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

Dec 25, 2016

$\left\mid 5 + 12 i \right\mid = 13$

#### 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

$\left\mid z \right\mid = \sqrt{z \overline{z}}$

So:

$\left\mid 5 + 12 i \right\mid = \sqrt{\left(5 + 12 i\right) \left(5 - 12 i\right)} = \sqrt{25 + 144} = \sqrt{169} = 13$

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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:

$\left\mid x + i y \right\mid = \sqrt{{x}^{2} + {y}^{2}}$

In our example:

$\left\mid 5 + 12 i \right\mid = \sqrt{{5}^{2} + {12}^{2}} = \sqrt{25 + 144} = \sqrt{169} = 13$

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]}

Dec 25, 2016

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

#### Explanation:

Absolute value of a complex number $a + b i$ is $\sqrt{{a}^{2} + {b}^{2}}$

hence absolute value of $5 + 12 i$ is $\sqrt{{5}^{2} + {12}^{2}} = \sqrt{25 + 144} = \sqrt{169} = 13$