How Limitless Is Infinity? Does Infinity End? If So, Where Is the End?

How limitless is infinity? Does infinity end? If so, where is the end?

Zimmerman, Samantha

000698-0116

Session: May 2016

        Again and again, throughout the span of our whole lives, we’ve been around the phrase that states, “all good things come to an end” which emphasizes the idea that most things in our lives do end. However, all our lives we have been taught that some things are infinite meaning that there are some things that never come to an end, it continues forever and these two ideas contradict each other. Based on this idea, representing infinity on a number line would be similar to illustrating the end of the universe. It seems nearly impossible, right? This is because infinity is the idea of something that consists of no end point (we also do not see the universe as having an endpoint). We assume there is no endpoint because of the large quantities that this idea represents. This means infinity is something that is seen to be limitless. This topic seemed relevant because infinity is a representation of something unknown and I was interested in whether or not I’d be able to gain more knowledge about what infinity actually is. Becoming more aware of what infinity is would allow me to understand more abstract concepts throughout my constant battle with this subject. In addition, I wanted to question an idea that is rarely questioned.

        Although infinity is considered as something that is endless and never ends, it does not grow. You are never going to be able to say there are two infinities once you have added two together. This is because it just does not seem to make any sense because two infinities is still one infinity. Infinity remains the same in each solution. For example, when you graph a ray, which is a line with a starting point and no end point, you are able to see an ending point in one direction, but in the other direction all you see is an arrow.

[pic 1]

In the image above, the arrow on the upward end represents that the line has no ending point in this direction. Because of this arrow, you are aware that the line is endless. However, this line is always going to be illustrated this way. This is because you are never going to be able to represent the line as any larger that it’s been shown as. This is because since you are unaware of the length of the line, you are not able to assume its size. Any guess at where the line ends would be an approximation. This is the same reason that infinity is described as one quantity rather that multiple quantities.

        Infinity is limitless, or without an end, but we create restrictions. You are able to see this idea in multiple examples. Some of these include the idea that 500 + ∞ does not have a set sum as 500 + 1 does. You are also able to represent this idea using π. Unlike infinity, pi represents a specific number, even if it is an approximation. When evaluating expressions, pi is usually substituted with the 3.14159, although pi is described as an infinite decimal^[1]. You are able to solve equations using pi because of your ability to substitute in 3.14159... but how do we solve equations using infinity?

This is a challenge because infinity is not a number, it is simply an idea, or a theory. Pi, as an approximation allows mathematicians to approximate an answer, whereas when evaluating infinity, the solution remains general. This is where we are met with this problem:

π²  = 9.86

∞²  = ∞

You are able to solve this equation using pi because pi is always going to represent the same quantity, which cannot be similarly said about infinity. Infinity alone represents millions of possibilities. This being said also explains why you are not able to divide infinity.

 may be equal to [pic 2][pic 3]

Evaluating the equation:

  [pic 4]

When evaluating this equation using GEMDAS^[2], the two infinities in the numerator are multiplied first leaving the equation as so:

[pic 5]

You are unable to cancel the two infinities that are squared, in both the numerator and the denominator because the solution is undefined^[3]. However,  may be equal to   because we are unsure of what quantity infinity represents. Infinity is able to represents a massive amount of quantities for which we have not accounted, or created numbers to represent due to the size of these quantities.[pic 6][pic 7]

If you replace infinity with a real number; for example, let's use 12.. this is what happens:

= 1 and = 1[pic 8][pic 9][pic 10]

It is easy to see how these two expressions would not be equal to each other. When using infinity, you are restricted by concepts that account for infinities “incompleteness.” When using the term incompleteness, the idea that infinity is unknown is what is being referred to. Mathematicians have limited how we evaluate expressions using infinity, so why do we describe infinity as limitless when it has all these limits?