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Absolute Value of Complex Numbers

Complex numbers are in the form of \(a+bi\), Where \(i\) is the Imaginary Unit, defined as the square root of minus 1.

Absolute Value is often viewed as the "distance" a number is away from 0, the origin. In the domain of Real Numbers, This is just the positive of that number, so the absolute value of -5 will be 5 and the absolute value of 5 will also be 5.

When it comes to the absolute value of Complex Numbers, we have to consider the Complex Plane. The Complex Plane is similar to the xy-Coordinate Plane, only instead of the x and y axis you have the real and imaginary axis. The Number \(3+4i\) will have coordinates (3,4) on the complex plane.

How will we find the distance a number is from 0 on the complex plane? We can use the Pythagorean Theorem! The Absolute Value of a complex number is defined as:

\[|a+bi| = \sqrt{a^2 +b^2}\]

For Example, the Absolute Value of the complex number \(3+4i\) is equal to:

\[|3+4i| = \sqrt{3^2 +4^2} = 5\]

Note by Yan Yau Cheng
2 years, 10 months ago

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Also, note that the "absolute value" of a complex number is also called the "magnitude." I myself tend to use magnitude for non-real numbers, and absolute value for real numbers. Michael Tang · 2 years, 10 months ago

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@Yan Yau Cheng Can you add this to the Wiki page of Complex Numbers - Absolute Values? Thanks! Calvin Lin Staff · 2 years ago

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Problem: What is the absolute value of the product of two complex numbers? Use specific examples to aid in your findings. Can you generalize your statement in any way? Bob Krueger · 2 years, 10 months ago

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