# May be a stupid question, but bothering me for long time

## Main Question or Discussion Point

When thinking of what's the philosophy behind the "magic" complex i, i*i=-1, this seems the connection between real world and imaginary world, but what this imaginary world stands for? why with its help, many functions get resolved? and we can use the Euler formula, which is the basic of many other formulas..
Who found i? how did he find it? what kind of problems he tried to solve lead to the discovering of i?

## Answers and Replies

just go to wikipedia and look up complex numbers. you will get an entire history of i.

arildno
Homework Helper
Gold Member
Dearly Missed
Imaginary numbers were noted as occasional intermediate numbers when extracting cube roots in the 16th century.

mathwonk
Homework Helper
what if i call i the purple number? does that make it purple? think about it.

as to its history, it seems to have been nopticed that if one assumes X^2 +1 = 0 has a solution and gives it a name, that oner can then express the answer to many other problems using it.

More interestingly, those answers sometimes elqad to easier solutions of problems which in the end do not involve i. Thsi may have been what really made it popular. you can bring it in, use it, get your answer, and take it back out again.

then eventully it began to be realized there was nothing imaginary about it, or nothing more imaginary than 0, or -1, or sqrt(2) say, which were also pretty revolutionary in their day.

HallsofIvy
Homework Helper
Mathwonk's response is best: "imaginary" is just a label. i is no more "imaginary" than 1.

thanks a lot, but

I was told this is an imaginary label the first day I learned it, but any label has its corresponding meaning in real world, eg: 1 means one object, 0 means no object, -1 means taking away one object, while this i, it says: to make the x^2+1=0 get solved, we imagine i, then tell me what x^2+1=0 stands for?

chroot
Staff Emeritus
Gold Member
There is absolutely no reason to require that anything in mathematics have any physical meaning in the "real world." Sorry, that's just nonsense. Mathematics exists in only one place: inside our own heads.

- Warren

As I believe that the maths exists before mankind discovered it, and the complex number is used in many fields, such as control theory, signal analysis... I doubt it only exist in human brain, here is a description I copy from http://www.answers.com/topic/complex-number
The words "real" and "imaginary" were meaningful when complex numbers were used mainly as an aid in manipulating "real" numbers, with only the "real" part directly describing the world. Later applications, and especially the discovery of quantum mechanics, showed that nature has no preference for "real" numbers and its most real descriptions often require complex numbers, the "imaginary" part being just as physical as the "real" part.
When looking at Euler formula, I can't help thinking how can it be so delicate, if i is only inside human being's head, if human in history not thinking in this way, how can another form of Euler formulas look like?

Gib Z
Homework Helper
The mathematics is there to be discovered. If we find out it has an application, well thats unrelated. It exists only in our minds, and thats one of the reasons why its beautiful.

The mathematics is there to be discovered.
Or is it only there after it's discovered?

matt grime
Homework Helper
Whether you're platonist or not, it is almost certainly the case that for any person, the symbol i represents something as real as the symbol 1. And no, 1 doesn't mean 1 object. Something that is 1 object displays oneness. That does not mean an object is the number one.

chroot
Staff Emeritus
Gold Member
To repeat matt's response in a more prosaic way: keep in mind that the number two is not "physical" either. Certainly, no one ever walks out their front door and stubs their toe on the number two. No one ever sees the number the two falling from the sky in a rainstorm.

The number two, the concept itself, exists only in the mind. In the same respect, the number i exists only in the mind. The two are thus on equal footing, even if one happens to bear the unfortunate name "imaginary."

- Warren

HallsofIvy
Homework Helper
I remember reading a review of Hawking's "A Brief History of Time". The reviewer said he could follow it until the last half when it started using negative numbers. I did a "double take"- at first I read it as "imaginary numbers" because I couldn't imagine a modern person having trouble with negative numbers.

The ancient Greeks, as good as they were with geometry, couldn't conceive of negative numbers- they had to be "defined" as a new kind of number. And, of course, the Pythagoreans were notoriously opposed to "irrational numbers". It wasn't until the 19th century that real numbers were properly defined. Complex numbers are just an expansion of real numbers. Nothing more "magical" about them than any other numbers.

mathwonk
Homework Helper
lets give him some physical experiment:

let squaring mean doubling the angle.

then an arrow located at 90 degrees to the x axis, when squared, is then located at 180 degrees to the x axis, i.e. at minus one!

so i is an arrow AT 90 DEGREES to the x axis.

hows them pomegranates?

or better, multiplication by i means rotation by 90 degrees. thus i^2 is rotation by 180 degrees, or multiplication by -1.

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Chris Hillman

Hi, christinaz,

When thinking of what's the philosophy behind the "magic" complex i, i*i=-1, this seems the connection between real world and imaginary world, but what this imaginary world stands for?
Using "philosophy", "magic", and "imaginary world" in the same sentence, especially the first sentence, can give mathtypes the unfortunate first impression that you are not "mathy". Fortunately, I think your real question was this:

why with its help, many functions get resolved? and we can use the Euler formula, which is the basic of many other formulas..
Lest we forget just how magical is the formula [itex]\exp(i \, \pi) = -1[/tex], I should explain some variant such as Steenrod twist algebras which are likely to be unfamiliar to many readers...

Who found i? how did he find it? what kind of problems he tried to solve lead to the discovering of i?
Some nice books which should answer some of these questions include:

Tignol, Galois's Theory of Algebraic Equations, World Scientific, 2001 (reprint).

van der Waerden, A History of Algebra, Springer, 1985.

I haven't seen this, but it was well reviewed at a publishing house associated with the National Academy of Science (of the U.S.A.) http://www.nap.edu/catalog/11540.html so it might be perfect for you:

Derbyshire, Unknown Quantity : a Real and Imaginary History of Algebra, Joseph Henry Press, 2006.

