3Blue1Brown - e^(iπ) in 3.14 minutes, using dynamics (2024)

The Real Case

One way to think about the function $e^t$ et is to ask what properties it has. Probably the most important one, from some points of view the defining property, is that it’s a function which is equal to its own derivative. Together with the added condition that inputting $0$ 0 returns $1$ 1, it’s the only function with this property.

You can illustrate what that means with a physical model: Think of $e^t$ et as your position on a number line as a function of time. The condition $e^0 = 1$ e0=1 means you start at $1$ 1. The equation above is saying that your velocity, the derivative of position, is always equal to your position. In other words, the farther away from 0 you are, the faster you move, with the very specific constraint that your velocity vector is perpetually identical to your position vector.

3Blue1Brown - e^(iπ) in 3.14 minutes, using dynamics (2)

So even before knowing how to compute $e^t$ et exactly, this ability to associate each position with the velocity you must have at that position paints a very strong intuitive picture of how the function must grow. You know you’ll be accelerating, at an accelerating rate, with an all-around feeling of things getting out of hand quickly.

3Blue1Brown - e^(iπ) in 3.14 minutes, using dynamics (3)

If we add a constant to this exponent, like $e^{2t}$ e2t, the chain rule tells us the derivative of our function is now $2$ 2 times the function itself.

So at every point on the number line, rather than attaching a vector corresponding to the number itself, first double the magnitude, then attach it. Moving so that your position is always $e^{2t}$ e2t is the same thing as moving in such a way that your velocity is always twice your position. The implication of that $2$ 2 is that our runaway growth feels all the more out of control.

The complex case

What about if the constant was $i$ i, the imaginary unit? If your position was always $e^{it}$ eit, how would you move as that time, $t$ t, ticks forward?

Well, assuming there’s any way to make sense out of $e^{it}$ eit, and assuming that derivatives still work the way we’d expect when extending to complex numbers, the derivative of your position $e^{it}$ eit would now always be $i$ i times itself. Multiplying by $i$ i has the effect of rotating numbers 90-degrees, and as you might expect, things only make sense here if we start thinking beyond the number line and in the complex plane.

3Blue1Brown - e^(iπ) in 3.14 minutes, using dynamics (8)

Geometrically, which of the following describes the point $i \cdot (a + bi)$ i⋅(a+bi) on the complex plane?

A 90-degree clockwise rotation of the point $a + bi$ a+bi about the origin.

A 90-degree counterclockwise rotation of the point $a + bi$ a+bi about the origin.

A reflection of the point $a + bi$ a+bi about the real axis.

A reflection of the point $a + bi$ a+bi about the imaginary axis.

So what does $e^{it}$ eit mean?

So even before you know how to compute $e^{it}$ eit, you know that for any position this might give for some value of t, the velocity at that time will be a 90-degree rotation of that position.

3Blue1Brown - e^(iπ) in 3.14 minutes, using dynamics (9)

Drawing this for all possible positions you might come across, we get a vector field, where, usually with vector fields we shrink things down to avoid clutter.

3Blue1Brown - e^(iπ) in 3.14 minutes, using dynamics (10)

At time $t=0$ t=0, $e^{it}$ eit will be 1. There’s only one trajectory starting from that position where your velocity is always matching the vector it’s passing through, a 90-degree rotation of position. It’s when you go around the unit circle at a speed of 1 unit per second.

3Blue1Brown - e^(iπ) in 3.14 minutes, using dynamics (11)

So after $\pi$ π seconds, you’ve traced a distance of $\pi$ π around; $e^{i\pi} = -1$ eiπ=−1.

3Blue1Brown - e^(iπ) in 3.14 minutes, using dynamics (12)

After $\tau = 2\pi$ τ=2π seconds, you’ve gone full circle; $e^{i\tau} = 1$ eiτ=1.

3Blue1Brown - e^(iπ) in 3.14 minutes, using dynamics (13)

And more generally, $e^{it}$ eit equals a number $t$ t radians [1] around this circle.

Nevertheless, something might still feel immoral about putting an imaginary number up in that exponent. And you’d be right to question that! For values of $x$ x which are not real numbers, what we write as $e^x$ ex has very little to do with repeated multiplication, and its relation to the number $e$ e feels frankly more incidental than definitional. To dig in a little bit deeper, the next lesson covers more general exponents, with a focus on matrices.

[1] - radian(s), SI unit of angle, 1 radian is equal to an angle at the center of a circle where the arc length of the circular sector is equal to the radius of the circle

3Blue1Brown - e^(iπ) in 3.14 minutes, using dynamics (2024)

FAQs

How to calculate e-IPI? ›

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.

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Why 3blue1brown? ›

The channel name and logo reference the color of Grant's right eye, which has blue-brown sectoral heterochromia. It also symbolizes the channel's visual approach to math.

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What is e-power i? ›

The value of e to the power of 1 is 2.718281828459045…

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What is the significance of Euler's identity? ›

Euler's identity is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in a proof that π is transcendental, which implies the impossibility of squaring the circle.

What is the most beautiful equation in the world? ›

Euler's pioneering equation, the 'most beautiful equation in mathematics', links the five most important constants in the subject: 1, 0, π, e and i.

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Why is e so important? ›

'e' is the base for Natural Logarithms and has profound implications in calculus. 'e' holds significance in calculus, probability theory, geometric progression, and wave mechanics. The unique number 'e' significantly links exponential growth and calculus.

What is the use of Euler's number in real life? ›

It frequently appears in problems dealing with exponential growth or decay, where the rate of growth is proportionate to the existing population. In finance, e is also used in calculations of compound interest, where wealth grows at a set rate over time.

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What is the real life application of Euler's identity? ›

Applications of Euler's Formula: It influences various fields such as engineering, physics, and computer science by simplifying complex calculations, transforming differential equations into algebraic ones, and allowing representations of infinite series and relations between trigonometric functions and exponentials.

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What does Euler's formula tell us? ›

Euler's formula in geometry is used for determining the relation between the faces and vertices of polyhedra. And in trigonometry, Euler's formula is used for tracing the unit circle.

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What is the e in the pert formula? ›

A = P × e^rt

A = Amount of money after a certain amount of time. P = Principle or the amount of money you start with. e = Napier's number, which is approximately 2.7183. r = Interest rate and is always represented as a decimal.

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How is e-ipi =- 1? ›

The mystery lies in the definition of i and what it means to have it in the exponent. It just turns out that with the definitions we have of those elements one ends up with the eiπ=−1 equation. You can think of eiθ as a rotation on complex plane. a rotation of π will take you from 1 to -1.

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What is the value of e to the power ipi? ›

Euler's formula: e^(i pi) = -1.

What is the formula for Euler's theorem? ›

Euler's formula, either of two important mathematical theorems of Leonhard Euler. The first formula, used in trigonometry and also called the Euler identity, says e^ix = cos x + isin x, where e is the base of the natural logarithm and i is the square root of −1 (see imaginary number).

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