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What is the absolute value of 4i.

The main objective of this question is to find the absolute value for the given expression, which is:

\[\space 4i \]

This question uses the concept of Cartesian coordinate system. In a plane, a Cartesian coordinate is a method to describe each point with a unique pair of numbers. These numbers are indeed the signed distances from two fixed, perpendicular lines to the point, analyzed in the same length unit. The origin of each reference coordinate line, which is located at the ordered pair, is referred to as a coordinate axis or simply an axis of the system (0, 0).

Expert Answer

We are given:

\[\space 4i \]

We have to find the absolute value for the given expression.

The given point in the complex plane is represented as:

\[(0 \space ,  \space 4)\]

Now we have to use the distance formula. We know that:

\[\space d \space = \space \sqrt{(x_2 \space – \space x_1 )^2 \space + \space (y_2 \space – \space y_1 )^2} \]

By putting the values, we get:

\[\space d \space = \space \sqrt{(0 \space – \space 0 )^2 \space + \space (0 \space – \space 4 )^2} \]

\[\space d \space = \space \sqrt{(0 )^2 \space + \space (0 \space – \space 4 )^2} \]

\[\space d \space = \space \sqrt{(0 )^2 \space + \space (- \space 4 )^2} \]

\[\space d \space =  \space \sqrt{0 \space + \space (- \space 4 )^2} \]

\[\space d \space =  \space \sqrt{0 \space + \space 16} \]

\[\space d \space =  \space \sqrt{16} \]

By taking the square root results in:

\[\space d \space = \space 4\]

Numerical Answer

The absolute value of $ 4i $ is $ 4 $.

Example

Find the absolute value for $ 5i $ and  $ 6i $ .

We are given that:

\[\space 5i \]

We have to find the absolute value for the given expression.

The given point in the complex plane is represented as:

\[(0 \space ,  \space 5)\]

Now we have to use the distance formula. We know that:

\[\space d \space = \space \sqrt{(x_2 \space – \space x_1 )^2 \space + \space (y_2 \space – \space y_1 )^2} \]

By putting the values, we get:

\[\space d \space = \space \sqrt{(0 \space – \space 0 )^2 \space + \space (0 \space – \space 5 )^2} \]

\[\space d \space = \space \sqrt{(0 )^2 \space + \space (0 \space – \space 5 )^2} \]

\[\space d \space = \space \sqrt{(0 )^2 \space + \space (- \space 5 )^2} \]

\[\space d \space =  \space \sqrt{0 \space + \space (- \space 5 )^2} \]

\[\space d \space =  \space \sqrt{0 \space + \space 25} \]

\[\space d \space =  \space \sqrt{25} \]

By taking the square root results in:

\[\space d \space = \space 5\]

Now we have to find the absolute value for $ 6i $.

We are given that:

\[\space 6i \]

We have to find the absolute value for the given expression.

The given point in the complex plane is represented as:

\[(0 \space ,  \space 6)\]

Now we have to use the distance formula. We know that:

\[\space d \space = \space \sqrt{(x_2 \space – \space x_1 )^2 \space + \space (y_2 \space – \space y_1 )^2} \]

By putting the values, we get:

\[\space d \space = \space \sqrt{(0 \space – \space 0 )^2 \space + \space (0 \space – \space 6 )^2} \]

\[\space d \space = \space \sqrt{(0 )^2 \space + \space (0 \space – \space 6 )^2} \]

\[\space d \space = \space \sqrt{(0 )^2 \space + \space (- \space 6 )^2} \]

\[\space d \space =  \space \sqrt{0 \space + \space (- \space 6 )^2} \]

\[\space d \space =  \space \sqrt{0 \space + \space 36} \]

\[\space d \space =  \space \sqrt{36} \]

By taking the square root results in:

\[\space d \space = \space 6\]

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