Basic Technical Mathematics, 11th Edition

12.1 Basic Definitions

• Imaginary Unit • Rectangular Form of a Complex Number • Equality of Complex Numbers • Conjugate

Because the square of a positive number or a negative number is positive, it is not possible to square any real number and have a negative result. Therefore, we must define a number system to include square roots of negative numbers. We will find that such numbers can be used to great advantage in certain applications.

If the radicand in a square root is negative, we can express the indicated root as the product of $\sqrt{\hspace{0.17em}-\hspace{0.17em}1}$ and the square root of a positive number. The symbol $\sqrt{\hspace{0.17em}-\hspace{0.17em}1}$ is defined as the imaginary unit and is denoted by the symbol j. Therefore, we have

$$j\hspace{0.17em}=\hspace{0.17em}\sqrt{\hspace{0.17em}-\hspace{0.17em}1}\text{and}{j}^2\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}1$$ (12.1)

Generally, mathematicians use the symbol ...