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:
eiθ = cos(θ) + i sin(θ)
When we set θ = π, we get the classic Euler's Identity:
eiπ + 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
- Choose whether your angles will be expressed using decimals or as multiples of π.
- 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.)
- You can drag point P to change the angle θ.
- When you set θ = π, you'll see (the equivalent of) Euler's Identity: eiπ = − 1
Angles: decimals multiples of π
Imaginary symbol: i j
Rectangular and Polar equivalents
For some background information on what's going on, and more explanation, see the previous pages,
See also the polar to rectangular and rectangular to polar calculator, on which the above is based:
Next, we move on to see how to calculate Products and Quotients of Complex Numbers