Possible Math Breakthrough

[Dark]

The worst place to be posting this is on YCM. But screw it. ;D

 

For all my life (16 yrs.), I have been learning about math. I love it to death. But I haven't seen rules for the concept of infinity. You can add i to i to get 2i, even though i is an imaginary number. But I don't remember seeing anything about the concept of infinity. So, here is my shitty speculation.

 

Infinity = inf

 

Addition

 

A non-inf number added to inf equals inf.

Inf added to inf also equals inf.

 

Subtraction

 

A non-inf number minus inf equals negative inf.

Inf minus inf equals zero.

Inf minus a non-inf number equals inf.

 

Multiplication

 

Inf times a number > or = to 1 equals inf.

Inf times a number < or = to -1 equals negative inf.

Inf times zero equals zero.

 

Division / Multiplication By Fractions

 

Inf divided by a number > 0 equals inf.

Inf divided by a number < 0 equals negative inf.

Inf divided by inf equals 1.

Inf divided by negative inf equals -1.

 

This is where it gets a little confusing for me.

 

A non-inf number > 0 divided by inf equals 0.

A non-inf number < 0 divided by inf equals -0 (which is 0).

 

Inf divided by 0 equals divide by zero error.

 

Exponents and Radicals

 

Inf to the power of a number > 0 equals inf.

Inf to the power of a number < 0 equals 0.

Inf to the power of 0 equals undefined.

 

Rule 1 also applies to fractional exponents, which are radicals. Any root of infinity would be infinity.

 

---

 

This is probably extremely flawed.

 

So, point out any errors I made in this.

 

And point out anything I missed which can be applied to infinity.

 

But be ready to prove your statements.

 

:D

[Blamonchesix]

At least you tried. (^_^)

 

But we can't prove if its right nor wrong so... good bob. XD

[Dark]

TBQH, the concept of infinity doesn't have too many real life purposes, as most things that would go on 'till infinity in the math world have a domain restriction in the real world.

[Exiro]

1. You are not the first one to think of this. This seems common logic.
   
2. There are already multiple mathematical modelings for this. One goes a lot further; it explains numbers higher than infinity.
   
3. Any number other than 0 divided by [inf] should either be higher or lower than 0, depending on whether it was a positive or negative number.
   
4. Any number to the power of 0 equals 1. [inf] should not be an exception.
   
5. [inf] to the power of a number below 0 should equal [inf].
   

[Dark]

1. You are not the first one to think of this. This seems common logic.
   
2. There are already multiple mathematical modelings for this. One goes a lot further; it explains numbers higher than infinity.
   
3. Any number other than 0 divided by [inf] should either be higher or lower than 0' date=' depending on whether it was a positive or negative number.
   [*']Any number to the power of 0 equals 1. [inf] should not be an exception.
   
4. [inf] to the power of a number below 0 should equal [inf].
   

 

1.) Yeah, but I wanted to write it down and get +1 post count.

2.) Never seen this written down before, curse those math teachers!

3.) Let's think about it this way.

 

Divide 1 by 1. 1.

1 by 2. .5

1 by 3. .333

1 by 4. .25

1 by 5. .2

1 by 6. .166

 

So, in the function 1/x, as x gets bigger, the function's value gets smaller. As x approaches infinity, y approaches zero. In normal math, 1/inf. does not exist because inf. is a concept and not a number. In this math, 1/inf. would be 0.

 

4.) 0 to the power of 0 does not equal 0. :/

5.) Unless I'm mistaken, negative powers work like this:

 

x^-1 = 1/x

x^-2 = 1/(x^2)

x^-3 = 1/(x^3)

 

So inf. to the xth power when x is less than zero would be:

 

1/(inf^x)

 

Infinity to any positive power (because it's now in the denom) is infinity.

 

And 1 over infinity is zero.

[Raylen]

Dark. Then what is

 

0 divided by 0?

[Chiri Tsukikawa]

That's comlicated, I only understood the addition and subtraction.

[Dad]

Dark. Then what is

 

0 divided by 0?

 

You can't divide by zero. D:

 

You'll kill us all.

[✗Panda✗]

Inf. minus inf. would equal -inf. too.

[Dark]

Dark. Then what is

 

0 divided by 0?

 

Undefined' date=' due to it being multiple answers at once.

 

Inf. minus inf. would equal -inf. too.

 

x - x = 0

 

I already found a contradiction.

 

1/inf. = 0

 

So, multiply each side by inf.

 

1 = 0

 

But inf. times zero must equal zero, so I have no way to fix that.

[Raylen]

Let y = 1 / 0

 

We are given by Dark's last axiom, 0 = 1/inf.

