Calculus How To

Cartesian Form

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Cartesian form (also called Cartesian representation) can refer to any function or relation written using (x, y) or (x, y, z) coordinates. Most of the time though, the term refers to a specific way of writing complex numbers.

Cartesian Form of a Complex Number

Complex numbers have two forms: Cartesian and polar form. The “usual” way (at least, the first way you learn to express complex numbers) is in Cartesian form: z = x + yi, where:

  • z = a complex number,
  • x = the real part of z,
  • i = the imaginary part of z.

The right hand part of that equation, x + yi, is called the Cartesian form.

The other way complex numbers can be written is in polar form, which are made up of two parts, the modulus and argument. Polar form looks like this:
z = r∠θ
In Cartesian form, complex numbers can easily be plotted on an Argand diagram.

Example Question 1: What is the Cartesian form of the complex number (3 + i)(2 – i)2 – i ?.


  1. Use FOIL to expand the squared term (2 – i)2, giving 3 – 4i.
  2. Use algebra to rewrite the newly expanded equation (3 + i)(3 – 4i)2 – i
    1. Expand using FOIL: 3 · 3 + 3(-4i) + i · 3 + i (-4i) -i
    2. Simplify and combine like terms:: 9 – 12i + 3i + 4 = 13 – 9i -i = 13 – 10i

The solution (in Cartesian form) is 13 – 10i.

Example: Polar Form to Cartesian Form

Example question 2: What is the Cartesian Form of z = 3∠40°?

Remember SOHCAHTOA from trigonometry? You can use that to convert to Cartesian form. First, a graph might help you visualize where the various parts are:

cartesian form

The complex number z = 3∠40°


cos 40° = ON / 3

Solving for ON gives:

ON = 3 cos 40° = 2.298


sin 40° = NP / 3

Solving for NP gives:

NP = 3 sin 40° = 1.928

The Cartesian form is z = 2.298 + 1.928i


Centre for Excellence. Sigma. The polar form of a complex number. Retrieved January 12, 2020 from:
Warner, S. Pure Mathematics for Pre-Beginners: An Elementary Introduction to Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra Paperback – September 29, 2019. Get 800.
Thomas, R. Complex Numbers. Retrieved January 12, 2020 from:

Stephanie Glen. "Cartesian Form" From Calculus for the rest of us!

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