by Ron Kurtus (updated 18 January 2022)
The number √−1 is considered an imaginary number since there is no number that multiplied by itself will equal −1.
When combined with a real number, it is called a complex number. You can multiply and divide complex numbers.
Just as with real numbers, there is a conjugate of a complex binary number.
Questions you may have include:
- What is an imaginary number?
- What is a complex number?
- What is a complex conjugate number?
This lesson will answer those questions.
An interesting property of the square root concerns the square root of −1.
The number √−1 is considered an imaginary number since there is no number multiplied by itself that equals −1. Imaginary numbers are usually designated by the letter i.
Raising i to various powers results in:
i2 = −1
i3 = i2*i = −i
i4 = (i2)(i2) = (−1)(−1) = +1
and so on.
Real numbers combined with imaginary numbers are called complex numbers. Examples of complex numbers include:
3 + 5i
Multiplying complex numbers
You can multiply complex number the same as you do with any polynomial.
In multiplying monomials:
(7i)(6i) = 42i2 = −42
In multiplying binomials, you can use the FOIL method:
(3 + 5i)(2 + 3i) =
3*2 + (3*3i + 5i*2) + 5i*3i =
6 + (9i + 10i) + 15I2 =
6 + 19i − 15 =
19i − 9 or − 9 + 19i
Dividing complex numbers
You can also divide complex numbers as you would divide polynomials.
Conjugate complex numbers
The conjugate of a binomial x + y is another binomial with one factor negated: x − y. A major feature of a conjugate is when you multiply the two expressions together, you get the difference of the squares of the terms:
(x + y)(x − y) = x2 − y2
The same holds for complex numbers:
The conjugate of 3 + 5i is 3 − 5i.
Thus (3 + 5i)(3 − 5i) = 9 − 25i2.
But since i2 = −1:
9 − 25i2 = 9 + 25 = 34
Another example is:
(a + bi)(a − bi) = a2 + b2
The number √−1 is defined as an imaginary number since no number that multiplied by itself will equal −1. When combined with a real number, it is called a complex number. You can multiply and divide complex numbers. Just as with real numbers, there is a conjugate of a complex binary number.
Follow your dreams
Resources and references
(Notice: The School for Champions may earn commissions from book purchases)
Share this page
Click on a button to bookmark or share this page through Twitter, Facebook, email, or other services:
Students and researchers
The Web address of this page is:
Please include it as a link on your website or as a reference in your report, document, or thesis.
Where are you now?