Complex Numbers Add, Subtract, Multiply, Divide

Summary

This 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

• 🔢 A complex number is composed of a real part and an imaginary part, represented in the standard form as 'a + bi'.
• 📐 The imaginary unit 'i' is defined as the square root of -1, which is denoted by 'i^2 = -1'.
• 🤹 Adding complex numbers involves adding the real parts together and the imaginary parts separately, resulting in a new complex number in 'a + bi' form.
• 🔄 Subtracting complex numbers is similar to addition, but involves subtracting the real and imaginary parts of the second number from the first.
• 🤝 Multiplication of complex numbers uses the FOIL method (First, Outer, Inner, Last), and the result is simplified by recognizing 'i^2 = -1'.
• 🚫 Division by a complex number is not allowed with 'i' in the denominator. It must be rationalized by multiplying by the complex conjugate.
• 🔄 The complex conjugate is used to eliminate 'i' from the denominator by changing the sign of the imaginary part of the denominator.
• 📉 When dividing, both the numerator and the denominator are multiplied by the complex conjugate of the denominator to simplify the expression.
• 📈 After division, the result should be expressed in the standard 'a + bi' form, separating the real and imaginary parts.
• 🔍 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'.
• 📚 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

00:00

🔢 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.

05:03

📚 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.

Keywords

💡Complex Number

A complex number is a number that consists of a real part and an imaginary part. In the video, the complex number is introduced as a combination of real and imaginary numbers, which are fundamental to understanding the theme of the video. For example, the script mentions '3 plus 2i', where '3' is the real part and '2i' is the imaginary part.

💡Imaginary Part

The imaginary part of a complex number is the component that involves the square root of a negative number and is represented by 'i'. The script explains that 'i' is the square root of negative one, which is a key concept in understanding how complex numbers are formed and manipulated.

💡Real Part

The real part of a complex number is the component that does not involve the imaginary unit 'i'. It is the fixed numerical value in a complex number. In the script, when discussing the complex number '3 + 2i', '3' is the real part, which is essential for performing arithmetic operations on complex numbers.

💡Standard Form

The standard form of a complex number is represented as 'a + bi', where 'a' is the real part and 'b' is the coefficient of the imaginary part. The script emphasizes this form as the conventional way to express complex numbers, which is crucial for performing arithmetic operations.

💡i (Imaginary Unit)

The imaginary unit 'i' is defined as the square root of negative one. It is a fundamental concept in the script, as it is the basis for the imaginary part of complex numbers. For instance, when the script mentions 'I squared equals negative 1', it is referring to the property of the imaginary unit.

💡Addition

Addition of complex numbers involves adding the real parts together and the imaginary parts together. The script provides an example: '3 + 4 + 2i + (-5i)', which results in '7 - 3i', demonstrating the process of adding two complex numbers.

💡Subtraction

Subtraction of complex numbers is similar to addition but involves subtracting the real and imaginary parts. The script illustrates this with '3 + 2i - (4 - 5i)', which simplifies to '(-1) + 7i', showing the subtraction process.

💡Multiplication

Multiplication of complex numbers involves the distributive property, also known as FOIL (First, Outer, Inner, Last). The script demonstrates this with '(3 + 2i) * (4 - 5i)', resulting in '20 - 7i' after simplifying, which shows how to multiply two complex numbers.

💡Division

Division of complex numbers requires eliminating the imaginary part from the denominator by multiplying by the complex conjugate. The script explains this process with '(3 + 2i) / (4 - 5i)', which is transformed to have a real number in the denominator, and then simplified to '2 + (23/41)i'.

💡Complex Conjugate

The complex conjugate is used to simplify division of complex numbers by changing the sign of the imaginary part in the denominator. The script uses the conjugate in the division example, changing '(4 - 5i)' to '(4 + 5i)', which helps in eliminating the imaginary unit from the denominator.

💡I Squared

When the imaginary unit 'i' is squared, the result is negative one, as stated in the script 'I squared equals negative 1'. This property is used in the multiplication and division of complex numbers to simplify expressions involving 'i'.

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