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.

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
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!


Product Rule

where a 0, b≥ 0
a ≥ 0, b < 0
a < 0, b < 0

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.
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)

divided dash

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.

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!


divided dash

Properties of Complex Numbers:
(Specifically those complex numbers involving "i ".)

bullet1. Conjugates: When we worked with radicals and with binomials in general, we discovered expressions referred to as "conjugates". A binomial expression, when multiplied by its conjugate, results in the difference of the squares of each term (with the resulting "middle terms" dropping away). This "dropping away" proved very useful when working with radicals, and became our strategy for rationalizing denominators. In a similar fashion, when a complex number is multiplied by its conjugate, the "middle terms" will drop away giving us a purely real number as the product. This is yet another strategy which will prove useful.

conjugates list

a + bi) • (a - bi) = a2 + b2
(a + bi)•(a - bi) = a2 - abi + abi - b2i2 = a2 - b2i2 = a2 - b2(-1) = a2 + b2
The squares of real numbers are positive.
The sum of the squares of two real numbers creates another real number.
Therefore, a2 + b2 is a positive real number, and we have the following rule:

Rule: The product of a complex number and its conjugate is a positive Real Number.


bullet2. Equality: Determining if two complex numbers are equal is exactly as you would think:

a + bi = c + di when a = c and b = d

Rule: Two complex numbers are equal if and only if their real parts and their imaginary parts are respectively equal.


bullet3. Closure: The complex numbers are closed under addition, subtraction. multiplication and division - when not considering division by zero.

Rule: If two complex numbers are added, the sum is a complex number.

Rule: If two complex numbers are multiplied, the product is a complex number.


bullet4. Commutative, Associative, Distributive Properties:
All complex numbers are commutative, associative, and multiplication distributes over addition.

Commutative: For all complex numbers z1 and z2,
 z1 + z2 = z2 + z1 (under addition)
Ex. (3 + 2i) + (5 - 4i) = (5 - 4i) + (3 + 2i)
z1• z2 = z2 z1 (under multiplication)
Ex. (3 + 2i) • (5 - 4i) = (5 - 4i) • (3 + 2i)

Associative: For all complex numbers z1, z2, z3,
z1 + (z2 + z3) = (z1 + z2) + z3 (under addition)
Ex. (7 + 5i) + ((3 + 2i) + (5 - 4i)) = ((7 + 5i) + (3 + 2i)) + (5 - 4i)
1• (z2 z3) = (z1 z2) • z3 (under multiplication)
Ex. (7 + 5i) + ((3 + 2i) + (5 - 4i)) = ((7 + 5i) + (3 + 2i)) + (5 - 4i)

Complex multiplication distributes over addition:
For all z1, z2, z3,     z1• (z2 + z3) = z1z2 + z1z3
Ex. (7 + 5i) ((3 + 2i) + (5 - 4i)) = (7 + 5i)• (3 + 2i)) + (7 + 5i)•(5 - 4i)


bullet5. Arithmetic Operations: For detailed information regarding the arithmetic of complex numbers see:
Arithmetic of Complex Numbers (Add, Subtract, Multiply) and
Arithmetic of Complex Numbers (Divide).


bullet6. Tidbit of Info:

Rule: The square of any complex number and the square of its conjugate are also conjugates.



Does anyone ever really use complex numbers?

Complex numbers offer a means of finding the roots of polynomials, and polynomials are used as theoretical models in a variety of fields. Consequently, complex numbers enjoy prominence in several specialized areas, including general engineering, electrical engineering and quantum mechanics.  Such topics as electrical current, the design of dynamos and electric motors, liquid flow in relation to obstacles, wavelength, analysis of stress on beams, the study of resonance of structures, the movement of shock absorbers in cars, and the manipulation of large matrices used in modeling utilize complex numbers. 

For example, electrical engineers discovered, when studying alternating current (AC) circuits, that the quantities of voltage, current and resistance, called impedance in AC, were not the familiar one-dimensional scalar quantities (regular numbers) that are used when measuring direct current (DC) circuits. These quantities which now alternate in direction and amplitude possess other dimensions (frequency and phase shift) that must be taken into account. In order to represent the two dimensions of frequency and phase shift, complex numbers were used with j representing the imaginary unit i to avoid confusion with the symbol for electric current which is I.
The form a + jb is used with the formula E = I • Z, where E is voltage (volts), I is current (amps) and Z is impedance (ohms).

AC circuit components such as resistors, inductors and capacitors all oppose the flow of current. The opposition to current is referred to as resistance for resistors and reactance for inductors and capacitors. The total opposition to current flow in a circuit is called impedance, Z, measured in ohms, ohms.

(Note in the table below that impedance referred to as resistance is represented with a Real Number while
impedance referred to as reactance is represented with an Imaginary Number. Impedance from Inductors is positive, and from Capacitors is negative.)

Circuit Component
Impedance (Z)
resistor(5 ohms)
Z = 5
inductor(6 ohms)
Z = 6i
capacitor(7 ohms)
Z = -7i

In a series circuit, the impedance is the sum of the impedances for the individual components.
In a parallel circuit, there is more than one pathway through which the current can flow. To find the total impedance, ZT, first calculate the impedances Z1 and Z2 of the pathways separately, by treating each pathway as a series circuit.
Then apply the formula zformula.

You will put this information to use on the Complex Numbers and Electricity Practice page.


How to use your
calculator with
complex numbers.
Click here.


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