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)
2. √-5 = √-1√5
= i√5 (Replace √-1 with i)
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)
4. √(-36/81) = √-36 /√81 (Rewrite as a division of two radicals)
= 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.
1. √-49x2y4 = √-1 ∙√49∙√x2∙√y4 (Rewrite as product of radicals)
= i∙7xy2 (Simplify all radicals and replace √-1 with i)
2. √-7x3y2 = √-1∙√7∙√x3∙√y2 (Rewrite as product of radicals)
= i∙√7∙x∙√x∙y (Simplify all radicals and replace √-1 with i)
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
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)
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)
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)
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.