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The absolute value of a complex number, sometimes called the modulus, is an essential concept that helps us measure the size or length of a complex number from the origin of the complex plane.

For any complex number of the form \(a + bi\), the absolute value is calculated using the formula:

• \(\sqrt{a^2 + b^2}\)

This formula might remind you of the Pythagorean theorem, and indeed, it is used in a similar way to determine the distance from the origin \((0,0)\) to the point \((a,b)\) on the complex plane.

In the specific case of the number \(4i\), which is written as \(0 + 4i\), the real part \(a\) is \(0\) and the imaginary part \(b\) is \(4\). Plugging these into our formula, we get \(\sqrt{0^2 + 4^2} = \sqrt{16} = 4\). This means the complex number \(4i\) lies at a distance of 4 units from the origin on the complex plane.

The complex plane is a graphical representation that helps us visualize complex numbers. It is similar to the traditional coordinate plane but designed to accommodate complex numbers.

Here’s how it works:

• The horizontal axis is called the real axis, representing the real part \(a\) of a complex number \(a + bi\).
• The vertical axis is called the imaginary axis, representing the imaginary part \(b\) of the complex number.

For example, the complex number \(4i\) can be written as \(0 + 4i\). Here, the real part is 0, meaning its position on the real axis is \(0\). The imaginary part is 4, placing it 4 units above the origin on the imaginary axis at the coordinates \((0, 4)\).

When you need to graph a complex number, just think of it as plotting a point on this unique plane where each point has a specific location based on its real and imaginary components.

In a complex number \(a + bi\), the imaginary part is represented by \(b\) in conjunction with \(i\), the imaginary unit. The imaginary unit \(i\) is defined as \(\sqrt{-1}\), and it's fundamental to the concept of complex numbers.

The imaginary part of a complex number provides information about its position along the vertical axis of the complex plane.

Let's look more closely:

• The imaginary part affects how far up or down the point representing the complex number will be placed. A positive imaginary part moves the point upwards, while a negative takes it downwards.
• For the complex number \(4i\), the imaginary part is 4, meaning it moves the number 4 units up the imaginary axis.

Understanding the behavior of the imaginary part is crucial for working with complex numbers, especially when dealing with their graphical representation or calculations that involve their distance from other points on the plane.