I had a discussion yesterday with a skeptic that was extremely interesting to me. It was a very deep conversation that started from an extremely simple concept to understand. The number 0 (Zero).
I was asked to write a proof for 2×0=0. At first I thought this was a joke. Maybe it still is. That said really I think it serves as a good example for many of the points I have been making throughout my research.
The big take away from this conversation was the construction/structure of a proper “Mathematical Proof”. There are many different types of proofs and it is a complicated and rigorous process.
It was suggested I read “Principles of Mathematical Analysis” by Walter Rudin. This book was described in almost a God fearing way. It was really driven home that It will be extremely hard and I will get frustrated. I am not so easily deterred and I really don’t get frustrated. Of course I get stumped and there are times I do not fully understand what I am reading or looking at. I am not at all bothered by or deterred by that. I keep pushing until I do understand it. It is that simple.
At this point for me, it is not really about what I agree with or don’t agree with. I know the material I have been presenting will have to be formatted as a proper “Mathematical Proof”. This will be a requirement. I also know the skeptics will keep coming. This means I need to be fluent in the various types of proofs and how they are structured. Reading this book will no doubt help me in many ways.
So I accepted his “Challenge” with one caveat. That I would complete two proofs as a part of this exercise. One before I have read the Rudin book, and one after. This will make for a very interesting comparison at the end both in the proof itself and my opinion of the construct of Mathematical Proofs.
This may very well seem like a silly example to start with. Maybe it is. However there is a twist which I think many will not expect so I think there will be something for everyone to learn by going through this process.
I have made the argument in previous posts that all the complexity we observe both in the natural world and mathematics comes from simple numbers and the most basic of math. This ties in nicely with that. So without having read a single page of Rudin and only a general knowledge about the technical details of a proof lets begin version 1 and talk about 0. Understanding this is not a “Mathematical Proof” in the technical sense, I still feel this is a reasonable and valid proof in general and is mathematical in nature. This means it can be verified mathematically.
A Proof for Zero
Before we can prove any mathematical operation involving the number 0 we must first define and prove the existence of 0 as a number, quantity or lack there of. Further we must prove what a number is in order to classify zero itself as a number.
A number can be any natural number, integer, negative or complex. It is a notation or description of both construction and measure. A value, quantity, angle or amplitude. It is further the sum of it’s parts, prime factors or otherwise.
For example if we use the number 1 to describe an object of unit 1. Then, placing one object beside another object requires a different numerical description for the objects new plural nature. Thus we have the number 2.
Intuitively we can make the same argument for every new number being a result of the described previous numbers. This happens in a few ways. We know any number can be constructed with prime numbers. Further these described numbers create a natural number base and inherit finite properties from previous numbers. For example one number being a multiple of another.
Looking at the Fibonacci Sequence we can see this playing out.
Now we must define what we mean by number base and finite properties. The number base is a roll over position. You can think of it as a unit on a ruler. It defines a period of measure and creates powers of that measure. Think powers and multiples of 10 for a 10 base number system.
Next we must define what we mean by finite properties. By summing the digits of any number and reducing it to a single digit number in the context of it’s number base, we can look at what is called the digital root or digital sum of that number. This digital root is like a unique finger print or the DNA of a number. Even more so than primes. Any number will inherit unique behaviors and properties based on this digital root.
Looking at the digital roots of the Fibonacci Sequence
What is not so obvious when looking at the Fibonacci Sequence in digital root is that 0 and 9 are essentially the same number or position. In fact in digital root form you can start the Fibonacci Sequence with either 9 or 0 and the resulting repeating sequence will be exactly the same.
Looking at the digital roots of the Fibonacci Sequence starting with 9
9+1 = 10 (1)
1+1 = 2
1+2 = 3
2+3 = 5
3+5 = 8
5+8 = 13 (4) etc…
Further if we test this with any sequence following the same method as the Fibonacci Sequence we find there are only 9 possible digital root sequences all of which also have a 24 digit repeating pattern. These repeating sequences remain exactly the same whether you start the sequence with 0 or 9. The start position is also the only position that is not repeated. This start position can also be completely excluded and the resulting sequences are exactly the same.
These unique behaviors and properties create natural group classifications and well as polar pairs of numbers. Using a 10 base number system this looks like the following.
Any number acts according to it’s number group and polarity.
To prove this if we study these sequences we will find that each sequence reflects another. For example. If we look at sequence #1 and #8 you will note that the last 12 digits of sequence #1 is the same as the first 12 digits of the #8 sequence.
This pattern continues through out the sequences creating these polar relationships naturally.
By arranging these sequences according to their number group we find perfect mathematical alignment. Adding up the columns of digits in sequences #1,#4 and #7 in digital root produces the #3 sequence. Adding the #2, #5 and #8 sequences produces the #6 sequence. Adding all four sequences on either side will produce the #9 sequence.
These patterns take on very simple yet significant mathematic principals which apply to any number.
Principle Interactions of addition and subtraction are what allow us to define multiplication, division or any other mathematical operation beyond these 7. These 7 interactions must be empirically true for any operation to be true.
Numbers of the same group will also share factors. The digital root of any number itself can often being a factor of said number.
For example. 147 = 1+4+7 = 12 for a digital root of 3. If we divide 147 by 3 we get exactly 49.
For another example. No prime number no matter how large or number of digits will have a digital root of 3,6 or 9 with one single exception. The number 3. Put another way, all prime numbers are never a Group 9 number with the exception of the prime number 3. Further no number will ever have a digital root of 0.
We know by all of these definitions that 1+1 =2. By any description of number, value or quantity this empirically true. By default this means that 1+1+1 = 3*1 = 3 or that 3/1=3, so forth and so on.
Zero is a number absent of any value. It can be a starting point, end point or even a point between 2 or more other points. While it may not have a value it represents a state which by default has a value in any numerical or mathematical system. It is an intrinsic value that is either implied or not implied but always present. It manifests itself either through 9 or zero not metaphorically but literally and mathematically. Put another way it is the interplay between the start and end of any number base system.
Now we are ready to prove 2*0=0. Further we will prove f(n)=n*0 = 0.
With digital roots there are only 3 numbers that when squared produce no change to the value at all.
In fact these 3 numbers to any power will produce the exact same result. A unchanging digital root. This is the same as adding these numbers n number of times.
Zero specifically because it is a no value state can undergo any operation and remain unchanged digital root or not.
While is seems obvious and intuitive that 2×0=0, I feel these facts combined create an interesting space for discussion. As I pointed out at the beginning this is not a pure “Mathematical Proof” at least not in the sense of Rudin. It is a description of numerical observations which can be mathematically proven. Without any doubt a year from now when I have completed reading this book I will be able to formulate this in a way that is more in line with what is considered a Mathematical Proof.