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Chapter 8: Problem 3

Graph the complex number and find its absolute value. $$i$$

Short Answer

Expert verified
The complex number \(i\) is plotted at (0, 1) on the Argand plane and its absolute value is 1.

Step by step solution

01

- Identify the Complex Number

The given complex number is represented by the imaginary unit \(i\). In complex numbers, \(i\) is equivalent to \(0 + 1i\), where the real part is 0 and the imaginary part is 1.
02

- Plot the Complex Number on the Argand Plane

On the Argand plane (a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part), plot the point (0, 1). This point represents the complex number \(i\).
03

- Calculate the Absolute Value

The absolute value (or modulus) of a complex number \(a + bi\) is given by \(|z| = \sqrt{a^2 + b^2}\). For the number \(i\), \(a = 0\) and \(b = 1\). Therefore, \(|i| = \sqrt{0^2 + 1^2} = \sqrt{1} = 1\).

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Key Concepts

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

Argand plane
The Argand Plane is a visual way to represent complex numbers. Instead of just using the number line, the Argand plane uses a two-dimensional plane.
The horizontal axis (x-axis) represents real numbers. The vertical axis (y-axis) represents imaginary numbers.

To graph a complex number on the Argand plane, you plot it just like you would in a usual coordinate system in mathematics.
  • The real part of the complex number determines its position on the horizontal axis.
  • The imaginary part determines its position on the vertical axis.
For example, the complex number represented by the imaginary unit \(i\) is plotted at the point (0, 1), meaning 0 units along the horizontal axis and 1 unit up on the vertical axis.
imaginary unit
The imaginary unit, usually denoted as \(i\), is a fundamental concept in complex numbers. It is defined by the property that \(i^2 = -1\).
Imaginary numbers build upon this concept.
  • The imaginary unit is used to express numbers that would otherwise not have a real solution.
  • Every imaginary number can be written as a multiple of \(i\) (for example, 3i or -2i).
This concept extends the real number system to a new set of numbers, allowing us to solve equations and problems that do not have solutions within the real numbers.
absolute value of complex numbers
The absolute value (or modulus) of a complex number measures its distance from the origin in the Argand plane. It is like the length of the vector that represents the complex number.
To find the absolute value of a complex number \(a + bi\), you use the formula:
  • \(|z| = \sqrt{a^2 + b^2}\)
For instance, for the complex number \(i\), which can be written as \(0 + 1i\),
the absolute value is calculated as:
  • \(|i| = \sqrt{0^2 + 1^2} = \sqrt{1} = 1\)

This method helps to quantify how 'large' or 'small' a complex number is in a geometric sense.

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