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I think I first learned imaginary numbers in algebra 2 which was like 6th or 7th grade. I know they use x^2=1 as an example, but I don't know how useful that is

I think I first learned imaginary numbers in algebra 2 which was like 6th or 7th grade. I know they use x^2=1 as an example, but I don't know how useful that is. I think they really need to teach trig before they teach imaginary number. They also need to teach polar coordinates with imaginary numbers. Most all of its uses are for solving trig problems. You use it in EE and ME to solve sinusoidal problems, but I don't know how i=sqrt(-1) helps you understand what its used for. I think euler formula is most elegant formula in math. How did the guy even come up with it.

gjm11 8y ago
I don't know how Euler came up with it, but here's a fairly intuitive way: think about circular motion. If something moves around a circle at constant speed, then its velocity vector is perpendicular to the vector from 0 to the moving thing. Once you have the idea (which isn't terribly difficult) that complex numbers live on a plane and multiplication by i is rotation through a right angle, this gives you the differential equation dx/dt = ix. Solving that is easy: x = exp(it). Since it's also easy (by definition of the trig functions) to see that x = cos t + i sin t, we're done. Given the sort of thing Euler was good at, though, it seems just as likely that he looked at the power series for sin, cos, and exp, and said "aha!".
- kevin_thibedeau 8y ago
  The 2D analogy and use of rotating vectors was alien to Euler and contemporaries. They really did think of it as an "imaginary" abstraction to make the math work.
skybrian 8y ago
I don't know how he came up with it, but I did some slides giving some geometric intuition about why it's the natural choice: http://slesinsky.org/brian/misc/eulers_identity.html http://slesinsky.org/brian/misc/eulers_identity.html
chess19 8y ago
There are two very good ways of understanding Euler's formula and one is the "circular motion" explanation given by another comment. Both are very similar. The other is that "multiplication of complex numbers is rotation" (which can be demonstrated purely by algebraic manipulation) and that "exponentiation is repeated multiplication". If we know what e^(ix) is then we also know what e^(2ix) is. It is the same "vector" as e^(ix) but the length of the vector will be squared and the angle it makes with the real axis will be doubled. It is trivial to differentiate exponents like a^(x) and we get that the derivative is simply a constant multiple of itself (depending only on "a"). We choose "e" to be the choice of real number that makes the constant 1. (We can also rigorously justify that such a choice of real number exists.) Now, what is the value of e^(ix) for very small positive values of x? It is approximately the value of e^(ix) at zero plus x times the value of the derivative at zero. (This is just the Taylor series.) In other words, for small x, e^(ix) is essentially 1 + ix except we know our answer should have magnitude 1 so we interpret e^(ix) as having magnitude 1 and angle x for small x. The properties of exponentiation as repeated multiplication and multiplication of complex numbers being rotation justifies interpreting e^(ix) as having magnitude 1 and angle x for all x. This is not very rigorous but it is the gist of the matter. Many tools in modern analysis were created to make arguments like this rigorous so this could definitely be considered a good way to understand complex exponentiation.

gjm11 8y ago

I don't know how Euler came up with it, but here's a fairly intuitive way: think about circular motion. If something moves around a circle at constant speed, then its velocity vector is perpendicular to the vector from 0 to the moving thing. Once you have the idea (which isn't terribly difficult) that complex numbers live on a plane and multiplication by i is rotation through a right angle, this gives you the differential equation dx/dt = ix. Solving that is easy: x = exp(it). Since it's also easy (by definition of the trig functions) to see that x = cos t + i sin t, we're done. Given the sort of thing Euler was good at, though, it seems just as likely that he looked at the power series for sin, cos, and exp, and said "aha!".

kevin_thibedeau 8y ago
The 2D analogy and use of rotating vectors was alien to Euler and contemporaries. They really did think of it as an "imaginary" abstraction to make the math work.

skybrian 8y ago

I don't know how he came up with it, but I did some slides giving some geometric intuition about why it's the natural choice: http://slesinsky.org/brian/misc/eulers_identity.html http://slesinsky.org/brian/misc/eulers_identity.html

chess19 8y ago

There are two very good ways of understanding Euler's formula and one is the "circular motion" explanation given by another comment. Both are very similar. The other is that "multiplication of complex numbers is rotation" (which can be demonstrated purely by algebraic manipulation) and that "exponentiation is repeated multiplication". If we know what e^(ix) is then we also know what e^(2ix) is. It is the same "vector" as e^(ix) but the length of the vector will be squared and the angle it makes with the real axis will be doubled. It is trivial to differentiate exponents like a^(x) and we get that the derivative is simply a constant multiple of itself (depending only on "a"). We choose "e" to be the choice of real number that makes the constant 1. (We can also rigorously justify that such a choice of real number exists.) Now, what is the value of e^(ix) for very small positive values of x? It is approximately the value of e^(ix) at zero plus x times the value of the derivative at zero. (This is just the Taylor series.) In other words, for small x, e^(ix) is essentially 1 + ix except we know our answer should have magnitude 1 so we interpret e^(ix) as having magnitude 1 and angle x for small x. The properties of exponentiation as repeated multiplication and multiplication of complex numbers being rotation justifies interpreting e^(ix) as having magnitude 1 and angle x for all x. This is not very rigorous but it is the gist of the matter. Many tools in modern analysis were created to make arguments like this rigorous so this could definitely be considered a good way to understand complex exponentiation.