1.3 Argand Diagrams

• An Argand diagram is a graphical representation of complex numbers on a two-dimensional plane
• Complex numbers in the form $a +$$bi$ are plotted with $a$ on the real axis and $b$ on the imaginary axis.
• How would the complex number $3 +$$4i$ be plotted on an Argand diagram?


  (3, 4)
• Match the axis with its representation:

  Horizontal Axis ↔️ Real axis

  Vertical Axis ↔️ Imaginary axis
• Steps to plot a complex number on an Argand diagram:

  1️⃣ Identify the real part

  2️⃣ Identify the imaginary part

  3️⃣ Plot the point (a, b) on the plane
• The real axis on an Argand diagram represents the imaginary part of a complex number.

  False
• Where is the complex number $2 +$$3i$ plotted on an Argand diagram?


  (2, 3)
• What is an Argand Diagram used for?

  Representing complex numbers
• In an Argand Diagram, the real part of a complex number is plotted on the horizontal
• The imaginary axis in an Argand Diagram corresponds to the y-axis in a Cartesian Plane.
• What is the vertical axis in an Argand Diagram called?

  Imaginary axis
• Match the components of an Argand Diagram with their counterparts in a Cartesian Plane:

  Horizontal Axis ↔️ x-axis

  Vertical Axis ↔️ y-axis
• Where is the complex number $3 +$$4i$ plotted on an Argand Diagram?


  (3, 4)
• To plot a complex number on an Argand Diagram, you must identify its real and imaginary parts.
• Steps to plot a complex number on an Argand Diagram:

  1️⃣ Identify the real and imaginary parts.

  2️⃣ Locate the point on the diagram.

  3️⃣ Mark the point to represent the complex number.
• The real axis in an Argand Diagram is horizontal.
• What are the two axes in an Argand Diagram called?

  Real and imaginary axes
• In an Argand Diagram, complex numbers are plotted using their real and imaginary parts.
• What are the axes of an Argand Diagram called?

  Real and imaginary axes
• The imaginary axis in an Argand Diagram is vertical
• Match the diagram type with its axis names:

  Argand Diagram ↔️ Real axis ||| Imaginary axis

  Cartesian Plane ↔️ x-axis ||| y-axis
• The complex number $3 +$$4i$ is plotted as (3, 4) on an Argand Diagram.
• How do you plot a complex number $a +$$bi$ on an Argand Diagram?


  Locate point (a, b)
• The modulus of a complex number z = a + bi</latex> is its distance from the origin
• The formula for the modulus of $z =$$a +$$bi$ is $|z| =$ \sqrt{a^{2} + b^{2}}.
• What does the modulus of a complex number represent graphically?

  Distance from the origin
• The modulus of a complex number is always a positive real number.
• The argument of a complex number $z =$$a +$$bi$ is the angle $\theta$ calculated as \theta = \tan^{ - 1}\left(\frac{b}{a}\right)</latex>.origin
• What is the argument of $z =$$3 +$$4i$ in terms of $\tan^{ - 1}$?


  $\tan^{ - 1}\left(\frac{4}{3}\right)$
• The argument of a complex number is measured from the positive real axis.
• What is the argument of $z =$$3 +$$4i$?


  $\tan^{ - 1}\left(\frac{4}{3}\right)$
• What is the argument of a complex number defined as?

  The angle $\theta$ between the positive real axis and the line connecting the origin to $z$ on an Argand Diagram
• The argument of a complex number z = a + bi</latex> is calculated as $\theta =$$\tan^{ - 1}\left(\frac{b}{a}\right)$
• On which diagram is the argument of a complex number represented graphically?

  Argand Diagram
• For $z =$$3 +$$4i$, the argument is $\theta =$$\tan^{ - 1}\left(\frac{4}{3}\right)$, which is the angle in radians
• What does the calculation of the argument of a complex number depend on?

  The quadrant of $z$
• In the first quadrant, if $a > 0$ and $b > 0$, then $\theta =$$\tan^{ - 1}\left(\frac{b}{a}\right)$
• What is the formula for $\theta$ in the second quadrant when $a < 0$ and $b > 0$?


  $\theta =$$\pi +$$\tan^{ - 1}\left(\frac{b}{a}\right)$
• What is the argument of a complex number defined as?

  Angle between positive real axis and z
• The argument for $z =$$3 +$$4i$ is calculated as $\tan^{ - 1}\left(\frac{4}{3}\right)$, which simplifies to \theta