Its all wrong...ALL OF IT...

[RQ]

As I'm not really sure with your virus theory, I do think that it's not "OK" to divide by zero.

[elpenmaster]

of course you can divide by zero. dividing by zero equals infinity. infinity is just the opposite of zero, i havent yet figured out why people say that you cant divide by it
the answer to the virus thing is that the virus cant kill anything if it doesnt exist because that would be an effect without a cause.

[RQ]

The thing is elpen, that infinity is as much opposite as it is 0. Can you count an object that is 0 in number? What about infinity? The virus will not kill because it never existed? With 0 time anything is possible. If you see a bicycle racer in a freezeframe, can you determine his/her speed from merely one picture? No. The answer is that it is undefined. I do believe it is 0, but it's not OK to use it as an application, therefore it's just a mathematical abstraction.

[Watters]

The idea behind dividing bye zero is that the answer existce in a place that can't be reached, It doesn't give you an aswer, it gives you a paradox that can be used in theoretical purposes. and it doens't give you infinity it gives you "I" the inaginary number.

[RQ]

How?

[Watters]

No clue. Not a dr. in math or physics. Just read a book called "My little book of infinity". It was writin by some mathimetition while he was in a mental isilum, he commited himself. It is all ideas, Like seeing how you can never reach infinity then you have no actual prof that two parrale lines might meet at a spot that can't be reached. that was the idea of one chapter. Just regurgitating stuff in my own words. Hell to me if i know how to solve thous math problems

[elpenmaster]

in your opinion, is it okay to divide by infinity, and get zero as an answer?

[chitspa]

I have a very very hard problem out there for all of you mathematicians.
Don't just give me the answer, but show me the work.

Here it is:

i=square root of -1

i^i=?



Good Luck!

[pat]

i^i=?

iⁱ = e^{i ln i}

As such, the first task is to determine ln i. And, we know that e^it = cos t + i sin t. We want an exponent it such that the right-hand side is i. That is all values of t for which sin t = 1. That is t = ( (^π/₂) + 2πk) for all integers k. This shows that the ln i = i ( (^π/₂) + 2πk) for all integers k.

Plopping that value (/those values) of ln i into e^{i ln i}, we end up with iⁱ = e<sup>-( (^π/₂) + 2πk) for all integers k.

[PWrong]

and it doens't give you infinity it gives you "I" the inaginary number.

"i" is the square root of minus 1

0/0 has infinite solutions. This doesn't mean the answer is infinity, it means there are infinitely many answers. Every real or complex number is a different answer.

Dividing anything else by zero has no solutions. It doesn't equal anything.


That i^i question is really interesting. My graphics calculator just returns 0.2078... (which equals e^(-pi / 2))

But I can see that the answer is actually going be a set of numbers. I can't wait to learn this stuff properly

[elpenmaster]

how about negative numbers? since 0 is halfway between 1 and -1, doesnt it have to have a value?
i see 0 as having the value of "none of anything"
the number line should be a circle: o at the bottom, infinity at the top, 1 halfway up at the right side of the circle, and -1 halfway up on the left side, directly opposite 1.
this would be based on multiplication rather than addition, because multiplication is a more "true" function (i dont know what that means but it makes sense) :wink: :?

[Keiji]

Let's just clear some things up here.

1. i^2 is -1, i^3 is -1i, i^4 is 1, i^5 is 1i, i^a is 1 if a is an even positive number greater than 3; or is 1i is a is an odd positive number greater than 3.

2. n/0 is +infinity and -infinity at the same time, if n does not equal 0.

3. 0/0 is every number.

Now, considering meanstotheend's theory that 2=1, I worked that out step-by-step and it works. It is exceedingly strange though. I'll have to ask my math teacher about that one after the hols.

[PWrong]

not quite right Bob.

1. i^6= -1, which contradicts your statement.

i^a, where "a" is a real number, isn't relevant to the question anyway.

You can draw the graph of i^t on an argand diagram. It's just a circle.
I did this on my TI-83 graphics calculator in parametric mode:

X1T=real(i^T)
Y1T=imag(i^T)

You can probably do the same thing with a computer program, or if you know more calculus than I do, you could just do it that way.

2. n/0 doesn't equal infinity.

infinity multiplied by 0 gives 0, just like everything else.

The correct answer is "there are no solutions to n/0, where n doesn't equal 0"
This means no solutions at all, not even infinity.

3. This one is correct. 0/0 is the set of all numbers, including real numbers, complex numbers and infinity.

The 2=1 proof is wrong, because it involves dividing by zero, which gives no solutions.

