**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:

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 ?.

**Solution**:

- Use FOIL to expand the squared term (2 – i)
^{2}, giving 3 – 4i. - Use algebra to rewrite the newly expanded equation (3 + i)(3 – 4i)
^{2}– i**Expand using FOIL:**3 · 3 + 3(-4i) + i · 3 + i (-4i) -i**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:

So:

Solving for ON gives:

And,

Solving for NP gives:

The Cartesian form is z = 2.298 + 1.928i

## References

Centre for Excellence. Sigma. The polar form of a complex number. Retrieved January 12, 2020 from: http://www.mathcentre.ac.uk/resources/sigma%20complex%20number%20leaflets/sigma-complex10-2009-1.pdf

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: https://home.cc.umanitoba.ca/~thomas/Courses/ComplexRSDT.pdf

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