**Imaginary Numbers**

Mathematicians symbolize the positive square root of —1, called the *unit imaginary number,* by using the lowercase italic letter *i* . Scientists and engineers symbolize it using the letter *j,* and henceforth, we will too.

Any imaginary number can be obtained by multiplying *j* by some real number *q* . The real number *q* is customarily written after *j* if *q* is positive or zero. If *q* happens to be a negative real number, then the absolute value of *q* is written after – *j* . Examples of imaginary numbers are *j* 3, – *j* 5, *j* 2.787, and – *j* π.

The set of imaginary numbers can be depicted along a number line, just as can the real numbers. The so-called *imaginary number line* is usually shown standing on end, that is, vertically (Fig. 6-3). In a sense, the real-number line and the imaginary-number line are fraternal twins. As is the case with human twins, these two number lines, although they share similarities, are independent. The sets of imaginary and real numbers have one value, 0, in common. Thus:

*j* 0 = 0

**Complex Numbers**

A *complex number* consists of the sum of some real number and some imaginary number. The general form for a complex number *k* is:

*k* = *p* + *jq*

where *p* and *q* are real numbers.

**Fig. 6-3.** The imaginary number line is just like the real number line, except that it is “stood on end” and all the quantities are multiplied by *j* .

Mathematicians, scientists, and engineers denote the set of complex numbers by placing the real-number and imaginary-number lines at right angles to each other, intersecting at the points on both lines corresponding to 0. The result is a rectangular coordinate plane (Fig. 6-4). Every point on this plane corresponds to a unique complex number, and every complex number corresponds to a unique point on the plane.

**Fig. 6-4.** The complex number plane portrays real numbers on the horizontal axis and imaginary numbers on the vertical axis.

Now that you know a little about complex numbers, you might want to examine the solution to the following equation again:

–3x ^{2} – 4x – 2 = 0

Recall that the solution, derived using the quadratic formula, contains the quantity (-8) ^{1/2} . An engineer or physicist would write this as *j* 8 ^{1/2} , so the solution to the quadratic is:

*x* = (4 ± *j* 8 ^{1/2} )/(-6)

This can be simplified to the standard form of a complex number, and then reduced to the lowest fractional form. Step-by-step, the simplification process goes like this:

*x =* (4 ± j8 ^{1/2} )/(—6)

= 4/6 ± *j* [8 ^{1/2} /(—6)]

= 2/3 ± *j* [2 x 2 ^{1/2} /(—6)]

= 2/3 ± *j* [2/(—6)x 2 ^{1/2} ]

= 2/3 ± *j* (—1/3 x 2 ^{1/2} )

= 2/3 ±[—j(1/3 x 2 ^{1/2} )]

= 2/3 ± *j* (1/3 x 2 ^{1/2} )

This might not look any “simpler” at first glance, but it’s good practice to state complex numbers in standard form, and reduced to lowest fractions. The last step, in which the minus sign disappears, is justified because adding a negative is the same thing as subtracting a positive, and subtracting a negative is the same thing as adding a positive.

The two complex solutions to the equation can be stated separately this way:

*x* = 2/3 + *j* (1/3 x 2 ^{1/2} )

or

*x* = 2/3 - *j* (1/3 x 2 ^{1/2} )

**Imaginary and Complex Numbers Practice Problems**

**Practice 1**

Solve the following equation using the quadratic formula:

*x* ^{2} + 9 = 0

**Solution 1**

In standard form showing all three coefficients *a* , *b* , and *c* , the equation looks like this:

1 *x* ^{2} + 0 *x* + 9 = 0

Thus the coefficients are:

*a* = 1

*b* = 0

*c* = 9

Plugging these numbers into the quadratic formula yields:

*x* = {-0 ± [0 ^{2} – (4 x 1 x 9)] ^{1/2} }/(2 x 1)

= ± (-36) ^{1/2} /2

= ± *j* 6/2

= ± *j* 3

**Practice 2**

Write out the quadratic equation from the preceding problem in factored form.

**Solution 2**

This would be tricky if we didn’t already know the solutions. But we do, so it’s easy:

( *x* + *j* 3)( *x* - *j* 3) = 0

You can verify that this works by “plugging in” the solutions derived from the quadratic formula in the previous problem.

Find practice problems and solutions for these concepts at: Quadratic, Cubic and Higher-Order Equations Practice Test.

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