An imaginary number in a computer system is typically represented as a complex number, which comprises two components: the real part and the imaginary part.

More specifically, a complex number can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit that satisfies the equation $i^2 = -1$. In this notation, $a$ represents the real part, while $b$ denotes the imaginary part of the complex number.

In computer systems, complex numbers are commonly represented as pairs of floating-point numbers. The first number in the pair corresponds to the real part, and the second number corresponds to the imaginary part. For instance, the complex number $3 + 4i$ would be represented by the pair of floating-point values $(3.0, 4.0)$.

Most programming languages offer built-in support for complex numbers. For example, in Python, you can create a complex number using the ‘complex’ function as follows:

This statement creates a complex number with a real part of $3$ and an imaginary part of $4$. Alternatively, you can use the ‘j’ notation to define complex numbers, like this:

In C++, you can represent complex numbers using the ‘std::complex’ class from the Standard Library. To create a complex number, you would write:

This initializes a complex number with a real part of $3.0$ and an imaginary part of $4.0$.

In Java, the Apache Commons Math library provides a ‘Complex’ class for representing complex numbers. You can create a complex number in Java with the following code:

This also results in a complex number with a real part of $3.0$ and an imaginary part of $4.0$.

In all these examples, the real and imaginary parts of the complex number are stored as separate values in the computer’s memory. This representation allows the computer to perform various operations on complex numbers, such as addition, subtraction, multiplication, and division, in the same way it handles real numbers.