Complex Numbers Add, Subtract, Multiply, Divide
Summary
TLDRThis video script introduces complex numbers, which consist of a real part and an imaginary part, denoted as 'a + bi'. It explains the concept of 'i' as the square root of negative one, and how to perform basic arithmetic operations with complex numbers, such as addition, subtraction, multiplication, and division. The script also covers the importance of using the complex conjugate to eliminate 'i' from the denominator during division, and emphasizes the need to express the result in the standard 'a + bi' form. The explanation includes step-by-step examples to illustrate the process clearly.
Takeaways
- π’ A complex number is composed of a real part and an imaginary part, represented in the standard form as 'a + bi'.
- π The imaginary unit 'i' is defined as the square root of -1, which is denoted by 'i^2 = -1'.
- π€Ή Adding complex numbers involves adding the real parts together and the imaginary parts separately, resulting in a new complex number in 'a + bi' form.
- π Subtracting complex numbers is similar to addition, but involves subtracting the real and imaginary parts of the second number from the first.
- π€ Multiplication of complex numbers uses the FOIL method (First, Outer, Inner, Last), and the result is simplified by recognizing 'i^2 = -1'.
- π« Division by a complex number is not allowed with 'i' in the denominator. It must be rationalized by multiplying by the complex conjugate.
- π The complex conjugate is used to eliminate 'i' from the denominator by changing the sign of the imaginary part of the denominator.
- π When dividing, both the numerator and the denominator are multiplied by the complex conjugate of the denominator to simplify the expression.
- π After division, the result should be expressed in the standard 'a + bi' form, separating the real and imaginary parts.
- π An example is given where dividing by a monomial with 'i' in the denominator is simplified by multiplying by 'i/i', resulting in a real number times 'i'.
- π The script provides a comprehensive overview of arithmetic operations with complex numbers, including addition, subtraction, multiplication, and division.
Q & A
What is a complex number?
-A complex number is a number that consists of a real part and an imaginary part, typically written in the form a + bi, where 'i' represents the square root of -1.
What is the significance of 'i' in complex numbers?
-'i' is used to denote the square root of -1. It is the imaginary unit and is fundamental in forming complex numbers.
How do you represent the square root of a negative number?
-The square root of a negative number is represented using 'i'. For example, the square root of -16 is 4i.
What is the result of i squared (i^2)?
-When you square 'i' (i^2), the result is -1, since 'i' is the square root of -1.
How do you add two complex numbers?
-To add two complex numbers, you add their real parts together and their imaginary parts together separately. For example, adding (3 + 2i) and (4 - 5i) results in (7 - 3i).
What is the process for subtracting complex numbers?
-Subtracting complex numbers involves subtracting the real parts and the imaginary parts separately. For instance, (3 + 2i) minus (4 - 5i) equals (-1 + 7i).
How do you multiply complex numbers using the FOIL method?
-When multiplying complex numbers, you use the FOIL method (First, Outer, Inner, Last), multiplying each term in the first binomial by each term in the second, and then combining like terms. For example, (3 + 2i) times (4 - 5i) results in 12 - 15i + 8i - 10i^2, which simplifies to 22 - 7i after considering i^2 = -1.
Why is it necessary to eliminate the imaginary unit from the denominator when dividing complex numbers?
-Having an imaginary unit in the denominator is considered improper. To eliminate it, you multiply both the numerator and the denominator by the complex conjugate, which helps to rationalize the denominator.
What is the complex conjugate and how is it used in division of complex numbers?
-The complex conjugate of a complex number is formed by changing the sign of the imaginary part. It is used to multiply the numerator and denominator to eliminate the imaginary part from the denominator, making it a real number.
How do you simplify the result of dividing two complex numbers?
-After multiplying by the complex conjugate, you simplify the result by performing the multiplication in the numerator and denominator, and then separating the real and imaginary parts into their respective terms, ensuring the result is in the form a + bi.
Can you provide an example of dividing a complex number by a monomial in the denominator?
-Sure, for example, dividing 3 by 2i involves multiplying by i over i, which results in -3/2 times i, or -3/2i. This is because multiplying by i over i is equivalent to multiplying by -1, which simplifies the expression.
Outlines
π’ Introduction to Complex Numbers
This paragraph introduces the concept of complex numbers, which are numbers composed of a real part and an imaginary part. It explains that the imaginary part is represented by the letter 'i', which is the square root of negative one. The standard form of a complex number is expressed as 'a + bi'. The paragraph also covers basic operations with complex numbers, such as addition, subtraction, multiplication, and division. It illustrates how to add and subtract complex numbers by combining their real and imaginary parts separately. Multiplication is explained using the FOIL method, and division is handled by multiplying by the complex conjugate to eliminate the imaginary part in the denominator.
π Further Exploration of Complex Number Operations
This paragraph continues the discussion on complex numbers, focusing on the division operation. It explains the process of dividing a complex number by another complex number, emphasizing the need to eliminate the imaginary part in the denominator by multiplying by the complex conjugate. The example given demonstrates how to multiply both the numerator and the denominator by the conjugate to simplify the expression. The result is then simplified further by separating the real and imaginary parts into two fractions, maintaining the standard form 'a + bi'. Additionally, a simpler example of dividing by a monomial in the denominator is provided, showing how multiplying by 'i over i' results in the simplification of the expression.
