Question

Compute:
|3+4i|+|3-4i|+|-3+4i|+|-3-4i|

Answer 1

20

Explanation:

`|a+bi|` is a value that represent the distance between the points
`(a,b)` and
`(0,0)`.

The horizontal distance is
`a-0=a`.

The vertical distance is
`b-0=b`.

The distance between
`(a,b)` and
`(0,0)` by Pythagorean Theorem is:
`√(a^2+b^2)`.

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Conclusion:
`|a+bi|=√(a^2+b^2)`.

You can use that to find each of your | | terms in your problem.

However,
`(3,4,5)` is a well remember Pythagorean Triple.

A Pythagorean Triple are three natural numbers satisfying
`a^2+b^2=c^2`.

`3^2+4^2=5^2` is true since
`3^2+4^2=9+16=25` while
`5^2=25`. Both sides are 25.

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If you don't like, just use the formula I mentioned once.

`|\pm 3+\pm 4i|`

`\sqrt{(\pm 3)^2+(\pm 4)^2`

`√(9+16)`

`√(25)`

`5`

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The problem becomes 5+5+5+5=4(5)=20.

Answer 2

20

Explanation:

This is just finding the magnitudes of each of the 4 complex numbers.

The magnitude of a complex number a + bi is found by the formula:
`√(a^2+b^2)`.

Looking at all four numbers, we see that a is always either 3 or -3, and the b is always either 4 or -4. However, we see that the sign of positive or negative doesn't matter because we'll be squaring a and b anyway. So, the magnitudes of all four will be the same.

Now, we can just find the magnitude of one and multiply it by 4. Arbitrarily, we can solve the magnitude of the first complex number 3 + 4i:

|3 + 4i| =
`√(3^2+4^2) =√(9+16) =√(25) =5`

So, the magnitude is 5. Then, the sum of all four is 5 * 4 = 20.

Hope this helps!