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To GibZ
The mathematics is there to be discovered. If we find out it has an application, well thats unrelated. It exists only in our minds, and thats one of the reasons why its beautiful.
I think maths is the abstract of the real world, it has a lot of relation with its applications, that's why its beautiful, it's not a mind toy, it's a powerful tool that we use to think and manipulate the real world, I'm always trying to find out the cast from the real world to labels in math or vise verse. Once I asked this i question of a friend, he said that this stands for an unexplainable world, then I started to think what this unexplainable world looks like? I also remember long time ago I read something like: if the 4th dimension exists, then people can find a way to disappear like Harry Porter's hiding gown.

To Christ
Thank you for your suggestion, you have very keen eyesight seeing that I'm not mathy, actually I'm very very unmathy.
I've glided roughly through the history of i, starting from Egypt to 18th century, and then the explaining of i from different standpoint, as what mathmonk has mention: taking i as rotating 90 degree, you know what's my feeling of all this explanation? I feel like before I know what a knife is, someone gives me one and told me: this is called knife, it's one member of the tool kit, you can use it to cut meat, or cut grass... how to "cut"? oh, let me show you...ok, now I know with a knife I can do cut, but why I can use the knife to cut, can I use it to peel an apple or do something else? I don't know.
Can any mathy ones here tell me of his feeling after reading or knowing the i history?
What's your reaction after you dial a telephone number and then hear something like: you are dialing an imaginary number, please rotate 90 degree and dial again? laughing or take it as normal?

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An explanation of Complex Numbers by Prof M. Alder at UWA. It's quite funny.

"There is a certain mysticism about complex numbers, much of which comes
from meeting them relatively late in life. The student has it explained that
there is an imaginary number, i which is the square root of −1 and this is
certainly imaginary because it is obvious to any fool that −1 doesn’t have a
square root. When much younger, you might have been troubled by 4−7 =
−3 which is just as obviously impossible and meaningless because you can’t
take seven from four. Since you met this when you were young and gullible
and the teacher assured you that −3 is a perfectly respectable number, just
not one which is used for counting apples, you went along with it and bought
the integers. Now, when your brains have nearly ossified you are asked to
believe in square roots of negative numbers and those poor little brains resist
the new idea with even more determination that when you first met negative
numbers.
The fact is all numbers are imaginary. The question is, can we find a use for
them? If so, we next work out what the rules are for messing around with
them. And that’s all there is to it. As long as we can devise some consistent
rules we are in business. Naturally, when we do it, all the practical people
scream ‘you can’t do that! It makes no sense!’. After a few years they get
used to it and take them for granted. I bet the first bloke who tried to sell
arabic notation to a banker had the hell of a time. Now try selling them the
Latin numerals as a sensible way to keep bank balances. Up to about 1400
everybody did, now they’d laugh at you.
It’s a silly old world and no mistake.
We got from the natural numbers (used for counting apples and coins and sheep and daughters, and other sorts of negotiable possessions) to the integers
by finding a use for negative numbers. It certainly meant that we could
now subtract any two natural numbers in any order, so made life simpler
in some respects. You could also subtract any two integers, so the problem
of subtraction was solved. The extra numbers could be used for various
things, essentially keeping track of debts and having a sort of direction to
counting. They didn’t seem to do any harm, so after regarding them with
grave suspicion and distrust for a generation or so, people gradually got used
to them.
We got from the integers to the rational numbers by finding that although you
could divide four by two you couldn’t divide two by four. So mathematicians
invented fractions, around five thousand years ago. While schoolies all around
were saying Five into three does not go, some bright spark said to himself But
what if it did? Then you could divide by pretty much anything, except zero.
Dividing by zero wasn’t the sort of thing practical folk felt safe with anyway,
so that was OK. And people found you could use these new-fangled numbers
for measuring things that didn’t, like sheep and daughters, naturally come
in lumps. Then the Greeks found (oh horror!) that the diagonal of the unit
square hasn’t got a length- not in Q anyway. The Pythagorean Society, which
was the first and last religious cult based on Mathematics, threatened to send
out the death squads to deal with anyone spreading this around. The saner
mathematicians just invented the real numbers (decimals) which of course,
to a Pythagorean, were evil and wicked and heretical and the product of the
imagination- and a diseased one at that.
So there is a long history of people being appalled and horrified by new numbers
and feeling that anyone spouting that sort of nonsense should be kicked
out if not stoned to death. Square root of minus one indeed! Whatever rubbish
will they come up with next? Telling us the world isn’t flat, I shouldn’t
wonder.
Meanwhile back in the mathematical world where we can make up whatever
we want, let’s approach it from a different angle.
The real line is just that– a line. We can do arithmetic with the points on
the line, adding and subtracting and multiplying and dividing any pair of
points so long as we don’t try to divide by zero.
Suppose you wanted to do the same on something else. Say a circle. Could
you do that? Could you make up some rules for adding and multiplying
points on a circle? Well you could certainly regard the points as angles from
some zero line and add them up. If you did it would be rather like the clock
arithmetic of the last chapter except you would be working mod 2 instead
of mod 12. You could also do multiplication the same way if you wanted.
The mathematician yawns. Boooooring! Nothing really new here.
Could you take R2 and make up a rule for adding and multiplying the points
in the plane? Well, you can certainly add and subtract them because this
is a well known vector space and we can always add vectors. Confusing the
vectors with little arrows and the points at the end of them with the arrows
themselves is fairly harmless."

If you want me to post the rest, let me know. Otherwise, that's all I will copy and paste since no-one will read very much.