How do you find the absolute value of the complex number #-4+i#?

1 Answer
Brian E.
Oct 9, 2015

Take the square root of the sum of the real part squared plus the imaginary part squared.

Explanation:

For real numbers, we find the distance of the number from 0 on the x-axis.

In this case, we have a complex number with a real part and an imaginary part. On a coordinate plane, we would use the y-axis as the imaginary axis '#i#' and plot this complex number at #(-4, 1)#.

Next, find the distance of that point from zero. To do that, we'd use the Pythagorean Theorem. Imagine the x-axis (from #(0,0)# to #(0, -4)# as one leg of the triangle, #a#, and the straight line parallel to the y- (or #i#-axis from #(-4, 0)# to #(-4, 1)# as the second leg, #b#. The hypotenuse, #c#, will then be the straight line going from #(0,0)# to #(4, -1)#.

So, solving for #c# in the equation #a^2 + b^2 = c^2#, we take the square root of both sides.

#sqrt(c^2) = sqrt(-4^2 + 1^2)#

#c = sqrt(16 + 1)#

#c = sqrt(17)#

Therefore, #abs(-4 + i) = sqrt(17)#.

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