Question

How do you find the absolute value of `5+12i`?

Answer 1

`abs(5+12i) = 13`

Explanation:

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Quick method

The first couple of Pythagorean triples are:

`3, 4, 5`

`5, 12, 13`

So a right angled triangle with legs `5` and `12` will have hypotenuse `13`

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Standard formula using Complex conjugate

`abs(z) = sqrt(zbar(z))`

So:

`abs(5+12i) = sqrt((5+12i)(5-12i)) = sqrt(25+144) = sqrt(169) = 13`

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Two dimensional distance formula

The absolute value of a Complex number is its distance from `0`, which is given by the distance formula:

`abs(x+iy) = sqrt(x^2+y^2)`

In our example:

`abs(5+12i) = sqrt(5^2+12^2) = sqrt(25+144) = sqrt(169) = 13`

graph{y(x+0.0001y-5)(5y-12x)sqrt(-((x-5/2)^2+(y-6)^2-169/4))/sqrt(-((x-5/2)^2+(y-6)^2-169/4)) = 0 [-12.22, 16.78, -1.6, 13]}

Answer 2

Absolute value of `5+12i` is `13`.

Explanation:

Absolute value of a complex number `a+bi` is `sqrt(a^2+b^2)`

hence absolute value of `5+12i` is `sqrt(5^2+12^2)=sqrt(25+144)=sqrt169=13`