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Euler's Identity: Definition & Example

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  • 0:03 Euler's Identity
  • 0:49 Why Euler's Identity…
  • 1:57 Euler's Formula
  • 3:39 An Example
  • 5:18 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn about Euler's Identity and why it is considered a mathematical jewel. Find out about the person who came up with it and also see an example of it in use.

Euler's Identity

Euler's identity is the famous mathematical equation e^(i*pi) + 1 = 0 where e is Euler's number, approximately equal to 2.71828, i is the imaginary number where i^2 = -1, and pi is the ratio of a circle's circumference to the circle's diameter approximately equal to 3.14. It is named after Leonhard Euler, a Swiss mathematician who discovered this formula in the 1700s.

Why is this worth remembering? It is worth remembering because it is the only equation that so simply links together the mathematical constants of pi, i, and e along with 0 and 1.

Why Is Euler's Identity Important?

Mathematicians love Euler's identity because it is considered a mathematical beauty since it combines five constants of math and three math operations, each occurring only one time. The three operations that it contains are exponentiation, multiplication, and addition. The five constants that this equation combines are the number 0, the number 1, the number pi, the number e, and the number i.

We know the numbers 0 and 1. We recall that the number pi is approximately 3.14 and it goes on forever. The number e, like the number pi, continues forever and is approximately 2.71828. The number i is our imaginary number where i^2 is equal to -1.

Why is this so beautiful for mathematicians? It is a beauty because it is such a simple equation that shows the relationship of so many constants of math. Can you think of other equations that are just as simple and that relate just as many constants together?

Euler's Formula

Euler's identity is actually a special case of Euler's formula, e^(i*x) = cos x + i sin x, when x is equal to pi. When x is equal to pi, cosine of pi equals -1 and sine of pi equals 0, and we get e^(i*pi) = -1 + 0i. The 0 imaginary part goes away, and we get e^(i*pi) = -1. Moving the -1 over to the other side by adding gives us Euler's identity. Looking at Euler's formula, e^(i*x) = cos x + i sin x, we see that e taken to an imaginary power equals a complex number consisting of a real part (the cosine part) and an imaginary part (the sine part).

I have to caution you when you use this formula. You might be tempted to use degrees for the x when calculating the cosine and sine, but this formula uses radians. Radians are a different way of measuring angles. So, when using your calculator to work out problems, make sure your calculator is set up for making calculations in radians and not degrees.

One other thing to note here is that sometimes mathematicians will abbreviate cos x + i sin x as cis x, so you might see Euler's formula written as e^(i*x) = cis x.

An Example

How do we use Euler's identity and Euler's formula? We use Euler's identity to show how the constants e, pi, and i are related. Euler's formula is used to help us calculate e to an imaginary power. Let's see how we go about calculating a sample number.

Let's use Euler's formula to help us calculate e^(i*2). Looking at what we want to calculate, we see that the 2 is in the place of our x, so that tells us to replace the x in our cosine and sine with 2.

e^(i*2) = cos 2 + i sin 2

Now, we need to calculate what cosine of 2 and sine of 2 equal. We turn to our calculator and remember to change our calculations to radians. Plugging in cosine of 2 in radians, we get -0.416. Sine of 2 gives us 0.909.

e^(i*2) = -0.416 + 0.909 i

Now that I've calculated by sine and cosine, I write it in complex number format with the real part first followed by the imaginary part. The imaginary part has the i following the value. That is why I have written the i after the 0.909. When we get to this point, we are done.

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