Complex Numbers Flashcards
What is the formula for multiplication of complex numbers?
If (a + bi) and (c + di) are to be multiplied, then their resulting multiplication is a complex number of the form:
(ac - bd) + (ad + bc)i
What is the general formula of the powers of “i”?
Every power of i can be expressed as:
i4n + k, where k can be 0, 1, 2, or 3.
When k = 0, the result is 1,
When k = 1, the result is +i
When k = 2, the result is -1
When k = 3, the result is -i.
What is 1/i ?
1/i = -i, from surds.
What are the properties of the conjugate of complex numbers?
What is the polar form of complex numbers?
For z = a + bi, its polar form is—
z = r(cosθ + isinθ), where:
r = √(a² + b²)
a = rcosθ , b = rsinθ
tanθ = b/a
θ = argument of z
When are two Nonzero complex numbers in polar form equal?
They are equal only if their modulus and arguments are the same.
What range of values does the principal argument take?
-π < θ < π
What is the formula for the nth power of a complex number in polar form?
Zn = rn(cosnθ + isinnθ)
What is the formula for finding the nth root of a complex number (in polar form, though even in regular form you still have to convert to polar)?
Nth root – the bars don’t mean anything
Where k = n - 1
How does Euler express complex numbers?
re(ix) = r(cosx + isinx),
Where x = tan^-1(b/a), of a + bi;
r = √(a² + b²)
What is Euler’s magic formula?
eiπ = –1
