Question

Which of the following statements about imaginary numbers are true? The square root of -1 is an imaginary number. is an imaginary number. i is an imaginary number. If an imaginary number is combined with a real number, the result is a complex number. Imaginary numbers aren’t practical

Answer 1

All of those statements are true about imaginary numbers

Answer 2

Answer:

The square root of -1 is an imaginary number: TRUE.

i is an imaginary number: TRUE.

If an imaginary number is combined with a real number, the result is a complex number: TRUE.

Imaginary numbers aren't practical: FALSE.

Step-by-step explanation:

First Part: $\\ i = \sqrt{-1}$

Imaginary numbers were the result of finding solutions for equations that require taking the square roots of negative numbers:

$\\ \sqrt{-36} = \sqrt{-1*36} =\sqrt{-1} * \sqrt{36} = \sqrt{-1} *6 = 6i$.

Thus, the square root of -1 is an imaginary number, and so $\\ i$ since they are, in fact, the same: $\\ i = \sqrt{-1}$, which represents the imaginary unit. A consequence of that is $\\ i^{2} = -1$.

Second Part: a complex number is the result of combining an imaginary number with a real number.

A complex number is a quantity having both a real and an imaginary part, and are of the form $\\ a + ib$, since imaginary numbers cannot be represented on the axis of real numbers. They actually have their own axis called imaginary axis.

Both axes, then, form what is known as complex plane or Argand diagram, where complex numbers are represented.

In fact, a complex number with no real part is a pure imaginary number, and, conversely, a complex number with no imaginary number is a pure real number.  

Third Part: imaginary numbers are practical

Historically, Heaviside (1850-1925) "adapted complex numbers to the study of electrical circuits" [Wikipedia, 2019].

As a result, imaginary numbers are use to solve problems regarding electrical alternating current theory in Electrical Engineering, and they are represented by $\\ j$ instead of $\\ i$, since $\\ i$ is the symbol for electrical current.

There are many other applications on Quantum Mechanics, Dynamic Equations, signals analysis, control theory, fractal geometry, and so on.

Thus, imaginary numbers are use in practice to solve many problems in Science, Engineering, and Mathematics.