Most people would say that infinity is the biggest number there is. While it could be called a close guess by some, mathematicians say that it actually has nothing to do with that. There is no “biggest number”. You can add 1 to any number and it will give you a bigger number than that.
Instead, infinity is the amount of numbers there is. It’s also the space there is. That is a better definition of infinty. Srinivasa Ramanujan was an Indian mathematician who lived in the late 19th century. “He made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions,” (Wikipedia) two of which are related to infinity. He’s mostly known for his Sums, the Ramanujan Sums.
One of the Ramanujan Sums is the addition of all positive integers. Almost everybody would think that it would equal to infinity, since it always adds a bigger number than before. But mathematicians think differently, and give the answer of “-1/12”. But it doesn’t make any sense, does it? No fractions are added, no negative numbers are added, yet mathematics’ answer is a negative fraction, something no one would agree to, if there was no proof. Mathematicians do have a proof, which hasn’t yet been disproved. To prove this, two other crazy divergent series are used, which are sums that continue forever and don’t have a limit.
The first one is “1-1+1-1+…”. Every person that I asked about this said that it is equal to either 0 or 1. Mathematicians say it is “?”.
After I gave them that answer, they asked if that’s because it’s the average of the two possible answers. It doesn’t have anything to do with the average. That answer is actually irrefutable and isn’t debatable. All the steps used in the solving of that are ‘legal’. The second series that is needed is “1-2+3-4+5-.
..”. This series uses the first one to prove itself. The answer is just as crazy as the answer of the other series.
The answer is “?”. This series is then used for the series that I mentioned in the beginning: “1+2+3+4+….” This isn’t the only insane thing that infinity has given us.
There are other paradoxes that it creates. One of those paradoxes is Zeno’s paradox. It is about a race between Achilles and a turtle. Achilles gives the turtle a 100 meter head start, which is what creates a problem. Of course, Achilles will reach the first 100 meters much faster. But when he reaches the 100 meter line, the turtle has also moved a little more, let’s say he moved 20 meters.
After Achilles reaches those other 20 meters, the turtle has also moved a bit more, 4 meters. When Achilles reaches those other 4 meters, the turtle has moved another 80 centimeters. And so it goes on. With this logic, the turtle is always ahead of Achilles and will win the race. While that is logical in the mathematical world, it is wrong. The distance that the turtle will make before Achilles passes him equal to the sum of all number of meters that he is ahead of Achilles, so “100+20+4+0.
8+…”. It seems that he will be ahead of Achilles for infinite meters, but since the numbers that are added are becoming smaller than the previous number, that sum has a limit. While that sum can’t be found just by adding those numbers, it is possible to be found by using calculus.
And it gives us the answer of 125 meters. So the turtle will be ahead of Achilles for the first 125 meters only. Another paradox related to infinity is Olber’s paradox. With the assumption that the universe is infinite, it says that the night sky should be filled with stars, making the night bright. But that’s of course wrong, because after a certain distance, the light of the stars becomes so dim it can’t be seen.
But people may think, what could infinity be used for, if it creates so many problems. Well, the sum of all the positive integers is used in many fields, like complex analysis, quantum field theory, and string theory. Many infinite series are very useful in other concepts of physics. Infinity is a complicated concept that creates many problems if it’s used in the wrong way, but mathematicians have methods to avoid these problems. It is a useful concept, different from what most people would think. It even applies in very important concepts of physics.
The absence of infinity would create a lot more problems than what infinity creates itself.