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You discovered, during your study of quadratic equations in Algebra 1, the existence of square roots of negative numbers. Such square roots are called "imaginary numbers".
See the Refresher page.

definition
The imaginary unit, i, is the number whose square is negative one.  
i 2 = -1
 

i 2 = -1     and     isquareroot
The imaginary unit i possess the unique property that when squared, the result is a negative value.
Consequently, when simplifying the square root of a negative number, an "i " becomes part of the answer.

Product Rule(s) for Square Roots
Up to this point, our work with square roots and Real numbers has involved only positive values under the radical. Thus, our Product Rule for Square Roots dealt only with positive Real number values.
Product Rule
radthm
where a 0, b≥ 0

But, when working with imaginary numbers, we need to extend our Product Rule for Square Roots to allow for ONE negative value under the radical.
Either "a" is negative OR "b" is negative, but NOT BOTH!

nottrue
trueG

Product Rule
(extended)

radthm
where a 0, b≥ 0
OR
a ≥ 0, b < 0
but
NOT
a < 0, b < 0


bulletnote
The secret to dealing with the square root of a negative value
is to deal with the i-part first!

Pull out the factor of -1 first, and then simplify the remaining portion of the square root.
simpex1
Be careful when you hand write your answer that "i " does not appear to be "under" the radical symbol.

You may see instances where the "
i " appears in front of a remaining radical symbol to avoid any confusion.
simprule (where a 0)

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definition
An imaginary number, is any number that contains the imaginary unit, i. It takes the form of a + bi where a and b are real numbers, but b ≠ 0.
Examples: 3i, -5i, πi, 6 + 3i, -2 - 4i, cnlist
When a = 0, the number may be referred to as purely imaginary, such as 3i, -5i, and πi.

When you put the Real Numbers together with the Imaginary Numbers,
you get the set of Complex Numbers.

definition
A complex number is a combination of a Real Number and an Imaginary Number, written as a + bi (where a and/or b may equal zero).
(a and b are real numbers and i is the imaginary unit)

• If we have a + bi with a = 0, we have 0 + bi which gives bi, a purely imaginary number.
• If we have a + bi with b = 0,  we have a + 0i which gives a, a real number.
In this manner, we can see that real numbers and imaginary numbers are also complex numbers.

In "a + bi ", the "a" is referred to as the "real part" of the complex number and
"b" is referred to as the "imaginary part" of the complex number with "b" being
referenced as the number of multiples of i.
Real Numbers: 3, 0, rad3, 0.125, π, -42.1, e, ... swirl Complex Numbers: 2 + 3i, -3 - i ,
8 + 0i, 0 + 4i, cnlist
(Includes All Reals and All Imaginaries)
Imaginary Numbers: i, -3i, 2 + i, πi, -5.1i, ...

Complex numbers are generally referenced by the letter "z" such as z = a + bi.

All Real Numbers are Complex Numbers!

complex

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ti84
How to use your
TI-83+/84+
calculator with
complex numbers.
Click here.

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