Find the absolute value of a complex number.
Understand the Problem
The question is asking how to find the absolute value of a complex number, which requires applying the formula to calculate the magnitude of the complex number based on its real and imaginary parts.
Answer
The absolute value of a complex number $z = a + bi$ is $$ |z| = \sqrt{a^2 + b^2} $$.
Answer for screen readers
The absolute value of a complex number $z = a + bi$ is given by $$ |z| = \sqrt{a^2 + b^2} $$.
Steps to Solve
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Identify the real and imaginary parts First, find the real part $a$ and the imaginary part $b$ of the complex number. A complex number can be represented as $z = a + bi$, where $a$ is the real part and $b$ is the imaginary part.
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Apply the formula for magnitude To find the absolute value (or magnitude) of the complex number $z$, use the formula: $$ |z| = \sqrt{a^2 + b^2} $$ This formula is derived from the Pythagorean theorem, viewing the complex number as a point in the Cartesian plane.
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Calculate the squares and sum them Square both the real part and the imaginary part: $$ a^2 $$ $$ b^2 $$ Then, add these two results together to get $a^2 + b^2$.
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Take the square root Finally, take the square root of the sum calculated in the previous step to find the magnitude: $$ |z| = \sqrt{a^2 + b^2} $$
The absolute value of a complex number $z = a + bi$ is given by $$ |z| = \sqrt{a^2 + b^2} $$.
More Information
The absolute value (magnitude) of a complex number reflects its distance from the origin in the complex plane. This concept is fundamentally important in complex analysis and has applications in various fields, including engineering and physics.
Tips
- Confusing the real and imaginary parts when plugging values into the formula. Always double-check which coefficient corresponds to the real and imaginary parts.
- Forgetting to square the values before adding them together, which is essential for applying the formula correctly.
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