Nature of Mathematics - What is a number?

If I asked you "What is a number?", you would probably laugh at me. You would say "that's an easy one." That's what me and many of my classmates thought when we were posed this question in our capstone math course. When we actually thought about the question, we realized we didn't really have a good answer. We all felt like this guy:

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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.

Doing Math: The Riddle of Diaphantus

In class we discussed a riddle and I wanted to discuss the way I thought about and solved the riddle.

The riddle: 'Here lies Diophantus,' the wonder behold. Through art algebraic, the stone tells how old: 'God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.'

Mathematician Diophantus

The only foreseeable strategy I could see was to use algebra. I started off by assuming that "x" was going to equal the length of Diophantus' life. Then I broke the riddle down:

boyhood was 1/6 of life = (1/6)x
grew whiskers was 1/12 of life = (1/12)x
marriage was 1/7 of life = (1/7)x
after 5 years he had a son = 5
child reached half of father's age = (1/2)x
he died 4 years later = 4

Then I added all of these values together and set them equal to x which is the length of Dionphantus' life. Thus we get:

(1/6)x + (1/7)x + (1/12)x +(1/2)x + 5 + 4 = x
(25/28)x + 9 = x
9 = (3/28)x
x = 84

So I believe that Diophantus lived to be 84 years old.

A question posed was: can you solve this problem without using algebra. After discussing with my classmates we came to the conclusion that no it would not be possible. The only way to possibly get an answer would be to assume a length of childhood and extrapolate form there. But even then you will not be able to get a correct answer when using algebra.

An example would be to assume that boyhood was 12 years. Based on that you can assume that life was 72 years long. But now lets add up the values.

boyhood was 1/6 of life = 12
grew whiskers was 1/12 of life = 6
marriage was 1/7 of life = 10.28
after 5 years he had a son = 5
child reached half of father's age = 36
he died 4 years later = 4

This all adds up to 73.28 which is not equal to 72. Therefore the only way to solve this riddle is to

What is Math?

On the first day of my capstone math class at GVSU, our professor asked us "What is Math?" Initially I didn't know how to answer it. There are so many different aspects of math that there is no real way to give a clear cut definition. 

But I will make an attempt. To me "Mathematics is using formulas, equations, and processes to solve and analyze numerical problems".

That is my best attempt to capture all of mathematics. I know that many people have their own definitions, even my classmates had much different definitions. But we all had the same general idea which was to try and capture all of mathematics in one sentence.

With a subject as complex as math, I think we can all agree that there is no one clear cut answer to the question "What is Math?"

Mathematics has a great and wonderful history. Unfortunately I don't know a whole lot about the History of Math. In many of my classes we have learned about mathematicians and the work that they did. Those like Euclid who gave us Euclidean Geometry or Pythagoras who gave us the Pythagorean Theorem. Other names such as Archimedes, Euler, Fibonacci and Newton are some of the names that I can recall at this moment. I look forward to this semester and refreshing my knowledge about these great mathematicians. I also look forward to potentially learning new material that I may not have known about these mathematicians.