Lösung 3.2:4a

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The magnitude of the number
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The magnitude of the number <math>3+4i</math> is the number's distance to the origin in the complex number plane.
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<math>\text{3}+\text{4}i</math>
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is the number's distance to the origin in the complex number plane.
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If we treat the line from the origin to
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If we treat the line from the origin to <math>3+4i</math> as the hypotenuse in a right-angled triangle which has its legs parallel with the real and imaginary axes, then the Pythagorean theorem gives that the magnitude is
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<math>\text{3}+\text{4}i</math>
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as the hypotenuse in a right-angled triangle which has its sides parallel with the real and imaginary axes, then Pythagoras' theorem gives that the magnitude is
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<math>\left| \text{3}+\text{4}i \right|=\sqrt{3^{2}+4^{2}}=\sqrt{9+16}=\sqrt{25}=5</math>
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{{Displayed math||<math>|3+4i| = \sqrt{3^2+4^2} = \sqrt{9+16} = \sqrt{25} = 5\,\textrm{.}</math>}}
[[Image:3_2_4_a.gif|center]]
[[Image:3_2_4_a.gif|center]]
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NOTE: In general, the magnitude of a complex number
 
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<math>z=x+iy</math>
 
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is equal to
 
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Note: In general, the magnitude of a complex number <math>z=x+iy</math> is equal to
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<math>\left| z \right|=\left| x+iy \right|=\sqrt{x^{2}+y^{2}}</math>
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{{Displayed math||<math>|z| = |x+iy| = \sqrt{x^2+y^2}\,\textrm{.}</math>}}

Version vom 09:54, 29. Okt. 2008

The magnitude of the number \displaystyle 3+4i is the number's distance to the origin in the complex number plane.

If we treat the line from the origin to \displaystyle 3+4i as the hypotenuse in a right-angled triangle which has its legs parallel with the real and imaginary axes, then the Pythagorean theorem gives that the magnitude is

\displaystyle |3+4i| = \sqrt{3^2+4^2} = \sqrt{9+16} = \sqrt{25} = 5\,\textrm{.}


Note: In general, the magnitude of a complex number \displaystyle z=x+iy is equal to

\displaystyle |z| = |x+iy| = \sqrt{x^2+y^2}\,\textrm{.}
Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_3.2:4a“