Finding the Absolute Value of Complex Numbers

Now that we're comfortable plotting complex numbers, let's talk about how to find the absolute value of a complex number.
So in the last video, we learned how to plot this complex number.
We look at the real and imaginary part and plot it on the complex coordinate plane.
So -3 is the real part, so we go over 3 on the real axis and up 4 on the imaginary axis.
So this complex number is plotted right there.
Now the absolute value of this complex number is actually the distance this point is from the origin of the complex graph.
So this distance right here.
What we can use now is the distance formula.
The distance formula, you may recall, looks like this, where we plug in the x-coordinates of the two points and the y-coordinates of the two points we're trying to find the distance between.
Let's do that for right here, and actually, I'll write this right above this point just so we know the coordinates.
Well, the coordinates are (-3,4), and the coordinates down here, the coordinates at the origin, are always (0,0).
So, back to the distance.
Let's plug in these into our distance formula, and we get x2 is -3 minus 0 squared, plus 4 minus 0 squared...
And we end up with (-3)^2 plus (4)^2.
Which is 9 + 16.
Which is 25.
And the square root of 25 is 5.
So the absolute value of this is 5.
Now, because we are always comparing our complex number to the origin, there's actually a shortcut for the distance formula.
We can actually, to find the absolute value of a complex number, just find the value of a^2 plus b^2, where a is our real part and b is our imaginary part.
That makes it really quick, and we can see here -3 squared is 9, plus 4 squared 16, 9 plus 16 is 25, the square root of 25 is 5.
So we could've done it that way as well.