00	01	11	10
-10	-2 even	-6 even	-14 even	-10 even
-11	-3 odd	-7 odd	-15 odd	-11 odd
-01	-1 odd	-5 odd	-13 odd	-9 odd
-00	-0 even	-4 even	-12 even	-8 even
00	0 even	4 even	12 even	8 even
01	1 odd	5 odd	13 odd	9 odd
11	3 odd	7 odd	15 odd	11 odd
10	2 even	6 even	14 even	10 even

The discrepancy in the way people tend to think about infinity and zero…

Elementary Definition of zero:

In most elementary math courses a student first learns that zero is nothingness. He learns the differences between integers and real numbers and fractions and so forth. These number systems include a zero which is said to be representative of when one has nothing of something. So, one can have an apple, or two, or three, or no apples. The last case is 0.

Elementary Definition of infinity:

At some time when a person is taught what infinity is for the first time, usually well before having taken Calculus, the definition given is “a very large number” or “a number too large to comprehend”

Calculus definition of infinity:

In calculus, the definition of zero stays the same, but the definition of infinity gets every so slightly modified implicitly. There is no clear statement of this new definition; it is merely assumed that everyone knows what infinity is. It is thereafter used again and again throughout all of it starting with limits. This implicit definition is not that you merely have an uncountable number of apples say, which is what most people think of when they think of infinite apples. This definition is that everything is an apple(s). This definition is that you cannot distinguish an apple(s) from the molecules around it.

The never thought about zero:

If, a sphere with a radius of 0 does not exist, then a sphere of infinite radius is all space itself, not merely a very large sphere, for else the analog zero would mean a sphere too small to be seen. But think about it, this is also a perfectly valid philosophy. That is, it can never be that something does not exist. It is just that when there is too little of a substance it cannot be caught in a measurement and that to us is zero. In that model it is fair to say infinity is “a number too large to comprehend.”

The two philosophies:

This gives rise to too valid and coherent philosophies. However, people tend to mix the two which is no longer a valid philosophy. People tend to think of elementary definition of infinity and elementary/calculus definition of zero. Now as we shall see there is further discrepancy in how zero and infinity are generally accounted for. Namely, people understand a separate negative and positive infinity; however, they try to excuse themselves from understanding a separate negative and positive zero. Assuming, the first valid philosophy that zero is nothing and infinity is everything, it does not matter whether zero is negative or positive and same goes for infinity. So in this first philosophy there is no such thing as a negative zero, however, there is also no such thing as a negative infinity. Take philosophy number two: infinity is “too large a number to comprehend” and “zero is too small a number to measure.” In this case, both negative and positive infinity make sense but so do both negative and positive zero. This might seem a little awkward at first but thinking about it long enough will make it very clear.

What is the opposite of “positive infinity?”

Is negative infinity the opposite of positive infinity? Or is it zero? Most people tend to answer negative infinity at first thought, but with some discussion or intuition might be a little more conscious and say its zero. The first answer results from the understanding that positive and negative are opposites. This answer is consistent with the idea that both negative and positive infinities and both zeros exist, however, it is not consistent with the philosophy that there is only one infinity and only one zero. Of course, that is because then positive and negative infinities are not opposites but the same. The second answer results from the understanding that large and small are opposites or nothing and everything are opposites. This second answer is consistent with both philosophies.

What is the opposite of “addition?”

This is the similar to asking what the opposite of addition is. Is subtraction the opposite of addition? Well, the first thing we need to realize is that addition is an operation on two numbers and subtraction “undoes” this operation. It does so by using the same operation on a number that has been negated. But really, the operation itself is the same. So subtraction is something that is the same as addition but is done ‘backwards’ or in ‘reverse.’ It is the same thing as taking ten steps forward and then ten steps back to undo what had been done. So let’s say that subtraction is a reverse process, this way the idea is consistent with both philosophies. If we take subtraction and addition to be opposite processes, then we would be consistent with the first philosophy but not the second where infinity is everything and zero is nothing. This is clearly because with the first philosophy positive and negatives can be taken to be opposites but not with the second. They are the same with the second.

Now, it is clear that integration is the addition of infinitely small pieces and differentiation is determining the slope of a curve at a given point. It is also clear that these two processes also undo each other. By analogy it must be then that differentiation is the subtraction of infinitely small pieces. Yet, it is not so. So, why is this analogy so obviously wrong? The reason is clear. The basis in which the opposites were defined has been changed. Subtraction and addition were opposites defined in the basis that positive and negative are opposites. However, one must realize that opposites are not discovered they are defined. And it has now been shown, two ‘opposites’ seen from a different view might indeed be the ‘same.’ In the second basis, addition and subtraction are the same, and differentiation and integration are opposites.

Opposite Subgroups

However, there seem to be another structure that forms through this defining of opposites. What I mean is take subtraction and addition to be opposites in the discussed basis. Now change the basis such that they become the same. Now this object that is now a combination of both addition and subtraction, namely integration seems to have an opposite as a whole and that opposite is differentiation.

So again, take for example, positive and negative infinity and group them together and call them object X. This object X now has an opposite of its own, and that is zero, in the sense that X is something very large and zero is something very small or everything/nothing as already discussed.

Now take another two processes that are thought of as opposites: Encoding and Decoding. Encoding is taking a set of objects and somehow merging it into fewer or usually one object. Decoding is taking an object or a small set of objects and somehow separating it into a larger set. So for example, our eyes sense all these visual cues and encode them into some kind of lets say chemical signals. These chemical signals are then decoded by the brain into the various objects so that we are able to then visualize all the objects around us. Now take encoding and decoding club them together and call them reverse processes. Consider them the same process, not two opposite processes because they are part of the input to the brain. Now clubbed together the opposite of object X becomes the output from the brain. Call this process of recall object Y and everything that happens in the middle object Z. So, object Z is the process of storing and processing this information within the brain, not including the decoding. In the case of infinity and zero, object Z would have been all the numbers between zero and infinity. Also, notice that object Z may be composed of the same reverse processes that object X was composed of. So, in this case positive and negative real numbers not including zero are reverse sets composing object Z. This brings me to another interesting thought. Why is zero a part of the real numbers but infinity is not? Shouldn’t they both either be in or both out to maintain consistency of thoughts? Doesn’t that suggest that people in general understand nothingness, but not everythingness, and yet pure logic treats them both equally?

Infinite Temperature

Often times we encounter a physics problem where the temperature tends to go to infinity. We all wonder what happens when the temperature is infinite. But how come we never seem to run into a problem where the temperature of an object seizes to exist? That is that it actually goes to zero or no temperature, not zero degrees Kelvin. If we really think about what happens at truly infinite temperature in light of the discussion so far we would realize it means everything in that space is heat and nothing else. At such a location we would not be able to measure anything because our instrumentation would also become heat, and therefore, we can only measure around it.

Similarly, we encounter the idea that probability zero does not mean impossible.

1’s complement? 2’s complement? Subtraction with 2’s complement…

Transformations characterizing a line and a circle

- This article is a stub I will be expanding and editing it with more material when I get time, if you have read it and have comments please post in the math->calculus->infinity thread in the forum: The forum is at http://scientificchess.proboards20.com. Also, please do not apply the information in this article in a math class just yet since it can result in a bad grade as this is NOT yet a well accepted idea….

Aditya Mittal