Euler Formula and Euler Identity Calculator plus Interactive Graph

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

e^iθ = cos(θ) + i sin(θ)

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

e^iπ + 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.

Euler's Formula

Euler's Identity

Euler's Identity states that for any complex number `z`:
`z^0 = 1`
`z^1 = z`
`z^2 = -1`
`z^3 = -z`
`z^n = (-1)^n*z^n`

Both the formula and the identity can be used to perform calculations, as well as to graph functions. The calculator can be used to input a complex number and calculate various powers of that number, as well as to graph the function. The graph can be zoomed in or out, and the complex number can be changed to see how it affects the graph.

Euler's Formula and Identity are both very useful in mathematics and engineering, as they can make many calculations much easier to perform. Try playing around with the calculator and graph to see what you can discover!

Euler's Formula and Identity Calculator and Graph

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 (the equivalent of) Euler's Identity: e^iπ = − 1

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

Complex Exponential Form

In mathematics, a complex exponential function is a function of the form `f(z) = ea(z)`, where `z` is a complex number, and `a` is an arbitrary complex constant. The function is entire, meaning that it is defined and differentiable for all complex numbers `z`. Unlike real-exponential functions, complex exponential functions are not one-to-one; that is, two different complex numbers may have the same image under the function.

The graph of a complex exponential function is a rotational symmetry about the origin with angle equal to the imaginary part of the constant `a`. If the real part of `a` is positive, then the graph will approach infinity as `|z|` approaches infinity; if the real part of `a` is negative, then the graph will approach zero as `|z|` approaches infinity.

The inverse of a complex exponential function is also a complex exponential function. That is, if `f(z) = ea(z)`, then `f^{-1}(z) = e^{-a}(z)`. This can be seen by solving for `z` in terms of `w` in the equation `f(z) = w`. We have `f(z) = e^{a}(z) = w`, so taking the natural logarithm of both sides we have `\ln(f(z)) = \ln(w)`. But since `\ln(f(z)) = a + bi` and `\ln(w) = ln|w| + i\arg(w)`, we have that `a + bi = ln|w| + i\arg(w)`. Therefore, `z = e^{-(ln|w| + i\arg(w))}= e^{-ln|w|}e^{-i\arg{w}}= |w|^{-1}e^{-i\arg{w}}= w^{-1}`. So `f^{-1}(w)= w^{-1}= e^{-(ln|w|+i\arg{w})}= e^{-(a+bi)}= e^{-a}e^{-bi}`.

Complex Functions

A complex function is a function of the form `f(z) = u(x, y) + iv(x, y)`, where `z = x + iy` is a complex variable, `u` and `v` are real-valued functions of the two real variables `x` and `y`, and `i` is the imaginary unit. This article will explore some of the properties of complex functions, with a focus on euler's formula and euler's identity.

Complex functions have many applications in physics and engineering, as well as in pure mathematics. In physics, they are used to model wave phenomena, while in engineering they are used to design electrical circuits. In pure mathematics, they are studied for their own sake as beautiful and intriguing objects.

Differential Equations

Differential equations are a subset of mathematical equations that allow for the calculation of rates of change. In other words, differential equations enable mathematicians to predict how a system will change over time.

Euler's formula is one of the most famous examples of a differential equation. This equation, which states that `e^(i*theta) = cos(theta) + i*sin(theta)`, is used in many branches of mathematics, including calculus and complex analysis. Euler's identity, another well-known differential equation, states that `e^(i*pi) + 1 = 0`. This equation is significant because it links together five of the most important numbers in mathematics: `e`, `pi`, `i`, `1`, and `0`.

Differential equations are immensely useful in mathematical modeling and have applications in a wide variety of fields, from physics to economics. Consequently, they are an essential tool for understanding and predicting change.

If you have a firm understanding of Euler's Formula and Euler's Identity, you can perform a wide range of calculations with relative ease. This makes the formula an invaluable tool for mathematicians, engineers, and other scientists. With that in mind, we encourage you to use the calculator above to explore how this identity can be used to simplify various equations.

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