Through basic mathematics, we are taught that we could not evaluate the square root of a negative number because the square root of a negative number is not a real number. For example, the solution to x² = -2 is not a real number solution because the square of a real number is always positive. In fact, when trying to find the square root of a negative number on a calculator, you will get a "syntax error", or simply "error" message. Therefore, to solve such equations, the number "i" was created such that i² = -1. The imaginary number "i" is defined as √-1 = "i". The imaginary numbers are part of another number system known as the complex number system.

To solve the square root of a negative number or expression, we use the same rules for multiplying and dividing radicals. We write the square root of a negative as "i" times a real number. In general, for any real number a > 0, √-a = i√a.

Examples: Simplify each of the following radicals.

1. √-16 = √-1√16
= i∙4 (Replace √-1 with i)
= 4i

2. √-5 = √-1√5
= i√5 (Replace √-1 with i)
=√5i

Note that i√5 and √5i are the same. Generally, we write the imaginary number with the radical after the "i". The reason is because it's easy to confuse √(5i) and √5i.

3. √-24 = √-1√24
= √-1√4√6 (Break √24 down into √4√6)
= i∙2√6 (Simplify √4 and Replace √-1 with i)
=2i√6

4. √(-36/81) = √-36 /√81 (Rewrite as a division of two radicals)
= (√-1√36)/√81
= i√36/√81 (Replace √-1 with i)
= 2i/3 (Simplify 6/9 to 2/3)

In the previous examples, we showed how to simplify the square root of negative real numbers. We can extend this to simplifying the square root of monomials containing a negative real number and variables.

Examples:

1. √-49x²y⁴ = √-1 ∙√49∙√x²∙√y⁴ (Rewrite as product of radicals)
= i∙7xy² (Simplify all radicals and replace √-1 with i)
= 7i(xy²)

2. √-7x³y² = √-1∙√7∙√x³∙√y² (Rewrite as product of radicals)
= i∙√7∙x∙√x∙y (Simplify all radicals and replace √-1 with i)
= xyi√(7x)

We learned how to simplify the square root of negative numbers with the use of the imaginary number "i". Next we use the imaginary number to define complex numbers. A complex number is any number of the form a + bi where a and b are real numbers and √-1 = "i". Some examples are complex numbers are as follows:

4 - 13i, -12 + 27i, 3 - 3i√2 and (5/2) + (3/2)i.

We can add, subtract, multiply and divide complex numbers. To add and subtract complex numbers, combine the real parts and the imaginary parts.

Examples: Add or subtract the following complex numbers.
1. (3 + 4i) + (10 + 6i)
3 + 10 = 13 (Add the real parts)
(4 + 6)i = 10i (Add the imaginary parts)
34

Therefore, (3 + 4i) + (10 + 6i) = 13 + 10i.

2. (-14 + 3i) - (4 + 7i)
-14 - 4 = -18 (Subtract the real parts)
(3 - 7)i = -4i (Subtract the imaginary parts)

Therefore, (-14 + 3i) - (4 + 7i) = -18 - 4.

Recall the product rule when multiplying two positive real numbers. If a and b are positive real numbers, then √a√b = √(ab). When trying to apply the same rule to multiplying the square root of two negative numbers, we get a wrong result. For example, using the product rule for √-3√-27 gives us √(-3)(-27) = √81 = 9. But this is incorrect. What we need to do is write each radical in terms of "i". When doing this we get i√3 ∙ i√27 = (i2)√81 = (-1)(9) = -9.

Examples: Multiply the following complex numbers.

1. √-40√-10 = √-1√40√-1√10
= i∙√40∙i∙√10 (Replace √-1 with i)
= i2∙ √400 (Multiply the i's and use product rule for radicals to multiply √40√10)
= (-1)(20) (Replace i² with -1 and √400 with 20)
= -20

2. √-36√-9 = √-1√36√-1√9
= i∙6∙i∙3 (Simplify the radicals and replace √-1 with i)
= i2∙18 (Multiply the i's and multiply 6 and 3)
= (-1)(18) (Replace i2 with -1)
= -18

The distributive property of multiplication and FOIL that apply to real numbers also apply to complex numbers. Sometimes a problem is in the form (a - bi)(a + bi). These complex numbers are known as complex conjugates and are used in division where the denominator is a complex number. It's used because multiplying complex conjugates results in a real number and that removes the complex number from the denominator. It's the same idea as removing radicals from the denominator. Therefore, to divide complex numbers, multiply the numerator and the denominator by the complex conjugate.

This guide should help anyone having difficulty understanding imaginary numbers and how to perform mathematical operations on complex numbers.