Complex Numbers: Real and Imaginary Parts

Once you meet complex numbers, a natural question follows: are they just a special notation, or can you actually compute with them the way you compute with everyday numbers? The short answer is that complex numbers are full participants in arithmetic. The usual arithmetic operations are supported for them, and the most common mathematical functions are supported too. This article explains what that support means in practice, how each operation is carried out, and where learners usually go wrong.

The two-part structure is the key to everything that follows. A complex number is not treated as a single opaque quantity. Instead, its real part and its imaginary part are kept distinct, and each operation is carried out by working with those parts. Real parts are combined with real parts, and imaginary parts are combined with imaginary parts. Keeping this two-part view in mind makes every operation in this article easier to follow.

Because the form a + bi includes ordinary numbers as a special case, the usual operations can be extended to the full set of complex numbers. A number whose imaginary part is zero is simply a real number written in complex form, so the extension does not discard anything familiar. The source states that complex numbers support the usual arithmetic operations, which is the reason they can be added, subtracted, multiplied, and divided.

Addition and subtraction are the most direct operations. For two complex numbers a + bi and c + di, you combine the real parts together and the imaginary parts together. The sum is found by adding the real parts and adding the imaginary parts, giving a single complex number in the same form.

Subtraction works the same way, with the second number's parts subtracted from the first. No product of imaginary units appears in either operation, so the calculation stays at the level of ordinary subtraction applied to each part.

Multiplication cannot be done one part at a time, because each part of the first number interacts with each part of the second. The method is to expand the product fully, as you would with brackets in algebra, and then replace i^{^2} with -1. The standard result has a real part and an imaginary part.

Division is handled by multiplying the top and bottom by the conjugate of the denominator. For a denominator c + di, the conjugate is c - di. This removes the imaginary unit from the denominator, leaving a real number there, so the result can be read off as a single real part and imaginary part.

The denominator simplifies because multiplying c + di by c - di produces c^{^2} + d^{^2} after the i^{^2} = -1 replacement. The imaginary terms cancel in the denominator, which is why the result has no imaginary unit there. The numerator is then expanded in the same way as any product.

Beyond the four arithmetic operations, the most common mathematical functions are also supported for complex numbers. This means that a complex value can be passed into a function, and the function's result is again interpreted as a complex quantity. The idea is the same as for arithmetic: the complex number is a legitimate input, not a special case that must be avoided.

The source introduces this support briefly and does not list the individual functions. The key point to take away is the general support rather than a particular catalogue of functions. When you meet a specific function, the question to ask is whether it accepts complex arguments and returns a complex result, and the general statement above tells you that the common functions do.

Most errors with complex numbers come from handling the parts incorrectly rather than from the operations themselves. The table below contrasts the correct handling of each step with the common error, so you can check your own work against it.