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Integrated math 2
Course: Integrated math 2 > Unit 5
Lesson 5: Multiplying complex numbersMultiplying complex numbers
Learn how to multiply two complex numbers. For example, multiply (1+2i)⋅(3+i).
A complex number is any number that can be written as
When multiplying complex numbers, it's useful to remember that the properties we use when performing arithmetic with real numbers work similarly for complex numbers.
Sometimes, thinking of
Multiplying a real number by a complex number
Example
Multiply
Solution
If your instinct tells you to distribute the
And that's it! We used the distributive property to multiply a real number by a complex number. Let's try something a little more complicated.
Multiplying a pure imaginary number by a complex number
Example
Multiply
Solution
Again, let's start by distributing the
At this point, the answer is not of the form
However, we know that
Using the commutative property, we can write the answer as
Check your understanding
Problem 1
Problem 2
Excellent! We're now ready to step it up even more! What follows is the more typical case that you'll see when you're asked to multiply complex numbers.
Multiplying two complex numbers
Example
Multiply
Solution
In this example, some find it very helpful to think of
In fact, the process of multiplying these two complex numbers is very similar to multiplying two binomials! Multiply each term in the first number by each term in the second number.
Since
Check your understanding
Problem 3
Problem 4
Problem 5
Problem 6
Challenge Problems
Problem 1
Problem 2
Want to join the conversation?
how do you make things so much easier than school does?(89 votes)
Sal explains the concepts rather than just gets you to memorise a method.(110 votes)
From what I understand, when added, multiplying, subtracting, and dividing, you think and act like the i is a variable. Is that correct?(31 votes)
"i" will act as the variable while you simplify the problem. But then you have to make sure to solve for "i" when possible. Like when you get i^2 or i^9 - make sure you solve for that instead of just leaving it as-is like you would a normal variable.(24 votes)
Is it necessary to practice some divisions on complex numbers? If so does it follow the same general division rule?(6 votes)
When dividing two complex numbers in rectangular form we multiply the numerator and denominator by the complex conjugate of the denominator, because this effectively turns the denominator into a real number and the numerator becomes a multiplication of two complex numbers, which we can simplify.
The complex conjugate of (𝑎 + 𝑏𝑖) is (𝑎 − 𝑏𝑖).
Example:
(2 − 16𝑖) ∕ (5 − 𝑖) =
= (2 − 16𝑖) ∙ (5 + 𝑖) ∕ ((5 − 𝑖) ∙ (5 + 𝑖)) =
= (10 + 2𝑖 − 80𝑖 + 16) ∕ (25 + 5𝑖 − 5𝑖 + 1) =
= (26 − 78𝑖) ∕ 26 =
= 1 − 3𝑖
– – –
Alternatively, we can let
(2 − 16𝑖) = (5 − 𝑖)(𝑎 + 𝑏𝑖) = 5𝑎 + 5𝑏𝑖 −𝑎𝑖 + 𝑏 = (5𝑎 + 𝑏) + (5𝑏 − 𝑎)𝑖
and then solve the system of equations
5𝑎 + 𝑏 = 2
5𝑏 − 𝑎 = −16
⇔
𝑏 = 2 − 5𝑎
5(2 − 5𝑎) − 𝑎 = 10 − 26𝑎 = −16
⇔
𝑎 = 1
𝑏 = −3(23 votes)
In challenge problem 2, isn't ( 1 + 3i )^2 equal to ( 1 + 9i^2 ) = ( 1 + 9*( -1 ) ) = ( 1 - 9 ) = -8?(2 votes)
No... to multiply ( 1 + 3i )^2, you must use FOIL.
It becomes 1 + 6i -9 = 6i - 8(30 votes)
- can I create my own complex numbers? like what if I created imaginary numbers called j(2 votes)
Good mathematical communication allows you to do any number of things. The convention is for imaginary numbers always to be represented by 'i'. However, if you began describing a mathematical model by saying, "Let 'j' equal the square root of -1 ..." everybody would understand what you meant. It can be just a variable, so long as you've defined it well, and it would continue to follow all the same rules of complex numbers.(3 votes)
Does any one know how to solve Challege Problems 1 and 2?? I'm a bit confused.(4 votes)
(a-bi)*(a+bi)
We multiply this like any other binomial: we apply the distributive property twice to get
(a-bi)*a +(a-bi)*bi
a^2-abi +abi -(b^2)*(i^2)
The middle terms cancel and we get a^2 -(b^2)*(i^2)
Remember i^2=-1 and we get a^2 -(b^2)*(-1)
a^2+b^2
For the second one, try expanding the squared term first, simplifying it, then multiplying the second term.(6 votes)
- How to solve (-2+I)(-2-I)(2 votes)
Once you expand the binomial, you will have two real terms and two imaginary terms (the i squared term is a real term since i^2=-1). THen you combine like terms. Since the two numbers you wrote are "conjugates" of each other, the imaginary terms will be opposites of each other and your answer will just be the real number (-2)^2 + 1^2 = 4 + 1 =5. You need to begin to recognize when you are multiplying conjugates as they result in a difference of squares (which for complex conjugates results in a sum of squares).(8 votes)
So every complex numbers form is a+bi ?(3 votes)- Yes... For a real number, in a+bi, b=0
For a imaginary number, in a+bi, a=0(2 votes)
- multiplying complex numbers is hard.(2 votes)
- Multiplying complex number uses FOIL (which you already know how to do). Treat i as a variable. Then, you have a new step. One term will include i^2. Since i^2=-1, you need to swap out the i^2. For example:
(2+3i)(5-2i)
-- FOIL: 10 - 4i + 15i -6i^2
-- Turn i^2 into -1: 10 - 4i + 15i -6(-1)
-- Simplify & combine like terms: 10 + 11i + 6 = 17+ 11i
Hope this helps.(4 votes)
problems fives anwser was 5 and not in from of a+bI but it said the answer was in a+bi(1 vote)- The problem accepts both "5" and "5+0i". Since the imaginary component is 0, it is not required.(5 votes)
