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Complex Numbers
What are complex numbers?
The idea of complex numbers arose from considering questions like, "What is the solution
of x2 = -1?" This question was answered by defining the 'number' i to have the property i2 = -1.
A complex number is defined as having two components: an ordinary number, and an ordinary number that is implicitly multiplied by i. These are called the real part and the imaginary part, respectively.
From this perspective, the ordinary numbers may be considered to be complex numbers whose imaginary parts happen to be zero.
To add (or subtract) two complex numbers, simply add the real parts and the imaginary parts separately to obtain the real and imaginary parts of the result.
Multiplication of complex numbers is straightforward, too. Consider the two numbers to
be a + i.b and c + i.d. Expanding the product of these two expressions
gives a.c + a.i.d + i.b.c + i2.b.d. Using the definition i2=-1 and rearranging the terms gives the product as (a.c - b.d) + i.(a.d + b.c).
For division, start with (a + i.b)/(c + i.d). Multiply both numerator and denominator by c - i.d to make the new denominator's imaginary part 0. The result
is: (a.c + b.d)/(c2 + d2) + i.(b.c - a.d)/(c2 + d2).
One property that complex numbers do not have is the concept of ordering. For example, is 3 - 10*i greater than, or less than, 10 + 2*i? There is no useful answer to this question.
Nomenclature
Mathematicians choose to use the name i for the number defined by i2=-1. In the fields of electricity and electronics, the symbol i has traditionally represented electric current. To avoid the otherwise inevitable confusion, people in these fields choose to use j to name what mathematicians call i.
You can choose the name to use by clicking on the 'Options' button and selecting the preferred name. You can also specify that complex numbers are not to be used, in which case i and j are both treated as ordinary symbols with no special meaning.
Functions used with complex numbers
real(expr)
This function returns the 'real' part of a complex number. When applied to a non-complex number, it returns that number. In all other cases, it does nothing.
Examples:
real((2 + 3*i)/(5 + 7*i) returns 31/74, having first simplified its argument.
real(a + i*b) returns real(a + i*b) if a and b have no values.
imag(expr)
This function returns the 'imaginary' part of a complex number. When applied to a non-complex number, it returns 0. In all other cases, it does nothing.
Examples:
imag((2 + 3*i)/(5 + 7*i) returns 1/74, having first simplified its argument. Note that the multiplier of i is not included in the result.
imag(a + i*b) returns imag(a + i*b) if a and b have no values.
abs(expr)
When applied to a complex number, this function returns its magnitude, namely the square root of the sum of the squares of the real and imaginary parts. When used on some other kind of number, this function returns the number.
Example:
abs((2 + 3*i)/(5 + 7*i) returns (13/74)^(1/2).
conj(expr)
This function returns the complex conjugate of the expression, i.e. with its imaginary part is negated. Numbers other than complex numbers are unaffected.
Examples:
conj(2 + 3*i) returns 2 - i * 3.
conj((a + b*i) * (c + d*i)) returns (a - i * b) * (c - i * d).