Absolute value of complex numbers

I have the complex number 3-4 I've plotted it on the complex plane we see that the real part is 3 we've so we've gone three along the horizontal axis or the real axis and the imaginary part is negative four so we've gone down four along the vertical axis so this right here is a point 3 minus 4i now what I want to think about is what the absolute value of 3 minus 4i is and just as a reminder absolute value literally means whether we're talking about a complex number or a real number it literally just means distance away from zero so the absolute value of 3 minus 4i is going to be the distance between zero between the origin on the complex plane and and that point and the point 3 minus 4i so this distance right over here is going to be the absolute value of 3 minus 4i so how can we think about that well we could literally just set up a right triangle and then use the Pythagorean theorem so let's think about it if we wanted to set up a right triangle we want to set up a right triangle the height here the height here well we went the distance between 0 and negative 4 well that distance is going to be 4 and then the base of this triangle the distance between 0 and 3 is just going to be 3 it's going to be 3 and this is definitely a right angle this is a horizontal line this is a vertical line we can now use the Pythagorean theorem to figure out the absolute value of 3 minus 4i the distance between this point and 0 it's the hypotenuse of this right triangle so we just use the Pythagorean theorem this side squared 3 squared plus this side squared plus 4 squared is going to be equal to it's going to be equal to the absolute value of 3 minus 4i 3 minus 4i squared the absolute value squared so 3 squared plus 4 squared that's 9 plus 16 which is 25 so you get 25 is equal to the absolute value of 3 minus 4i squared 3 my for I and we know if you take the absolute value something this is just a distance it's going to be positive so we want to take the positive square root the principal square root of both sides of this and so we're going to be left with well the principal square root the positive square root of 25 is 5 is equal to the absolute value of 3 minus 4i 3 minus 4i so another way of saying it this thing right over here is going to be equal to 5 this distance right over here is equal to 5 now without having to draw it one way you could just think about this is well have my real part I have my imaginary part I can literally take each of those parts square them take the sum and take the square root so another way of taking it if you didn't want to visualize all of this but this is really what we're doing you could say well this is just going to be equal to take the real part squared it's going to take the real part squared take the imaginary part squared so let me write this so the imaginary part squared add them together add them together and then take the square root so that's go over the principal root so that's going to be the principal roots just a positive square root so that's going to be the square root of 9 plus 16 plus 16 which once again is equal to 5