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

2 Answers
Dec 25, 2016

#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#

#color(white)()#
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]}

Dec 25, 2016

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#