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Euler Formula and Euler Identity interactive graph

Below is an interactive graph that allows you to explore the concepts behind Euler's famous - and extraordinary - formula:

e = cos(θ) + i sin(θ)

When we set θ = π, we get the classic Euler's Identity:

e + 1 = 0

Euler's Formula is used in many scientific and engineering fields. It is a very handy identity in mathematics, as it can make a lot of calculations much easier to perform, especially those involving trigonometry. We saw some of this concept in the Products and Quotients of Complex Numbers earlier.

Leonhard Euler was a brilliant and prolific Swiss mathematician, whose contributions to physics, astronomy, logic and engineering were invaluable. He was certainly one of the greatest mathematicians in history.

Interactive Graph - Investigating Euler's Formula

In the following graph, the real axis (labeled "Re") is horizontal, and the imaginary (`j=sqrt(-1)`, labeled "Im") axis is vertical, as usual. We have a unit circle, and we can vary the angle formed by the segment OP.

Point P represents a complex number. The angle θ, of course, is in radians.

Things to do

  1. Choose whether your angles will be expressed using decimals or as multiples of π.
  2. Choose i or j for the symbol for the imaginary number. (Many text books use i for the imaginary number symbol, but IntMath, like many engineering texts, uses j to try to reduce the confusion with the symbol for current in electronics.)
  3. You can drag point P to change the angle θ.
  4. When you set θ = π, you'll see Euler's Identity: e = − 1

Angles: decimals multiples of π

Imaginary symbol: i j

Euler's Formula

Rectangular and Polar equivalents

For some background information on what's going on, and more explanation, see the previous pages,

Complex Numbers and Polar Form of a Complex Number

See also the polar to rectangular and rectangular to polar calculator, on which the above is based:

Polar to Rectangular Online Calculator

Next, we move on to see how to calculate Products and Quotients of Complex Numbers


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