[Keiji]

1. i^6= -1, which contradicts your statement.

Yes, you are right. i^6 = i^4*i^2, which is -1.

2. n/0 doesn't equal infinity.

infinity multiplied by 0 gives 0, just like everything else.

The correct answer is "there are no solutions to n/0, where n doesn't equal 0"
This means no solutions at all, not even infinity.

Let me show you:

1/1 = 1; 1/-1 = -1.
1/0.1 = 10; 1/-0.1 = -10.
1/0.01 = 100; 1/-0.01 = -100.
1/0.0..._[r]...01 = 100..._[r]...0; 1/-0.0..._[r]...01 = -100..._[r]...0.
Thus 1/0 = +infinity, 1/-0 = -infinity. Since 0 and -0 are the same, 1/0 is both +infinity and -infinity at the same time.

[PWrong]

That's only the limit of 1/0
1/0 itself doesn't have a solution.

If 1/0 = infinity
Then 1 = infinity * 0
But anything * 0 = 0

so there's a contradiction

[Keiji]

n/0 = +infinity and -infinity at the same time as previously stated.
n*0 = 0 BUT infinity*0 can equal any number.

Infinity is a strange number, and it can often bend the rules of math. It's something that you just have to live with.

For example, infinity minus anything equals infinity. So infinity minus infinity equals infinity, which states that 0=1, which it is obviously not.

Another odd thing: Infinity multiplied by anything other than zero gives infinity as well. So infinity times infinity equals infinity. Now this is trying to say that infinity is 1, as if you do 1*1 you get 1. If infinity was one, then 1+1 must be 1, which is incorrect.

As I said before, it's a strange number. Don't kill your brain trying to think about it.

[elpenmaster]

to clear some things up (in my opinion):
infinity times 0 equals all numbers negative and positive
the reasoning for this is:
to deal with the number infinity, you must use a circular number line, with infinity at the top, zero at the bottom, -1 on the left, 1 on the right. -1 and 1 are both halfway up the circle. one quarter up the circle is 1/2, one eighth up the circle is 1/4, and so on. three quarters up the circle is2, seven eighths up the circle is 4, and so on. the placement of numbers is based on multiplication, not addition.
if you picture it this way, 1/2 x 2 =1, because 1/2 and 2 are equal distance from 1. any two numbers (if they are both negative or both positive) that are an eequal distance from 1 multiplied by each other =1
since 0 is at the bottom and infinity is at the top, and 1 is halfway up the number line, 0 and infinity are equal distances from 1. thus, infinityx0=1

since 0 times infinity equals 1, then logically 1/0 equals infinity

this all is logical if you use a circular number line.

[PWrong]

Actually, it does make more sense that way. I just have one question, and I think I've figured out the answer. It might even get us back on the subject of dimensions

If the real number line is a circle, what shape is the complex plane?

[pat]

If the real number line is a circle, what shape is the complex plane?

The real number line is not a circle. Positive and negative infinity are distinguishable. For some sorts of calculations, it can be convenient to consider it as a projection from a circle. Place a circle of unit diameter tangent to the real line at the origin. You can identify points on the real number line with points on the circle (minus the top point) by drawing a line from the top point of the circle to the point on the real line. That line will intersect the circle in one and only one place. You can use this technique to compress the whole real line into a finite range.

Similarly, the complex plane is a plane. There are a whole slew of distinguishable infinities. But, you can do the same sort of projection trick with a sphere now instead of a circle. It may give you a false sense that all of the infinities are really near each other though... so you have to be careful with this sort of projection.

[elpenmaster]

in what way are positive and negative infinity distinguishable?

[Keiji]

They are not. I have a theory which says numbers wrap at infinity, ie +infinity+1 = -infinity+1. (This doesn't mean that 6+6=-6+6, it only applies to infinity.)

I also think that the number line is a circle and I think your way of picturing it is the best way to do so.

The complex plane therefore would be a sphere. This means:

infinity*ni = -infinity*ni

whatever number n is.

[pat]

in what way are positive and negative infinity distinguishable?

Besides the fact that one is less than zero and the other is not?

Take the limit as x goes to infinity and then the limit as x goes to negative infinity of e^x or tanh x.

If one wants the number line to wrap around, then one has to have some reason for the discontinuity at infinity of functions like those. If one's just going to claim "well, it's because it's infinity" then they're going to have to explain what they mean by wrap around.

