confusedi'm confused the absolute value of 6-8i is the same with -6+8i...why?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The absolute value of a complex number of the form a + bi is given by sqrt(a^2 + b^2)

Here we have 6 - 8i and 6 + 8i

The absolute values of both of them are sqrt (6^2 + 8^2) = sqrt 100 = 10

The absolute value is the same because it is the distance of the point represented by the complex number on a plane where the x-axis is the real axis and the y-axis is the complex axis, from the origin (0,0)

pohnpei397's profile pic

pohnpei397 | College Teacher | (Level 3) Distinguished Educator

Posted on

This is because the absolute value of a number is simply how far it is from zero (like if you imagine it on a number line).

So, -6 and 6 are both the same distance from 0, right?  So they have the same absolute value.  The same goes for 8i and -8i.

To show this, let's just say i = 1.

6-8i = 6-8 = -2

-6+8i = -6+8 = 2

2 and -2 are the same distance from 0 and so they both have the same absolute value.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll write the formula of the absolute value of the imaginary number z.

|z| = sqrt[Re(z)^2 + Im(z)^2]

We'll identify the real part and imaginary part:

Re(z) = 6 and Im (z) = -8

|z| = sqrt [(6)^2 + (-8)^2]

|z| = sqrt (36 + 64)

|z| = sqrt 100

|z| = 10

Now, we'll identify the the real part and imaginary part of the complex number z = -6 + 8i.

Re(z) = -6 and Im (z) = +8

|z| = sqrt [(-6)^2 + (8)^2]

|z| = sqrt (36 + 64)

|z| = sqrt 100

|z| = 10

We notice that in the formula of absolute values, we'll use the squares of Real part and Imaginary part, so it doesn't matter if the values are positive or negative.

Since they are raised to square, they will be always positive.

And not to forget, the absolute value represents the distance vector or position vector from the point, whose coordinates are represented by the real part and imaginary part of complex number, to origin of the cartesian coordinate plane.

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