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Chapter 6: Problem 19

Graph each complex number, and find its absolute value. \(3+3 i\)

Short Answer

Expert verified
The complex number 3 + 3i is graphed at the point (3, 3) and its absolute value is 3\sqrt{2}.

Step by step solution

01

- Plot the complex number on the complex plane

Plot the complex number by identifying its real part and its imaginary part. For the given complex number 3 + 3i, the real part is 3 and the imaginary part is 3. On the complex plane, plot the point (3, 3).
02

- Use the distance formula to find the absolute value

The absolute value of a complex number is the distance from the origin to the point (a, b). For a complex number in the form a + bi, the absolute value is given by: \[|a + bi| = \sqrt{a^2 + b^2}\] For the complex number 3 + 3i, substitute a = 3 and b = 3 into the formula to get: \[|3 + 3i| = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18}\]
03

- Simplify the result

Simplify the square root of 18 to get the final absolute value. Remember that \sqrt{18} can be further simplified to 3\sqrt{2}. Therefore, \[|3 + 3i| = 3\sqrt{2}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex plane
The complex plane is a two-dimensional plane used to graph complex numbers.
It's similar to the Cartesian plane but with a twist.
Instead of just x and y coordinates, each point on the complex plane represents a complex number.
The horizontal axis (often called the real axis) represents the real part of the complex number.
The vertical axis (called the imaginary axis) represents the imaginary part.
For example, the complex number 3 + 3i is plotted at the point (3, 3).
This visually shows its position with 3 units along the real axis and 3 units along the imaginary axis.
Absolute value of complex numbers
The absolute value of a complex number tells us how far the number is from the origin on the complex plane.
For a complex number in the form of a + bi, the absolute value is calculated using the formula: \[|a + bi| = \sqrt{a^2 + b^2}\]
This formula is similar to the Pythagorean theorem and gives the length of the line segment from the origin (0, 0) to the point (a, b).
In our example with the complex number 3 + 3i, we substitute a with 3 and b with 3: \[|3 + 3i| = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18}\]
Therefore, the absolute value of 3 + 3i is \(3\sqrt{2}\).
Distance formula
The distance formula helps us find the distance between two points in a plane.
For complex numbers, we use this formula to find the absolute value.
The distance from the origin to the point (a, b) is given by: \[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
When finding the absolute value of a complex number a + bi, the points are (0,0) and (a,b).
This simplifies our formula to: \[|a + bi| = \sqrt{a^2 + b^2}\]
Applying this to our example of 3 + 3i, we get the distance from the origin (0, 0) to (3, 3) as \(\sqrt{18}\).
This method confirms the absolute value or distance from the origin is \(3\sqrt{2}\).

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