How to find the modulus of a complex number?
Understand the Problem
The question is asking how to calculate the modulus of a complex number, which involves finding the distance from the origin to the point represented by the complex number in the complex plane.
Answer
The modulus of the complex number is $5$.
Answer for screen readers
The modulus of the complex number ( z ) is $5$.
Steps to Solve
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Identifying the Complex Number First, identify the complex number given in the problem. 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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Using the Modulus Formula The modulus (or absolute value) of a complex number is calculated using the formula: $$ |z| = \sqrt{a^2 + b^2} $$ This formula represents the distance from the origin (0,0) to the point $(a,b)$ in the complex plane.
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Calculating the Real and Imaginary Parts Determine the values of $a$ and $b$ from the complex number. If the complex number is given as $z = 3 + 4i$, then $a = 3$ and $b = 4$.
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Plugging Values into the Formula Substitute the values of $a$ and $b$ into the modulus formula: $$ |z| = \sqrt{3^2 + 4^2} $$
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Performing the Calculation Calculate the squares of the real and imaginary parts: $$ |z| = \sqrt{9 + 16} $$
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Final Calculation of Modulus Now, simplify under the square root: $$ |z| = \sqrt{25} $$
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Result Finally, find the square root to get the modulus: $$ |z| = 5 $$
The modulus of the complex number ( z ) is $5$.
More Information
The modulus gives the distance of the complex number from the origin of the complex plane. It's often used in various applications in engineering and physics, especially in dealing with oscillations and waves.
Tips
- Forgetting to square both the real and imaginary parts before adding.
- Not taking the square root at the end of calculations.
- Confusing the real and imaginary parts of the complex number.
