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What is the number of distinct solution of the equation = ?
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Let us write the number z in Trigonometry form: z = .
Then = , and your equation is
= , or, which is the same,
= 0.
The last equation deploys in two independent equations
1) r = 0, which means simply z = 0.
2) (r*(cos(2a) + i*sin(2a)) + 1 = 0, which is the same as
r*(cos(2a) + i*sin(2a) = -1.
The last equation implies
r = 1, cos(2a) = -1 and sin(2a) = 0, which, in turn, implies
r = 1 and {2a = pi OR 2a = }.
It means that the solutions are
a) r = 1, a = , OR/AND
b) r = 1, a = .
In the rectangular form these solutions are z = i OR/AND z = -i.
Answer. The original equation has three solutions: z = 0; z = i, and z = -i.
There is a bunch of lessons on complex numbers
- Complex numbers and arithmetical operations on them
- Complex plane
- Addition and subtraction of complex numbers in complex plane
- Multiplication and division of complex numbers in complex plane
- Raising a complex number to an integer power
- How to take a root of a complex number
- Solution of the quadratic equation with real coefficients on complex domain
- Solved problems on taking roots of complex numbers
- Solved problems on arithmetic operations on complex numbers
- Miscellaneous problems on complex numbers
- Advanced problem on complex numbers
- A curious example of an equation in complex numbers which HAS NO a solution
in this site.
Also, you have this free of charge online textbook in ALGEBRA-II in this site
- ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic "Complex numbers".