If |u + iv| = 3, then u + iv is an arbitrary point on a circle of radius 3, centered at the origin in the complex plane. What effect does subtracting 3 from u + iv have on this circle?ivan_x3000 said:Homework Statement
Suppose that u and v are real numbers for which u + iv has modulus 3. Express the imaginary part of (u + iv)−3 in terms of a polynomial in v.Homework Equations
The Attempt at a Solution
|u+iv|=3 then sort(u^2+i^2) = 3 then
u = 3 and v=0 or u=0 and v=3(0+3i)^-3
i swear i am missing something with these equations we've only being doing a topic on roots but wow it's different what I've been seeing in class
Those are two solutions, but there are infinitely many more.ivan_x3000 said:Homework Statement
Suppose that u and v are real numbers for which u + iv has modulus 3. Express the imaginary part of (u + iv)−3 in terms of a polynomial in v.
Homework Equations
The Attempt at a Solution
|u+iv|=3 then sort(u^2+i^2) = 3 then
u = 3 and v=0 or u=0 and v=3
The way I'm interpreting the problem, u + iv - 3 doesn't represent just a single point.jbunniii said:Those are two solutions, but there are infinitely many more.
By the way, is there a typo? The imaginary part of ##(u+iv) - 3## is simply ##v##, which is already a polynomial in ##v##.
Mark44 said:The way I'm interpreting the problem, u + iv - 3 doesn't represent just a single point.
In the OP you changed it to (u + iv)-3 and above you have (u + iv)3. Which is it?ivan_x3000 said:Yes definitely a type it was meant to be (u+iv)^3
krnysus said:@jbunniii can you please explain?
So (u + iv)^-1 = (u-iv)/9 then how do i solve it?
Ray Vickson said:Can you really not see how to get ##(u + i v)^{-3}## if you know ##(u + iv)^{-1}##?
Polynomials are algebraic expressions that consist of variables and coefficients, combined using the operations of addition, subtraction, and multiplication. They can also include exponents, but the variables must have whole number exponents.
The degree of a polynomial is the highest exponent of the variable in the expression. For example, the degree of the polynomial 5x3 + 2x2 + 7x + 1 is 3, since that is the highest exponent of x.
To add or subtract polynomials, you must combine like terms. Like terms are terms that have the same variable with the same exponent. You can add or subtract the coefficients of these like terms, while keeping the variable and exponent the same.
Complex numbers are numbers that are expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is equal to the square root of -1. These numbers allow us to work with and solve certain equations that cannot be solved using only real numbers.
To multiply complex numbers, you can use the FOIL method, just as you would for binomials. To divide complex numbers, you must first rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator. Then, you can simplify the resulting expression.