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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.
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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.
Good topic, and I like how you try a definition and apply it to different types.
ReplyDeleteIt could use a little more content to be complete. Two ideas: dig into irrationals a little more - those ideas were a bit incomplete. Or find a counterpoint - someone who feels complex numbers are numbers and why.
C's: 4/5
Hi Marty! I thought that it was really interesting that you think of numbers as showing order. I've never thought of it that way, and it was interesting to read about your understanding of what a number is. One thing that you could look into is how you might expand your definition of a number to include complex and irrational numbers, since they are numbers. It might also be cool if you included some research about how the concept of a number changed throughout history, and maybe discuss some of the different definitions of a number that mathematicians had. Great post!
ReplyDeleteHi Marty! It was really interesting to read about your understanding of what a number is. I liked your idea that a number shows order, and I think you brought up some valid points. One thing you could spend some more time on might be to think about how you can expand your definition of a number to include complex and irrational numbers, since they are numbers. It might also be cool if you included some research about how the concept of a number evolved throughout history, and maybe discuss different mathematician's ideas about what a number is. Great post!
I enjoyed the memes throughout your blog! I would be interested in what your own definition of a number would be after going through the steps you did to try to make sense of the google definition. It was a good idea to use a definition from google because it makes it relatable to what a lot of other people might do in order to come up with a definition of a number! I enjoyed reading your post!
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