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.
Some of these tests require you to create new functions for the expr command.
Read the Tcl mathfunc
documentation to learn how.
Edit via GitHub
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