 

So, y = 1 / (1/inf)

 

This can be rewritten as

 

y = 1 / (inf^-1) [Dark's negative exponent axiom]

 

y = (inf ^ -1) ^ -1

 

y = inf ^ (-1 x -1)

 

y = inf ^ 1

 

y = inf [Dark's infinite exponent axiom]

 

So, therefore, 1/0 = inf.

 

However, what is 0/0? Is my question, this proof no longer works for 0.

[✗Panda✗]

Inf. minus inf. would equal -inf. too.

 

x - x = 0

 

I already found a contradiction.

 

1/inf. = 0

 

So' date=' multiply each side by inf.

 

1 = 0

 

But inf. times zero must equal zero, so I have no way to fix that.

[/quote']

 

That doesnt make sense though. If you take an infinite amount of numbers away from an infinite amount of numbers, youll get either +inf or -inf depending on which number is higher. Inf cant = inf.

[NeoDemonX]

It is like doing this with inf: (Inf + Inf) x 100 - 50 =? it can't be done.

[Dark]

Let y = 1 / 0

 

We are given by Dark's last axiom' date=' 0 = 1/inf.

 

So, y = 1 / (1/inf)

 

This can be rewritten as

 

y = 1 / (inf^-1) [Dark's negative exponent axiom']

 

y = (inf ^ -1) ^ -1

 

y = inf ^ (-1 x -1)

 

y = inf ^ 1

 

y = inf [Dark's infinite exponent axiom]

 

So, therefore, 1/0 = inf.

 

However, what is 0/0? Is my question, this proof no longer works for 0.

 

Ugh, fudge sticks.

 

I'll get back to you on this.

 

Inf. minus inf. would equal -inf. too.

 

x - x = 0

 

I already found a contradiction.

 

1/inf. = 0

 

So' date=' multiply each side by inf.

 

1 = 0

 

But inf. times zero must equal zero, so I have no way to fix that.

[/quote']

 

That doesnt make sense though. If you take an infinite amount of numbers away from an infinite amount of numbers, youll get either +inf or -inf depending on which number is higher. Inf cant = inf.

 

This is why infinity is a concept, and this is why infinity + 1 is still equal to infinity.

 

If you take away 1 from 1, you get 0.

Likewise for 2 and 2, 3 and 3, 56981 from 56981 and .151 from .151.

 

You are taking away an infinite amount of objects from the infinite that you have. Resulting in zero.

 

It doesn't make much sense literally, as infinity is a concept. But it makes sense mathematically.

---

It is like doing this with inf: (Inf + Inf) x 100 - 50 =? it can't be done.

 

2Inf still equals Inf.

 

Inf x 100 still equals Inf.

 

Inf - 50 still equals Inf.

 

So what's the question?

[CrabHelmet]

The problem is that you have not properly defined infinity here.

 

Also, if you study set theory you'll learn about cardinalities, which will tell you that some infinite sets are larger than some other infinite sets. For example, the natural numbers {1, 2, 3, ...} and the integers, {..., -2, -1, 0, 1, 2, ...} have the same cardinality, called aleph-naught, and so those two sets are the same size, even though the natural numbers are clearly a subset of the integers. However, the integers and the real numbers have different cardinalities - there's a very elegant proof of that using something called the Cantor Diagonal Argument - and so, even though there are an infinite number of real numbers and an infinite number of integers, the former infinity is strictly greater than the latter.

[Dark]

The problem is that you have not properly defined infinity here.

 

Also' date=' if you study set theory you'll learn about cardinalities, which will tell you that some infinite sets are larger than some other infinite sets. For example, the natural numbers {1, 2, 3, ...} and the integers, {..., -2, -1, 0, 1, 2, ...} have the same cardinality, called [i']aleph-naught[/i], and so those two sets are the same size, even though the natural numbers are clearly a subset of the integers. However, the integers and the real numbers have different cardinalities - there's a very elegant proof of that using something called the Cantor Diagonal Argument - and so, even though there are an infinite number of real numbers and an infinite number of integers, the former infinity is strictly greater than the latter.

 

And that makes perfect sense. I haven't learned this yet (and I'm not sure I ever well), but it's obvious that the integers have infinitely many more numbers than the naturals, even though both extend to +inf., the integers have all negative numbers (and zero).

 

The problem is, putting that into mathematical terms, it would result in:

 

Infinity (numbers in integers) - Infinity (numbers in naturals) = Infinity (all nonpositive integers).

 

Which, theoretically, doesn't make much sense. :/

 

And I don't have a way to define infinity at the moment. Or at least in math terms. D:

[Raylen]

I've seen that elegant proof. Although, Crab Helmet, I think we are using addition and subtraction not in an integer sense. Since infinity is not a real number, it does not belong to the set of N, Z, Q, or R, it can't be defined in a complex manner either (expressed in the form of a + bi). So I think what Dark is trying to do is simply define a new system of operations in which infinity can exist based on common logic.

