It's time to brush up our mathematical fundamentals and here are the list of topics that we need to
study.
In order to understand quantum computing and quantum machine learning in depth here, we are going to
start with complex numbers and then we'll move towards madrases, application of derivatives, probability
and basic integration techniques.
All of these we have done in our high school grades.
So.
You would basically go back to your school and remember all the concepts that you have already done
during your school days, so let's get back to school and let's get started with complex numbers.
Consider an equation for X squared plus one is equal to zero.
Now just imagine, can you resolve this particular equation in a real number system?
I don't think so.
It's not possible because what could be the square root of minus one?
The answer is not available in real numbers system, so we got to stretch.
The idea of real amber system and hence the concept complex, no.
In complex, no, apart from real number, you even have an imaginary partner.
And that imaginary part is denoted by eye.
I is equal to the square root of minus one.
So the powers of I, as you can see, I square is equal to minus one IQ is equal to minus I IRI's to
four is equal to one and on and on.
So now, bingo, we have the solution available, that complex number X Square is equal to minus one
and now we know that minus one is equal to a square.
Complex number, as I told you, consists of two parts, a real part and an imaginary part.
So consider a number A plus B, I hear it is the real part and B is the imaginary part.
Here are some of the examples of complex numbers, two plus three, four, minus three, minus five,
minus six I Cixi and one plus I.
So here, every number which is attached to EI is an imaginary quantity.
So let's take the example of two plus three, I hear two is the real number and three is imaginary.
In the second example, four minus three, four is the real number.
Minus three is the imaginary quantity.
And the same holds true for the rest of the examples.
Now, let's see the algebra of complex numbers, Ed. is very simple, as you can see, two plus three.
Plus three, plus two is equal to five plus five I.
Next example, ten plus two plus two minus I is equal to 12.
Plus I.
Subtraction.
Ten plus to a minus eight plus eye is equal to two plus eye.
Next example, two plus I, minus five plus I equals minus three, which is a real quantity.
Multiplication.
Multiplication is also pretty easy, but you have some more steps, let's say you want to multiply two
complex numbers, three plus two, I multiplied by one minus four, I.
Equals three minus dualeh, plus two, minus eight I square now, guys, you know that I square is minus
one.
Correct.
So three minus tennie, minus eight square root of minus one square.
Basically is nothing, but we have got a square is equal to minus one.
So I is equal to square root of minus one, and we are doing the squaring off it.
Yeah, so the final answer is 11 minus banai.
Division division also is pretty much the same, the only thing is in division, whatever is there in
the denominator.
Yeah, so let's say in this example, three minus four, I divide by three plus two, I v multiply.
The numerator and denominator by three minus two and three minus two I.
So basically, we just reverse the sign of the denominator and we multiply with the same number in the
numerator as well as denominator, and then the solution is very much similar to multiplication.
Yep, so you've got to remember that while dividing a complex number, you just need to inverse the
sign of the denominator and multiply the numerator and denominator with that quantity.
Now, what exactly is complex conjugate, complex conjugate is nothing but what you saw in Division
V. Just reverse the sign of imaginary quantity.
That's nothing but the complex conjugate of that denominator.
Here it is.
So you see a plus by its conjugate is a minus by.
A minus by its conjugate is a plus by.
Here are some of the examples of complex conjugate.
Very simple.
Now, what exactly is organ playing, organ playing is the same as complex plane, like for real numbers,
we had Cartesian plane for complex numbers, which has imaginary quantity.
We have organ plane or complex plane.
This is how organ plane looks like.
If we want to denote a complex number, a plus by air can be represented on X axis and B on Y axis.
If we want to represent the complex number in popular form.
Here is how it looks like.
Here, X can be said as our cost.
Peter.
And VI can be saved as our sign, Peter.
This is the polar representation of complex numbers.
I hope you guys got a crystal clear understanding on the basics of complex numbers.
Josh.