What is Euler's Identity?

Co je Eulerova identita?

e^iπ + 1 = 0

Pět základních konstant. Jedna rovnice. Nic dalšího není třeba.

Pět konstant

e

Eulerovo číslo≈ 2.71828…

Základ přirozených logaritmů. Řídí růst a rozpad.

i

Imaginární jednotka= √(−1)

Splňuje i² = −1. Základ komplexních čísel.

π

Pí≈ 3.14159…

Poměr obvodu kruhu k jeho průměru.

1

Jedna

Neutrální prvek násobení. Libovolné číslo × 1 = samo sebe.

0

Nula

Neutrální prvek sčítání. Libovolné číslo + 0 = samo sebe.

Eulerova identita plyne z Eulerova vzorce: e^ix = cos(x) + i·sin(x). Dosazením x = π dostaneme e^iπ = cos(π) + i·sin(π) = −1, tedy e^iπ + 1 = 0.

Krok za krokem

Eulerův vzoreceⁱˣ = cos(x) + i·sin(x)

Dosaď x = πeⁱπ = cos(π) + i·sin(π)

Vyhodnoťeⁱπ = −1 + 0i

Zjednodušeⁱπ = −1

Přičti 1eⁱπ + 1 = 0 ✓

Pohled na jednotkovou kružnici

e^iθ obíhá jednotkovou kružnici. Otočení o π končí v −1. Přičti 1 a získáš 0.

Proč ji matematici milují

Spojuje aritmetiku (0 a 1), algebru (i), geometrii (π) a matematickou analýzu (e) · čtyři různá odvětví matematiky · v jediné rovnici překvapivé jednoduchosti. Richard Feynman ji nazval „nejpozoruhodnějším vzorcem matematiky“.

Historie

Leonhard Euler (1707–1783) publikoval vzorec e^ix = cos(x) + i·sin(x) ve své knize Introductio in analysin infinitorum (1748). Identita je jeho zvláštním případem pro x = π. Euler zavedl nebo zpopularizoval zápis e, i, f(x), Σ a π.

Used in

∑Mathematics

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⚛Physics

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⚙Engineering

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🧬Biology

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💻Computer Sci

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📈Finance

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🎨Art

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🏛Architecture

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♪Music

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🔐Cryptography

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⚗Chemistry

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Více o e →Více o π →

Taylor series for e to the i pi showing it equals minus 1

The Taylor series for eˣ groups into cos(π) for the real terms and i·sin(π) for the imaginary terms. Since cos(π) = −1 and sin(π) = 0, we get e^(iπ) = −1, so e^(iπ) + 1 = 0.

Geometric meaning: rotation on the complex plane

The formula e^(i*theta) traces a unit circle on the complex plane as theta increases. e^(i*pi) is a rotation of exactly pi radians (180 degrees) from 1, landing at -1. Adding 1 brings you back to 0. This is why e^(i*pi) + 1 = 0: it is a half-turn of the complex plane expressed as an equation.

e^(iπ) is a half-turn: it sends every point to its opposite

e^(iθ) is a rotation operator. At θ=π you have rotated exactly half a circle. The point 1 on the real axis travels to -1. Adding 1 to both sides gives e^(iπ) + 1 = 0.

The five constants in Euler's identity

e^(iπ) + 1 = 0

e ≈ 2.71828 (natural growth) · i = √(−1) (imaginary unit)

π ≈ 3.14159 (circle ratio) · 1 (multiplicative identity) · 0 (additive identity)

Five fundamental constants, three operations (+, ×, exponentiation), one equation.

Related topics

E Pi Complex Numbers

Euler's Number e → Pi (π) → Complex Numbers → De Moivre's Theorem → Taylor Series →

Key facts about Euler's Identity

Euler's identity e^(i*pi) + 1 = 0 unites the five most important constants in mathematics: e (the base of natural logarithms), i (the imaginary unit), pi (the circle constant), 1 (the multiplicative identity), and 0 (the additive identity). It follows directly from Euler's formula e^(i*theta) = cos(theta) + i*sin(theta) by setting theta = pi. Since cos(pi) = -1 and sin(pi) = 0, we get e^(i*pi) = -1. First published by Euler around 1748. Voted the most beautiful equation in mathematics in multiple polls.

Used in

∑Mathematics

✓

⚛Physics

✓

⚙Engineering

✓

🧬Biology

–

💻Computer Sci

✓

📊Statistics

–

📈Finance

–

🎨Art

–

🏛Architecture

–

♪Music

–

🔐Cryptography

–

🌌Astronomy

–

⚗Chemistry

–

🦉Philosophy

–

🗺Geography

–

🌿Ecology

–

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Question

What does e^(iπ) represent geometrically?

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