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The absolute value of a complex number represents its distance from the origin on the complex plane. A complex number is in the form of \(a + bi\). Here, \(a\) is the real part, and \(bi\) is the imaginary part.

To find the absolute value of a complex number, use the formula: \[ |a + bi| = \sqrt{a^2 + b^2} \] Let’s understand this with an example. Given the complex number \(-1 - i\), the real part \(-1\) and the imaginary part is also \(-1\). Plug these values into the formula: \[ | -1 - i | = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \] So here, the absolute value is \sqrt{2}\. This tells us that the point \(-1 - i\) is \sqrt{2}\ units away from the origin (0,0) on the complex plane.

The complex plane is a graphical representation to visualize complex numbers. Think of it as a coordinate plane.

On the complex plane, the horizontal axis (x-axis) represents the real part of a complex number, while the vertical axis (y-axis) represents the imaginary part. Thus, any complex number \(a + bi\) can be represented as a point \((a, b)\) on this plane.

For example, the complex number \(-1 - i\) has real part \(-1\) and imaginary part \(-1\). So in the complex plane, you would find this number at point \((-1, -1)\).

Understanding this concept is crucial for plotting complex numbers and calculating their absolute values.

Plotting complex numbers on the complex plane helps visualize their positions and relationships. Each complex number \(a + bi\) corresponds to a unique point \((a, b)\) on the plane.

To plot \(a + bi\):

• Identify the real part \(a\) and the imaginary part \(bi\).
• Move \(a\) units along the x-axis.
• Next, move \(b\) units up or down the y-axis, depending on the sign of \(b\). Up for positive, down for negative.

Let’s plot \(-1 - i\):

• The real part is \(-1\), so move 1 unit to the left along the x-axis.
• The imaginary part is \(-1\), so move 1 unit down along the y-axis.

You will end up at point \((-1, -1)\). Make a dot there.

This method applies to all complex numbers and helps in understanding their spatial relations. Don't forget to always label your axes and points for clarity.