complex numbers?

Question

can someone explain "i" to me? i do not understand how comlpex numbers work at all. what do they stand for? what are they? help please?

Answer 1

At the lower levels, 'i' is simply a new number that satisfies i^2=-1. All the usual algebraic rules still apply, so it is possible to add, subtract, multiply, and divide (using conjugates). The question of consistency of this new number is never addressed. This is fixed at the upper levels, but the 'fix' is not an easy one. (If the next paragraph doesn'tmake sense to you, just ignore it.)

At the upper levels, we define the field of complex numbers, C, as
R[x]/(x^2+1). That is, the ring of polynomials over the real numbers moded out by the ideal generated by x^2+1. In this case, 'i' becomes the element x+(x^2+1) in the quotient. By the general theory, C becomes a field (so all algebraic operations work) and i^2+1=0, as needed. Also, every complex number can be written in the form a+bi where a and b are real numbers.

To ask what complex numbers signify is more complicated. If you consider real numbers to be points on a line, the complex numbers become points in a plane. Addition is defined using ideas from addition of vectors and multiplication turns out to correspond to rotations and stretches. To multiply two complex numbers, add the angles from the x-axis and multiply the distances from the origin. In this way, multiplication by 'i' corresponds to rotation by 90 degrees counterclockwise.

Answer 2

Different people tend to learn them in different ways, so I don't think it will be very useful to have an answer here, if the few answers above didn't help. You can try the sourced pages (below). Probably the best thing is to ask a Math teacher at your local high school, college or university. If your teacher doesn't explain it in a way you understand, don't be afraid to ask for more help, maybe from another teacher.

Answer 3

i, or sometimes j, is an imaginary number that is the square root of -1. It is essentially a hard to imagine place holder to allow more complex mathematical equations that will end up with -1 roots.

Answer 4

We want all algebraic equations to have solutions. So, what x solves
x^2+1=0?
x^2=-1
x = sqrt(-1)

Originally, we just call this i and use it like any other number except that i^2=-1. It works! Now all algebraic equations have solutions.

Answer 5

i is actually an imaginary number used to describe a problem likethe square root of -16 = 4i