Absolute Value of a Complex Number

Absolute Value of a Complex Number Concept Explanation:

The absolute value of a complex number ( z = a + bi ) represents the distance of the point ( (a, b) ) from the origin in the complex plane. It is given by the formula:
$|z| = \sqrt{a^2 + b^2}$

This is essentially the magnitude of the vector representing the complex number.

Diagram:

Here’s a diagram showing the complex number ( z = 3 + 4i ) in the complex plane:

The absolute value of ( z = 3 +

4i ) is ( |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 ).

Hard Questions:

Q1: Find the absolute value of ( z = 6 + 8i ).

Step-by-Step Solution:

$|z| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

Answer:

The absolute value of ( z = 6 + 8i ) is 10.

Q2: Calculate the absolute value of ( z = -7 + 24i ).

Step-by-Step Solution:

$|z| = \sqrt{(-7)^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$

Answer:

The absolute value of ( z = -7 + 24i ) is 25.

Q3: Find the absolute value of ( z = 5 – 12i ).

Step-by-Step Solution:

$|z| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$

Answer:

The absolute value of ( z = 5 – 12i ) is 13.

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