Argand diagrams

Argand Diagrams Revision Content

Basics of Argand Diagrams

  • Argand diagrams are a way to visually represent complex numbers.
  • They’re named after Jean-Robert Argand, a mathematician who first used them to understand complex numbers.
  • The real part of a complex number is represented along the horizontal axis (‘x’ axis).
  • The imaginary part of a complex number is represented along the vertical axis (‘y’ axis).

Plotting and Reading Argand Diagrams

  • To plot a complex number on an Argand diagram, treat the real part as the ‘x’ coordinate and the imaginary part as the ‘y’ coordinate.
  • For example, the complex number 2 + 3i would be plotted at the point (2, 3).
  • To read a complex number from an Argand diagram, simply reverse the process: the ‘x’ coordinate gives the real part and the ‘y’ coordinate gives the imaginary part.

Magnitude and Argument of a Complex Number

  • Each point in an Argand diagram can be represented by its magnitude and angle (also known as argument), similar to polar coordinates.
  • The magnitude (or modulus) of a complex number z = a + bi is given by ** z = √(a² + b²)**.
  • The argument (arg) of a complex number represents the angle it makes with the positive real axis, moving in an anti-clockwise direction.
  • It can be calculated using trigonometry, with arg(z) = tan⁻¹(b/a), but it’s important to consider the quadrant the number is in to determine the correct angle.

Operations on Argand Diagrams

  • Addition and subtraction of complex numbers can be represented graphically as a movement in the direction of the number being added or the reverse direction of the number being subtracted.
  • Multiplication by a real number scales the vector in the direction it’s already pointing.
  • Multiplication by an imaginary number rotates the vector 90° clockwise (for -i) or anti-clockwise (for i).
  • Multiplication of two complex numbers involves multiplying their magnitudes and adding their arguments.
  • Division of complex numbers can also be performed, involving dividing the magnitudes of the numbers and subtracting the arguments.

Complex Conjugates and Argand Diagrams

  • The complex conjugate of a complex number z = a + bi is the number z* = a - bi.
  • On an Argand diagram, this is represented as a reflection in the real axis.