 Posers and Puzzles

1. 18 May '05 22:00
I have often been intrigued by the concept of the number i. It is not part of the negative, positive, limitless/irrational, or nuetral numbers. Therefore I see it as a different root from 0 (there is 0, then there are the positive numbers, the negative numbers, (both which root from 0 (ie 0 is the limit))and then there's i and similar number groups). other roots from 0 besides i are cuberoot(-1) ithroot(i) etc. what do you guys think? Anybody have something to say? Maybe add something they think is very peculiar about mathematics? I realize the thread name isn't very accurate... oh well.
2. 18 May '05 22:21
Originally posted by bobbob1056th
I have often been intrigued by the concept of the number i. It is not part of the negative, positive, limitless/irrational, or nuetral numbers. Therefore I see it as a different root from 0 (there is 0, then there are the positive numbers, the negative numbers, (both which root from 0 (ie 0 is the limit))and then there's i and similar number groups). ...[text shortened]... is very peculiar about mathematics? I realize the thread name isn't very accurate... oh well.
I am also intrigued by the number i. A good way to visualize any complex number A + Bi, where A and B are both real, is to set-up the 'complex plane' where A is plotted on the x-axis, and B is plotted on the y-axis. Then any point (A,B) on the complex plane corresponds to the complex number A + Bi. Additionally, if you draw a vector from the origin of the plane to this point (A,B), then this vector is also a very useful construct. The magnitude of the vector is just (A^2 + B^2)^0.5, and the angle the vector makes with the x-axis is InverseTan(B/A). Thus this is a vector representation of the number A + Bi.

Using Euler's formula you can show that i raised to the i power (i^i) is actually purely real and is just exp(-Pi/2) = 0.2078795764....... Very peculiar results IMO.
3. 19 May '05 01:17
I wouldn't say "i" is any more strange than a negative number considering that both are equally imaginary as neither actually exist, both are just tools invented for maths. Zero is a bit odd too really...
4. 19 May '05 01:35
But functions with i and "normal" numbers are much stranger than functions with negative numbers, or zero, and "normal" numbers.
5. 19 May '05 02:05
Originally posted by davegage
I am also intrigued by the number i. A good way to visualize any complex number A + Bi, where A and B are both real, is to set-up the 'complex plane' where A is plotted on the x-axis, and B is plotted on the y-axis. Then any point (A,B) on the complex plane corresponds to the complex number A + Bi. Additionally, if you draw a vector from the origin of ...[text shortened]... s actually purely real and is just exp(-Pi/2) = 0.2078795764....... Very peculiar results IMO.
Along those lines, another strange result from Euler's formula is the following:

e^(i*pi) + 1 = 0

This equation contains 5 of the most important/useful/unique constants in mathematics. I think that's <buck> weird.
6. 19 May '05 02:151 edit
Originally posted by nickhawker
I wouldn't say "i" is any more strange than a negative number considering that both are equally imaginary as neither actually exist, both are just tools invented for maths. Zero is a bit odd too really...
I think "i" is a bit of a weird number. I'm not 100% positive, but I'm pretty sure it was invented to solve an equation involving the square root of -1. Before "i", that equation didn't make sense. "i" was like a big mathematical band-aid, almost a cheat in a sense.

Q "Rene, come off it, it just doesn't make sense!"

R "I know, but pretend it does..."

Q "But it doesn't, you back-pedalling old fro-"

R "Well, just pretend for now, OK? Imagine it makes sense. I know, I'll call it a...a...magic number! No, too gauche. How about an imaginary number?"

R "Well, let's see how I likes being in a equation, shall we?

(-1)^0.5 = i

There. Now "i" makes sense, don't "i"? Oui oui! Une jeste!"

Q "You can take your imaginary friend and shove it. I'm out of here. If anyone needs me, I'll be at the bar."

R "Well, "i" will see you there, won't "i"? Hee hee!"

Q "Shut up."

And thus math was fixed forever.
FIN
7. 19 May '05 04:11
i is by definition an 'imaginary' number, useful for solving certain kinds of equations.

I don't why on earth one of the posters thinks that NEGATIVE numbers don't exist, though.
8. 19 May '05 05:02
Originally posted by orfeo
i is by definition an 'imaginary' number, useful for solving certain kinds of equations.

I don't why on earth one of the posters thinks that NEGATIVE numbers don't exist, though.
That reminds me of a joke I saw somewhere (might have been here at RHP).

Two men, one of which happens to be a mathematician, are sitting in a park, watching a house across the road. At some point, someone enters the house, and a few minutes later, two people leave. Says the mathematician, "If another person should now enter the house, it will be empty."

9. 19 May '05 08:43
Originally posted by bobbob1056th
But functions with i and "normal" numbers are much stranger than functions with negative numbers, or zero, and "normal" numbers.
Actually, real functions are much harder to study than complex functions (= functions defined on the complex plane).

When dealing with reals you notice you cannot always take the root of a number. sqrt(-5) does not excist for example.

Mathematicians wondered if there was a number system in wich you could always take the root. They defined i as sqrt(-1) and went to see if R[i] met all the beautifull properties a number system should have.

It did, and R[i] was named C, the complex plane.

By the way... the cube root of -1 is simply -1, the cube root of i is -i
10. 19 May '05 08:46
i, like many things in mathematics, is a tool. Don't think of it in terms of simple reality.

In fact, negative numbers are a perfect analogy - in many real world situations, negative numbers have absolutely no meaning, but in others, they are as valid as positive numbers. The same applies to complex numbers. The discipline of electrical engineering wouldn't exist as we know it today if it weren't for i.
11. 19 May '05 10:48
Originally posted by orfeo
i is by definition an 'imaginary' number, useful for solving certain kinds of equations.

I don't why on earth one of the posters thinks that NEGATIVE numbers don't exist, though.
Give me an example of a negative number existing in reality then...
12. 19 May '05 11:13
Originally posted by nickhawker
Give me an example of a negative number existing in reality then...
Last time I checked, my bank balance 😕
13. 19 May '05 12:57
Originally posted by Alcra
Two men, one of which happens to be a mathematician, are sitting in a park, watching a house across the road. At some point, someone enters the house, and a few minutes later, two people leave. Says the mathematician, "If another person should now enter the house, it will be empty."

It's true. I've seen it happen.
14. 19 May '05 13:13
Originally posted by Bowmann
It's true. I've seen it happen.
Why were you sitting in a park with a mathematician?
15. 19 May '05 14:24
Originally posted by THUDandBLUNDER
Why were you sitting in a park with a mathematician?
I think Bowmann was distracting the guy while his wingman slept with the guy's wife.