A complex number is a number in the form a + b * i
where a
and b
are real and i
satisfies i^2 = -1
.
a
is called the real part and b
is called the imaginary part of z
.
The conjugate of the number a + b * i
is the number a - b * i
.
The absolute value of a complex number z = a + b * i
is a real number |z| = sqrt(a^2 + b^2)
. The square of the absolute value |z|^2
is the result of multiplication of z
by its complex conjugate.
The sum/difference of two complex numbers involves adding/subtracting their real and imaginary parts separately:
(a + i * b) + (c + i * d) = (a + c) + (b + d) * i
,
(a + i * b) - (c + i * d) = (a - c) + (b - d) * i
.
Multiplication result is by definition
(a + i * b) * (c + i * d) = (a * c - b * d) + (b * c + a * d) * i
.
The reciprocal of a non-zero complex number is
1 / (a + i * b) = a/(a^2 + b^2) - b/(a^2 + b^2) * i
.
Dividing a complex number a + i * b
by another c + i * d
gives:
(a + i * b) / (c + i * d) = (a * c + b * d)/(c^2 + d^2) + (b * c - a * d)/(c^2 + d^2) * i
.
Raising e to a complex exponent can be expressed as e^(a + i * b) = e^a * e^(i * b)
, the last term of which is given by Euler's formula e^(i * b) = cos(b) + i * sin(b)
.
Implement the following operations:
Assume the programming language you are using does not have an implementation of complex numbers.
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