[elpenmaster]

i think that positive infinity and negative infinity are the same. my basis for believing this is that positive and negative 0 are the same thing, and that infinity is the opposite of 0. positive and negative do not apply to infinity, because they are related to addition/subtraction, and there is no way that you can sensibly use addition or subtraction on the circular number line, and the circular number line is the only number line that contains infinity. infinity does not exist on a linear number line. if you use a linear number line, you cant actually put infinity or negative infinity anywhere. you can only determine their location on a number line when they are actually on the number line. since they are not on a linear number line, you cant use the inherent seperatness of +/- infinity to determine their location. you can only determine their location using a number line in which they actually exist, and they only exist on a circular
numbr line. on a circular number line they are the same, so they have to be the same because they only exist on a circula numbe line. you can say that +/- infinity are different on a linear number line, but they dont exist on a linear number line, so this is meaningless!

[pat]

positive and negative do not apply to infinity, because they are related to addition/subtraction, and there is no way that you can sensibly use addition or subtraction on the circular number line

So, you've got your work cut out for you if you want me to accept a number system in which "there is no way that you can sensibly use addition or subtraction".

[pat]

you can say that +/- infinity are different on a linear number line, but they dont exist on a linear number line, so this is meaningless!

But, I can say that arbitrarily large and positive is different from arbitrarily large and negative. The only way a number circle can work is that infinity does exist... but is unreachable. That's hardly different than just not being there. Given a number circle, what is the limit as x goes to infinity of e^x? Is it zero or infinity? Or does it just not make sense somehow?

If the numbers are circular, then -5 = ∞ + ∞ - 5. So, 25 = -5 * -5 = 4∞² -10∞ + 25. This implies that 2∞² = 5∞. A similar argument works for any number. This means, we either cannot multiply by ∞ or we can't divide by it or add with it or at the very least we can't square it. What good is this number then?

[elpenmaster]

infinity on the circular number line works very much like zero. any number (besides 0) times infinity equals infinity, just as any number times 0 equals 0 in teh linear number line.
infinity in the circular number line is unreachable, but it does exist. zero on this number line is also unreachable. if you take 1 and try to get to zero, you must go: 1, 1/10, 1/100, 1/1000, . . . you can never get there. to try to get to infinity, you must go 1, 10, 100, 1000, 10000, . . . you can never get there either.
the traditional linear number line is "supermonoly biased". this means that it considers numbers that are greater than 1 to be more fundamental than numbers less than 1. this is where the confusion comes from
on a circular number line, 1/10 and 10 are truly opposite, because they re multiplacatively opposite, and the circular number line is based on multiplication rather than addition.

[PWrong]

The complex plane therefore would be a sphere. This means:

infinity*ni = -infinity*ni

whatever number n is.

I think the complex plane might be better off wrapped around a torus.

If a complex number a+bi lies on a sphere, and we know what b is, then the number lies on a circular "slice" of the sphere. But this circle gets smaller as b approaches the side of the sphere, i.e. as b approaches 1. By this logic, the circle doesn't exist when the imaginary component equals 1. This means that 1+i =2+i, which is obviously not true. Also, the number a+0i is on the same part of the sphere as a+ infinity i. This implies that 0i = infinityi, so 0=infinity. 0 and infinity may have similar properties, but they are not equal.

To solve this problem, we just place the real number circle (a+0i) next to (a+i), then (a+2i) and so on, creating a complex plane wrapped around an infinite cylinder. This is based on multiplication for the real part, and addition for the imaginary part. Then we start using multiplication for the imaginary part, bend the cylinder around and make a torus.

This is all assuming that the real number line is circular, of course, but I think there are just two different ways of looking at it.

In the linear number line, addition is possible, integers are important, prime numbers are difficult to understand. 0 and infinity are confusing limits.

As the circular number line is based on multiplication, prime numbers are the basic units, but the integers are more difficult to understand. 0 and infinity are less confusing, although 1 may be a limit of some kind, because it is neither prime nor composite.

So both are equally useful, and probably both correct.

[Keiji]

Addition, multiplication. There must be an exponential number line somewhere too.

[Watters]

This means, we either cannot multiply by ∞ or we can't divide by it or add with it or at the very least we can't square it. What good is this number then?

When you divid by infinity then then a number becomes zero. the rule of thumb is that if you can't fit anotehr deciman place inbetween a number then it is that number ie. 3.9 repeater is 4. it is the same idea with dividing by infinity. 1/10 = .1 1/100 .01 so forth on to infinity eventualy .0000000000......... = 0. Timesing by infinity doesn't realy matter you can't get a bigger number tehn the bigest possible numebr.

[Keiji]

Watters, we all know that anyway.