Question

How do you find the absolute value of `10-7i`?

Answer 1

`abs((10-7i))=sqrt(149) approx 12.21`.

Explanation:

The absolute value of a number is better thought of as the distance that number is from the origin. For numbers in `RR`, this definition simplifies to "just take off the negative sign if there is one". But for complex numbers and multi-dimensional vectors, we have to do a little more calculation.

Since we use a 2-D plane to illustrate complex numbers, we will use the formula for the 2-D distance `d` between two points `(x_1,y_1)` and `(x_2,y_2)`:

`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

This can be simplified further for the use of absolute values, since the first point will always be the origin `(0,0)` and only the second point will be allowed to vary. Thus:

`d=sqrt((x_2-0)^2+(y_2-0)^2)`
`color(white)d=sqrt(x_2^2+y_2^2)`

or simply

`abs[(x, y)` `=sqrt(x^2+y^2)`

A complex number `a+bi` is drawn in the complex plane exactly the same way an `(x,y)` point is plotted in the `x"-"y` plane: `a` (or `x`) denotes where we are on the real (or horizontal) axis, and `b` (or `y`) denotes where we are on the imaginary (or vertical) axis.

That means the distance that a number `a+bi` is from the origin (i.e. its absolute value) is calculated as

`abs((a+bi))=sqrt(a^2+b^2)`

For the complex number `10-7i`, this is calculated to be

`abs((10-7i))=sqrt(10^2+("-7")^2)`
`color(white)(abs((10-7i)))=sqrt(100+49)`
`color(white)(abs((10-7i)))=sqrt(149)"             "approx 12.21`.