SOLUTION: What does it mean if your solution is a complex number? Explain what a complex solution means both algebraically and graphically.Thanks a ton!!
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Question 550693: What does it mean if your solution is a complex number? Explain what a complex solution means both algebraically and graphically.Thanks a ton!!
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
It simply means that replacing x or whatever variable with a complex number satisfies the equation. Graphically, this is more difficult since you would need 3D graphs or graphs with colors to represent values.
Here is an example of a well-known function, called the Riemann zeta function. It is often defined as
 = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + ...)
and uses a process called analytic continuation to extend the domain to complex numbers. If you copy/paste the link, you will see a graph of the zeta function to the right side:
http://en.wikipedia.org/wiki/Riemann_zeta_function
The Riemann Hypothesis claims that all "nontrivial" complex s satisfying  = 0)
have a real part of 1/2. Graphically, this means that all such solutions lie on a line. If you look at the graph, you will see that the black spots indeed appear to lie on that same line. Prove the Riemann Hypothesis, and you will be US$1,000,000 richer (seriously). Keep in mind that the problem was first posed over 150 years ago, and no one has solved it yet.
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