What are imaginary numbers? • Hacker News Board

Re: What are imaginary numbers?

#141

Post by esrauch » Mon, Sep 24, 2012, 1:08 AM UTC

gizmo686 wrote: post #125

esrauch wrote: post #68

Earlier quoted context omitted.

Regarding 4, don't believe anyone who claims it's an intuitive result; it is simply a properly of real numbers that you cannot have nonzero infinitesimals and that any two distinct numbers have a number between them that is not equal to either (infinitely many in fact). You can construct alternate number lines that do allow nonzero infinitesimals and then .9999... actually is not equal to 1 under that number line; th…

There do exist number systems where .999.../=1, however, they are not a strict superset of the reals. If it were, then any operation involving only real numbers would behave identicly to the real number system. Also, this property is not a mere convention, but rather a nessasary result of what we want the number line to be. For example, assume that X<Y. Consider Z=(X+Y)/2. Z=X/2+Y/2. X<Z<Y. I have just shown, using b…

Thanks for clarifying the strict superset part, I'm not actually a mathematician.

To clarify my convention comment; it is only convention that mathematicians have decided that any given set of properties are useful or interesting to be used pervasively and alternate number systems are not. To any person who is asking why .9 repeating is 1 those reasons are entirely outside the scope of their knowledge and so entirely unrelated to the question; is it not true that the answer is "based on the axioms of which this number line is created due to complex reasons that you can't possibly know at this stage, this is effectively decided to be true as an axiom".

If one of these alternate number lines was in common use and reals were only uncommonly used then I'm pretty sure there would be tons of people saying "why is .99.. NOT equal to 1 in ($reals-replacement)" and all the people who have been taught it would sigh condescendingly because they believe it is just an intuitively obvious feature of numbers and not simply an axiom of the number line they are choosing to use. I'm interested in whether you disagree with that sentiment.

Re: your proof, trivially could be refused based on the other number line not being closed over division, then there would simply be no such number Z=(X+Y)/2 for X=.999.. and and Y=1. Then the proof where be like those silly ones where someone uses a /0 and arrives at a contradiction. Actually it seems that reals themselves already are not closed over division since x/0 doesn't result in a real.

Re: What are imaginary numbers?

#142

Post by gizmo686 » Mon, Sep 24, 2012, 3:12 AM UTC

esrauch wrote: post #141

gizmo686 wrote: post #125

Earlier quoted context omitted.

There do exist number systems where .999.../=1, however, they are not a strict superset of the reals. If it were, then any operation involving only real numbers would behave identicly to the real number system. Also, this property is not a mere convention, but rather a nessasary result of what we want the number line to be. For example, assume that X<Y. Consider Z=(X+Y)/2. Z=X/2+Y/2. X<Z<Y. I have just shown, using b…

Thanks for clarifying the strict superset part, I'm not actually a mathematician. To clarify my convention comment; it is only convention that mathematicians have decided that any given set of properties are useful or interesting to be used pervasively and alternate number systems are not. To any person who is asking why .9 repeating is 1 those reasons are entirely outside the scope of their knowledge and so entirely…

My point was that .999=1 is a direct result from the fact that the number line is continuas.

Imagine that you had a number system which does have holes in it (for example, the difference between .999... and 1). Now, take 2 object, one at mass .999..., and the other one of mass 1. In this hypothetical system, these objects would have a different mass. Now, take a third object, whose mass is between the 2 of them.

I consider it a critical property of our number system that we can be guaranteed to have a number representing the third mass.

As a historic note, the Greeks believed that any 2 numbers could be expressed as a multiple of some unit. For example 2 and .333... could both be expressed as multiples of 1/3. Eventually, they proved that this system could not accuratly moddel their world (the diagonal of a square for example).

I suspect that if we did have a number system such as you described, we would eventually discover that it did not work well enough for us, and switch to a continuas one.

Re: What are imaginary numbers?

#143

Post by john_gabriel » Mon, Sep 24, 2012, 1:27 PM UTC

stiff wrote: post #49

A complex "number" (don't think of it as of a number! think of it like you would think of a vector, group, ring or any other abstract structure) is just an ordered pair of real numbers that behaves in a certain predefined way when being added to another complex number or multiplied by it. For an introduction, to avoid unnecessary confusion, it is best to write such "numbers" as ordered pairs using the notation: (a,b)…

Complex numbers are ill-defined and real numbers don't exist either.

http://thenewcalculus.weebly.com