BIG Numbers

It is easy to be overwhelmed when we think of big numbers.  Not big numbers like 100, but REALLY big numbers. When I think about how to describe this overwhelming sensation, my mind goes to little kids.  It reminds me of when you're little, and you stretch your arms out as wide as you possibly can, and your left hand is as far as humanly possible.  You look up at your mom or dad and say "I love you THIIIS much!!"  and your parent looks back at you and says "I love you to the moon and back!"

...Do we ever think about just how much that is? I don't think we do, and I would argue that as we get older, we really don't understand huge distances any better than we did when we were 5, 6, or 7.

When a little kid stretches their arms as far as they can, that distance between their arms is their "forever" distance.  That is their standard for "the most in the whole wide world."  When we tell people that we love them to the moon and back, that is our "forever" distance.  Yes, we logically know that there exist distances larger than that, but we don't consciously comprehend that.  A little kid knows that there exists a distance larger than the space between their palms, but they don't consciously comprehend it.

This is all due to standards.  We created standards in our minds, and that is what we compare everything we discover after the standard is set to.  To test my theory, I asked a 7 year old girl, Maci, what the biggest number she knows is.  She said that 100,000 is the biggest number she knows.  100,000 is Maci's standard.  She cannot comprehend numbers bigger than that, because they are too big when compared to her standard.  When she thinks of small numbers, she compares them to her large standard of 100,000, and she is able to comprehend them.  As we get older, our standard gets larger, but when we hear big numbers we are still comparing them in the same way that 7 year old Maci does.
Everyone has different standards for their "forever" numbers, and that is why we individually comprehend different numbers in different ways.  An example of this is a runner.  Someone who runs 3 miles normally, cannot comprehend running 10.  This is because 3 is their standard.  Someone who runs 26 miles normally, can easily comprehend a 10 mile run.

Thus, it is all in what we are used to.  Standards are the key.

Transcendental Functions

I started by pulling up Webster's online dictionary and searching the word, "transcendental." I was directed to the definition of "transcendent". This is defined as "going beyond the limits of ordinary experience," or "far better or greater than what is usual."  Now, I don't think that reveals to me how the word is related to math, but I do think that I can think of it in such a way that it does.  For example, maybe we are studying transcendental functions because we want to take our minds to a place where they go beyond the limits of an ordinary mathematical experience, or because we want to study mathematics in a way that is far better or greater than what is usual.  These definitions and applications intrigued me.

Next, I found out what transcendentalism is.  It is a practice that I am not sure I am in agreement with, but that is interesting to ponder.  It is a practice of people who do not accept things as religious beliefs, but rather as a way of understanding life's relationships.  This also got my "math brain" thinking.  Maybe we are studying transcendental functions because we want to study the relationships between numbers and equations.  We want to discover how things work together in a mathematical context.

Finally, as I did research on transcendental functions themselves, I found my wonderings to be almost correct.  Transcendental functions have the name they do because they cannot be expressed simply.  They cannot be expressed using simple algebra.  They allow us, and force us, to go deeper into our mathematical thinking.