[Dark]

I've seen that elegant proof. Although' date=' Crab Helmet, I think we are using addition and subtraction not in an integer sense. Since infinity is not a real number, it does not belong to the set of N, Z, Q, or R, it can't be defined in a complex manner either (expressed in the form of a + bi). So I think what Dark is trying to do is simply define a new system of operations in which infinity can exist based on common logic.

[/quote']

 

But even that seemingly causes a mass of contradictions, considering that you are mixing the concept of infinity with real numbers.

 

1/inf. = 0

 

So, multiply each side by inf.

 

1 = 0

 

Since we are mixing infinity, a concept, with real numbers, 1 cannot equal 0. Meaning that 1/inf. cannot equal 0.

 

But I've already shown that in the graph of 1/x, when x approaches infinity, y approaches 0. So in the world of infinity, when x equals inf., y equals 0.

[Raylen]

Maybe we can define 1 and 0 differently? I'm kinda confused too. Also the entire deal with my proof that n / 0 = infinity.

[Larxene]

Inf times a number > or = to 1 equals inf.

Inf times a number < or = to -1 equals negative inf.

 

Are you saying there are no numbers between 1 and 0?

[NeoDemonX]

It is like doing this with inf: (Inf + Inf) x 100 - 50 =? it can't be done.

 

2Inf still equals Inf.

 

Inf x 100 still equals Inf.

 

Inf - 50 still equals Inf.

 

So what's the question?

 

Their is no question I was just messing around with math and it was = ? means the answer.

[CrabHelmet]

The problem is that you have not properly defined infinity here.

 

Also' date=' if you study set theory you'll learn about cardinalities, which will tell you that some infinite sets are larger than some other infinite sets. For example, the natural numbers {1, 2, 3, ...} and the integers, {..., -2, -1, 0, 1, 2, ...} have the same cardinality, called [i']aleph-naught[/i], and so those two sets are the same size, even though the natural numbers are clearly a subset of the integers. However, the integers and the real numbers have different cardinalities - there's a very elegant proof of that using something called the Cantor Diagonal Argument - and so, even though there are an infinite number of real numbers and an infinite number of integers, the former infinity is strictly greater than the latter.

 

And that makes perfect sense. I haven't learned this yet (and I'm not sure I ever well), but it's obvious that the integers have infinitely many more numbers than the naturals, even though both extend to +inf., the integers have all negative numbers (and zero).

 

The problem is, putting that into mathematical terms, it would result in:

 

Infinity (numbers in integers) - Infinity (numbers in naturals) = Infinity (all nonpositive integers).

 

Which, theoretically, doesn't make much sense. :/

 

And I don't have a way to define infinity at the moment. Or at least in math terms. D:

 

Again, the problem is that infinity is poorly-defined here. Subtracting infinity from infinity leads to major problems if you have not adequately specified what you mean by infinity. For example, suppose we take INF-INF=0. But we know INF=1+INF. So now we have 0=INF-INF=1+INF-INF=1+0=1. Therefore, 0=1. The same paradox arises if you define INF-INF to be any finite number. But if you define it to be infinite, then is it positive or negative infinity? Why should you choose one over the other?

 

Again, the problem is that your terms are not well-defined, and you are trying to apply operations designed for finite numbers to transfinite numbers.

 

There is actually an interesting case to be made for +INF=-INF. One can think of the real number line as a circle - and by "think of the real number line as a circle", I mean "construct a bijection between the real number line and a circle, making this more than just a silly analogy". In other words, one can set it up so that every real number corresponds to a specific point on a circle, and every point on the circle with one exception corresponds to a single real number. That exception point on the circle that corresponds to no real number is directly opposite the 0 point. As one approaches the exception point from one side, the numbers corresponding to the points become increasingly small; from the other, they become increasingly large. As such, that exception point serves as both +INF and -INF - mapping the real number line onto a circle makes them one and the same. This solves ugly questions like "What's infinity minus infinity?" Why, it's infinity. "Positive or negative infinity?" They're the same.

 

Of course, different definitions of infinity have different purposes. This definition is good for some things but not for others. Most problems with infinity arise from using the wrong definition of infinity - or, worse yet, applying multiple definitions of infinity to a single transfinite value and getting them mixed up.

 

Set theory also leads to some oddities if you try subtracting infinite sets from one another to find the size of the result. For example, if you remove the natural numbers (positive integers) from the integers, you still end up with an infinite set left over. However, you can set up a one-to-one correspondence between the natural numbers and the integers, and if you do that and then proceed to cancel every natural number with its corresponding integer, nothing will remain - that's why the sets are really the same size, even if this is not immediately obvious.

