Math Tutoring on Complex Numbers
If we assume then . This lead to a new set of numbers called Imaginary Numbers and then the set of Complex Numbers.
Imaginary Numbers are those in the form , where ‘a’ is a negative real number.
Complex Numbers are those numbers written in the form , where ‘a’ and ‘b’ are real numbers and .
The expression is called Standard Form of a Complex Number. Therefore
(C) Complex: and (I) Imaginary:
Addition and Subtraction of Complex Numbers in Math Tutoring:
Multiplying Complex Numbers:
To divide Complex Numbers following steps should be taken:
1. Write the problem in fractional form.
2. Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.
As a Complex Number form has two components, we can graph a Complex Number similar to how we graph vectors, except we use the Complex Plane instead of the Coordinate Plane.
The Complex Number and angle θ can be given in trigonometric form:
, where ‘r’ represents the Magnitude of ‘z’ called Modulus and ‘θ’ is often called the Argument.
Every nonzero Complex Number has exactly ‘n’ distinct complex roots. For any positive integer ‘n’, the Complex Number is a nth root of the Complex Number if . We can extend De Moivre’s Theorem to find all nth roots of a Complex Number. De Moivre’s Theorem makes raising complex numbers to powers very simple. If is a Complex Number and if ‘n’ is any real number, then
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1. What are Complex Numbers?
2. What is the trigonometric form of Complex Numbers?
3. State and prove De Moivre’s theorem?
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