times zero equals zero and any number times infinity equals infinity, it follows that zero divided by zero and infinity divided by infinity equal any number. Conventional mathematics indeed considers these to be undetermined. Unlike undefined, the result of infinity*0 or 0/0 or infinity/infinity can be anything.
What does all this mean? HaShem is the infinite, the eternal spiritual source. Only one thing can bring forth something from nothing. Infinity times zero does not equal zero or infinity; it equals any number. When there was nothingness, tohu vavohu, everything could and did come forth. The infinite can do that to the void. Moreover, dividing something by the infinite is the source of complete limitation, complete nothingness. Ain od milvado. Divide something by HaShem and there is nothing. Divide something by nothing and it is infinite.
Can this be defined? It was not meant to be. But numbers don’t lie.
By Nosson Dovid H