Mindmap
Keywords
π‘Complex Number
π‘Imaginary Part
π‘Real Part
π‘Standard Form
π‘i (Imaginary Unit)
π‘Addition
π‘Subtraction
π‘Multiplication
π‘Division
π‘Complex Conjugate
π‘I Squared
Highlights
Complex numbers consist of a real part and an imaginary part.
Imaginary numbers are combined with real numbers to form complex numbers.
The standard form of a complex number is a + bi, where 'i' equals the square root of negative one.
The square of 'i' equals negative one (i^2 = -1).
Adding complex numbers involves adding the real parts and the imaginary parts separately.
Subtracting complex numbers is done by subtracting the real parts and the imaginary parts.
Multiplication of complex numbers uses the FOIL method (First, Outer, Inner, Last).
When multiplying, 'i' squared is replaced with negative one.
Division of complex numbers requires eliminating the imaginary part from the denominator.
To remove 'i' from the denominator, multiply by the complex conjugate.
The complex conjugate changes the sign of the imaginary part of a complex number.
After multiplying by the complex conjugate, simplify the expression.
Separate the real and imaginary parts when expressing the result of a division of complex numbers.
An example is provided for dividing a complex number by a monomial in the denominator.
Multiplying by 'i' over 'i' results in 'i' squared, which is negative one.
The process of working with complex numbers is demonstrated with step-by-step examples.
Transcripts
so a complex number is a number that's
made up of a real part and an imaginary
part so up until this point you've
probably just been working with real
numbers now we're going to be working
with the imaginary numbers and real
numbers combined together they form a
complex number and they call this the
standard form of a complex number it's a
plus bi form so what does I equal I was
the square root of negative one okay so
if you're taking the square root of a
negative number say like negative 16
square root of 16 is 4 square root of
negative 1 we represent with the letter
I so if you put this together this comes
out to 4i
now if you multiply I times I you get I
squared but remember I is really the
square root of negative 1 times the
square root of negative 1 equals
negative 1 so you want a number that I
squared equals negative 1 and that when
you have a square root of negative 1 you
get I or if you're taking the square
root of a negative number you're going
to get I so now we're going to talk
about is how do you add subtract
multiply and divide complex numbers well
let's take these two here 3 plus 2i 4
minus 5 I say we want it to add those
two complex numbers together all you
have to do is add the real parts 3 plus
4 which equals 7 and then add the
imaginary parts you can almost imagine
that I as a variable like X Y or Z 2i
plus negative 5i is negative 3i and
that's it that's in a plus bi form that
standard form now say you wanted to
subtract the two complex numbers 3 plus
2i minus 4 minus 5i
same idea just subtracting 3 minus 4
which equals negative 1 to I minus a
negative 5i is like adding 5i two
negatives has become a positive so
that's going to be 7i now let's take a
look at multiplication so 3 plus 2i
times 4 minus 5i
here you've got a by knowing binomial
times a binomial you can foil you could
do the distributive property however you
like to work with a binomial times a
binomial so we're going to do the foil
first terms gives us 12 outer terms
gives us negative 15 I inner terms gives
us a tie and then the last terms gives
us negative 10 I squared okay because I
times I is I squared just like x times X
is x squared but remember I squared
equals negative 1 so this is negative 1
times negative 10 is positive 10 okay
so 12 plus 10 is 20 2 negative 15 I plus
8i is negative 7 I so that's these two
complex numbers multiplied together and
then the last one we're going to talk
about is dividing so 3 plus 2i divided
by 4 minus 5i
now you don't want I in the denominator
that's considered improper so how do we
get rid of the I in the denominator you
probably remember working with radicals
and square roots and what we did there
when we had a binomial two terms in the
denominator is we multiplied by the
conjugate here we're going to do a
similar thing it's we're multiplying by
the complex conjugate because this is a
complex number
you're gonna write these same two terms
but instead of - you're gonna put a +
here if this was adding you'd make this
subtracting so you just make these signs
the opposite of course whatever you
multiply the denominator by you want to
multiply the numerator by okay because
this is really like 1 1 times anything
doesn't change the value of this
fraction it just changes the way that it
looks and puts it into a nicer neater
standard form so let's go ahead and do
this we're going to multiply these
together let's do the denominator first
we have 4 times 4 which is 16
negative-20 i and positive twenty idols
cancel then the last term's gives us
negative 25 I squared remember I squared
is negative 1 times a negative is a
positive
so this is going to be 16 plus 25 which
equals 41 and the numerator we're going
to do the foil we get 12 outer gives us
15 I inner gives us a tie and the last
term's give us 10 I squared remember I
squared is negative 1 so this is
actually negative 10 plus 12 is 2 15 I
and 8i gives us 23 I and now here's the
part that sometimes students forget
looks like we're done but we actually
want to write this in a plus bi form we
want to separate the real and the
imaginary part so the way you do that is
you split this up into two fractions
okay so this is going to be 2 over 40 1
plus 23 over 41 hi so there's the real
part and there's the imaginary part a
plus bi form
I'll show you one other example real
quick say you just have 3 divided by 2 I
this is just a monomial in the
denominator just one term so what you're
gonna do here is you're just going to
multiply by I over I this will give you
2 I squared which is negative 2 and this
will give you 3 I so you can just write
this as negative 3 halves I so again I
times I gives you the I squared which is
negative 1 times 2 which is negative 2
and we just multiply the numerators
whatever you do to the denominator you
want to do to the numerator and that
gives you the negative 3 halves I so
this is in standard form so this has
been how to work with complex numbers
and what a complex number is I'll see
you in the next video
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