As we shared with the class, we all had similar but different definitions of what a number was. When I googled "What is a number?" the definition I got was:
Number: an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification.
When I looked at this definition it made me wonder about things like infinity, zero, fractions, complex numbers, rational, and irrational numbers. Are they really numbers?
To me there are only 2 categories that I listed which are not numbers. Those two are complex numbers and infinity. They words in the definition for me are SHOWING ORDER.
Infinity has no order. The idea of the word order in math is that if you take a number, add another number, then that new number is bigger than the original. An example would be if I have the number 1 and I add 0.2, I will get 1.2 which is larger than 1. For infinity that is not the case. Infinity + any number = infinity. Therefore infinity has no order and it not a number.
Complex numbers also have no order. Lets look at two complex numbers like 3i + 7 and 10i + 3. How do we know which one is larger? Do we assume the one that has a larger real part is larger? Do we assume the one with the larger imaginary part is larger? Do we add both imaginary and real parts together and that determines which is larger? WHO KNOWS!! Since we cannot order complex numbers I do not believe they are numbers.
As I was writing this post I became skeptical of irrational numbers being numbers. My initial thought was that irrational numbers are numbers because they can be put in an order. The problem is that irrational numbers go on forever without repeating therefore they can't be represented as a fraction which makes them nearly impossible to compare. If the first 200 decimals of 2 irrational numbers are the same, but we don't know the 201st decimal, how do we know which is bigger? This question has left me puzzled and feeling confused like Homer. I really can't say definitively whether I believe irrational numbers fit my definition. If I had to choose I would say they are numbers because they can still be ordered when looking at a set number of decimal places.
What is a number? It is not an easy question to answer. We all have our own definitions and beliefs to what a number is. I believe that showing order is the most important part to being a number. That is why I believe that complex numbers and infinity are not actually numbers. Irrational numbers have me puzzled. It is possible for them to show order if we knew every decimal place but they can't be represented in the real world. Hopefully one day we will have the perfect definition for what a number is. I'm not sure if that day will ever come.