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# Euler's formula

Euler's formula relates the complex exponential to the cosine and sine functions. This formula is the most important tool in AC analysis. It is why electrical engineers need to understand complex numbers. Created by Willy McAllister.

## Want to join the conversation?

• I was wondering how did Euler come up with this formula?
• The reasoning behind the formula requires calculus to understand.
• So if i want to be a electric and electronic egineering i have to know this by heart?
• Hello Miguel,

There are many fun and easy ways to get involved. You could say it depends on how you view the world. Some folks like to learn all the theory first. Others (myself included) start out slow building and experimenting. The maths and scarry stuff can come later.

Regards,

APD
• So, all these variables that he is using, are they interchangeable for other letters? For example, at , where he says e ^ jx , could he as easily have said, e^ nr?
• The "x" variable name is free to choose. The "j" variable is not. "j" is the imaginary unit, the square root of -1. (In engineering we use "j" instead of "i" for the imaginary unit.) So you have to keep the "j", but you can use anything in place of "x".
• At , he mentions a video of Sal, can anybody give me link of that video, I'm unable to find? Thank you
• what is the class code of this topic, AC circuit analysis?.....Like if we want to add it in our google classroom.
• Why can you treat "j" as a variable and manipulate it in the equation? Shouldn't "j" be a vector notation in dimension of imaginary number?
(1 vote)
• "j" is the variable name in engineering for the "square root of -1". It is treated like any other variable name in algebra. It is not a vector notation or "decoration".
• i couldnt understand 5 minutes into the video , can someone recommend a video about what exactly is d/dx thingy?
(1 vote)
• We use the notation dy/dx(where y is a function) to represent the rate of change of y with respect to x. This can be calculated using calculus.
(1 vote)
• I got a hard problem"prove that the sum of n nth roots of any complex number is zero."
I
(1 vote)
• Where does : "So Euler's Formula says that e to the jx equals cosine X plus j times sine x. Sal has a really nice video where he actually proves that this is true. "?
(1 vote)
• What would "sin(x) + jcos(x)" and "sin(x) - jcos(x)" equal?
(1 vote)
• Hmm. Perhaps e^j(x-pi/2)? It's some shifted way of wandering around the unit circle.

After checking with some expansions, the second one is indeed e^j(x-pi/2), I suspect the first one would then be e^-j(x+pi/2)
(1 vote)