Final answer:
Real numbers can be represented on a number line, complex numbers have a real and an imaginary part, pure imaginary numbers have only an imaginary part with a zero real part, and nonreal complex numbers have both nonzero real and imaginary parts.
Step-by-step explanation:
To differentiate between real, complex, pure imaginary, and nonreal complex numbers, let's first define each type:
- Real Numbers are the set of numbers that can be represented on the number line, including both positive and negative numbers, as well as zero.
- Complex Numbers are numbers that have both a real part and an imaginary part, and they are written in the form a + bi, where a and b are real numbers, and i is the imaginary unit (i2 = -1).
- Pure Imaginary Numbers are a subset of complex numbers where the real part is zero and the imaginary part is non-zero. They are written as bi, where b is a real number.
- Nonreal Complex Numbers are complex numbers that have both a nonzero real part and a nonzero imaginary part.
For example, if A is a complex number with real and imaginary parts (a + ib), then when multiplied by its complex conjugate (a - ib), the product A* A = (a + ib) (a - ib) = a2 + b2. This operation eliminates the imaginary part, leaving a real number result.