 

I've seen that elegant proof. Although' date=' Crab Helmet, I think we are using addition and subtraction not in an integer sense. Since infinity is not a real number, it does not belong to the set of N, Z, Q, or R, it can't be defined in a complex manner either (expressed in the form of a + bi). So I think what Dark is trying to do is simply define a new system of operations in which infinity can exist based on common logic.

[/quote']

 

Precisely - the usual binary operations are defined on some sets, but the sets on which they are normally defined do not usually contain infinity. In other words, to perform these operations on infinite values, one must first construct a new field in which infinity is defined.

 

I remember seeing an interesting field of, if memory serves me correctly, exactly eight elements: 0, 1, 2, 3, 4, 5, 6, and INF. The seven more easily recognizable elements - represented here by digits for their similarity to the integers, though not exactly the same, as shown by the addition equation below - had addition and multiplication defined in a manner similar to modular arithmetic via mod 7, so that, for example, 4+6=3. The "INF" element was another element like the other seven with operations well-defined; it was primarily created as a way of getting around problems like division by zero, and was treated as just another number. That's what all the numbers are, in the end: objects. 0 is an object. 1 is an object. 5 is an object. INF is an object.

 

I've seen that elegant proof. Although' date=' Crab Helmet, I think we are using addition and subtraction not in an integer sense. Since infinity is not a real number, it does not belong to the set of N, Z, Q, or R, it can't be defined in a complex manner either (expressed in the form of a + bi). So I think what Dark is trying to do is simply define a new system of operations in which infinity can exist based on common logic.

[/quote']

 

But even that seemingly causes a mass of contradictions, considering that you are mixing the concept of infinity with real numbers.

 

1/inf. = 0

 

So, multiply each side by inf.

 

1 = 0

 

Since we are mixing infinity, a concept, with real numbers, 1 cannot equal 0. Meaning that 1/inf. cannot equal 0.

 

But I've already shown that in the graph of 1/x, when x approaches infinity, y approaches 0. So in the world of infinity, when x equals inf., y equals 0.

 

Again, poor definitions and conflicting constructions are presenting problems here.

 

For example, the phrase "x approaches infinity" is used on the real numbers, but infinity is not itself an element of the real numbers. So how can x approach it? Here's how: we don't really mean x is approaching infinity. We mean "x becomes arbitrarily large", and say "x approaches infinity" as a shorthand.

 

Furthermore, you are assuming that, just because the limit as x approaches a of y is b, the value of y at x = a must be b, here taking a to be infinity and b to be 0. However, this is not necessarily the case even when infinity is not involved.

 

The 1=0 argument, again, comes from poor definitions. It relies on the fact that 0x = 0 for all x, but how do you know that that is true? That is a theorem, not an axiom - and the proof of that theorem relies on the axiom that x + -x = 0, which we already know causes problems for infinite values (which, in turn, causes the axiom regarding additive inverses to break down, since that axiom demands INF-INF=0).

 

Maybe we can define 1 and 0 differently? I'm kinda confused too.

 

If you redefine 1 and 0' date=' they are no longer 1 and 0; they are instead new objects bearing the same name.

 

The standard definitions of 1 and 0 in any field come from the field axioms. One axiom states that there exists at least one element "1" such that 1[i']x[/i] = x for all x in the field - in other words, at least one multiplicative identity exists. Another axiom states that there exists at least one element "0" such that x + 0 = x for all x in the field - in other words, at least one additive identity exists. It is also known that at least one "0" and at least one "1" are distinct - both roles are not uniquely held by the same single element. (In many fields, including the real numbers, one can then prove that there is only one 1 and only one 0.)

[Dark]

Inf times a number > or = to 1 equals inf.

Inf times a number < or = to -1 equals negative inf.

 

Are you saying there are no numbers between 1 and 0?

 

I consider the multiplication of fractions the same as the division of whole/fractional (greater than 1, 3/2 for example) numbers.

 

EDIT: Glad Crabby came in here, cause he is clearing a shitload of stuff up for me.

[Womi]

1/inf = 0.inf

[CrabHelmet]

Inf times a number > or = to 1 equals inf.

Inf times a number < or = to -1 equals negative inf.

 

Are you saying there are no numbers between 1 and 0?

 

I consider the multiplication of fractions the same as the division of whole/fractional (greater than 1' date=' 3/2 for example) numbers.

[/quote']

 

Not every real number can be expressed as a fraction. That's the whole point of distinguishing between the real numbers and the rational numbers, a smaller set consisting of ratios of integers - in other words, fractions whose numerator and denominator are both integers. (Note that integers are rational numbers, since they can all be written with themselves as the numerator and 1 as the denominator.)

 

Incidentally, another bijection shows that there are exactly as many rational numbers as natural numbers.

 

1/inf = 0.inf

 

"0.inf"? That's just nonsense.