BY SAMUEL ELLINGSON
STAFF WRITER
Infinity is bigger than you can imagine. It is the number at the end of the number line, and yet it doesn’t ever end. And as it turns out, when you place mathematics under the weight of an impossibly large number, things start to snap.
Infinity does show up in our math, but we don’t use it that often. An irrational number has an infinite number of non-repeating decimal places. A circle has an infinite number of sides. But in these scenarios, infinity can be ignored by rounding out what we don’t need. For example, we use the first few digits of pi, rounding the infinite complexity into one easy-to-use number. We even divide circles into 360 sections because degrees are easier to use than radians (which are infinitely precise). What happens if we embrace the explosive size of infinity instead of ignoring it?
Well, bad things happen. For example, pretend you have an infinite number of blocks. If you were to drop another block into the pile, how many blocks would you have? Well, that’s simple; the answer’s still infinity. However, this leaves us with an equation ∞+1=∞, and if we subtract infinity from both sides, we get 1=0. Obviously, this isn’t true.
Oh, the fun doesn’t stop there! What if you wanted to measure an infinite amount of centimeters? Its final length would be an infinite distance (this raises the question of whether space continues outside the boundaries of our universe, which is another matter altogether). Would an infinite amount of meters be the same distance or a hundred times further? Can it be a hundred times further if the first distance is infinite? Assuming they are the same distance, we would get the equation: ∞m=∞cm. If we divided by infinity on both sides, we would have some problems!
If we are not allowed to break math, we need to redefine infinity as a number that can be greater than itself. But how could we prove it? If you were to take 1+1+1+1+1 forever and ever, you would get infinity. If you were to take 1+2+3+4+5 and so on forever, you would also get infinity, but would the second infinity be infinitely more than the first? Another example illustrates this paradox more clearly. If you were to count how many whole numbers there are on the number line, you would assume that the final answer is infinity. However, if you included every decimal (down to the infiniteth place) between every number, you would also get infinity. For each integer counted in the first infinity, an infinite number of decimals are counted in the second. Is the second infinity infinitely times greater than the first? The answer is actually yes. While infinity itself is unendingly large, infinity squared is larger! It has to be! This solution seems to solve all of the inequalities above, yet it still has flaws that can be exploited.
Imagine a meter stick. How many points are there between both ends of the meterstick? As usual, the answer is infinity. Now imagine two meter sticks. How many points are there between both ends of the meter sticks? With our unequal infinity definition, we could say infinity plus infinity or 2∞. But what if I asked you the second question first? Would you define the number of points as infinity or two times infinity? Technically, since each point that is right next to another point has space in between, another point can fit, adding twice as many points. Does that change the answer to 2∞? What if you keep adding points in the spaces between ∞*∞? What if you use the points to fill a square? ∞^∞? A cube? When does it stop getting bigger?
Infinity is giving math a run for its money, but there is one possible solution to save it: Can a number that is larger than itself be considered a number? Infinity doesn’t fit on the number line; there is no number just before infinity. At what point when counting do we switch from normal integers to infinity? Never. Infinity must just be a concept developed by mathematicians to explain what happens when numbers increase forever. Treating infinity as a concept alleviates all of the mathematical problems above. We cannot plug infinity into our normal mathematical equations because it is an unattainable number that doesn’t have a set size. One hundred centimeters, meters, kilometers, and light years don’t have a distance that can be compared because they do not stop measuring. While it is fun to imagine owning infinite blocks, we cannot. Math is restored. Phew!