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Chapter 7: Problem 5

Graph each complex number, and find its absolute value. $$8 i$$

Short Answer

Expert verified
The absolute value of \(8i\) is 8, and it is plotted at the point (0, 8) on the complex plane.

Step by step solution

01

Understanding the Complex Number

The complex number given is purely imaginary, represented as \(8i\). In the complex plane, this corresponds to the point \((0, 8)\), where 0 is the real part and 8 is the imaginary part.
02

Plotting the Complex Number

To plot the complex number \(8i\), place a point at \((0, 8)\) on the complex plane. This means moving 8 units up on the imaginary axis from the origin.
03

Finding the Absolute Value

The absolute value or modulus of a complex number \(a + bi\) is given by \ \ \ 鈭鈭� = 鈭�(a虏 + b虏). For \(8i\), \(a = 0\) and \(b = 8\). Thus, the absolute value is 鈭�8i鈭� = 鈭�(0虏 + 8虏) = 鈭�(64) = 8.

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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 where we can graphically represent complex numbers. The horizontal axis (x-axis) is used for the real part of the complex number, while the vertical axis (y-axis) is used for the imaginary part. Each complex number corresponds to a unique point on this plane.
For example, the complex number '8i' has no real component (real part is 0) and an imaginary component of 8. To plot '8i' on the complex plane, we start at the origin (0,0) and move vertically upwards to the point (0,8). This is because the imaginary component of 8 corresponds to 8 units on the imaginary axis.
imaginary number
An imaginary number is a number that can be written as a real number multiplied by the imaginary unit 'i'. The imaginary unit 'i' is defined as the square root of -1. So, any number of the form 'bi', where 'b' is a real number, is an imaginary number.
For instance, in the complex number '8i', '8' is the coefficient and 'i' indicates that it鈥檚 an imaginary number. Imaginary numbers are plotted on the vertical axis of the complex plane. Numbers such as '3i' and '-5i' will also be plotted on the imaginary axis at points (0, 3) and (0, -5), respectively.
absolute value of complex numbers
The absolute value, also known as the modulus, of a complex number gives us a measure of its magnitude or distance from the origin on the complex plane. It鈥檚 similar to finding the length of a vector in Euclidean geometry. If the complex number is 'a + bi', its absolute value is defined as the square root of the sum of the squares of its real and imaginary parts.
Mathematically, it鈥檚 represented as: \(|z| = \sqrt{a^2 + b^2}\).
For '8i', since the real part 'a' is 0 and the imaginary part 'b' is 8, the absolute value is found as follows:
\(||8i|| = \sqrt{0^2 + 8^2} = \sqrt{64} = 8\).
modulus
The modulus of a complex number is another term for its absolute value. The modulus represents the length of the vector from the origin to the point representing the complex number in the complex plane.
It's a crucial concept for understanding the geometry of complex numbers and is used in many mathematical and engineering applications to find the magnitude of a complex number.
For example, the modulus of the complex number '8i' is calculated as:
\(|8i| = \sqrt{0^2 + 8^2} = 8 \).
This tells us that the point representing '8i' is 8 units away from the origin on the complex plane.

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