PM [B10] In comes Euler
0 likes491 views
The document discusses Leonhard Euler's contributions to mathematics, particularly his introduction of imaginary numbers in equations and the origins of Euler's formula. It highlights the elegance of the equation that relates fundamental mathematical constants (0, 1, e, i, and π) and its profound implications in various fields such as engineering and quantum mechanics. Euler's formula is described as a crucial tool for simplifying complex trigonometric problems into algebraic solutions, significantly impacting areas like signal processing and digitization.
![© ABCC Australia 2015 new-physics.com
IN CAME EULER
PM [B010]](https://image.slidesharecdn.com/pmb10incomeseuler-151020180420-lva1-app6891/85/PM-B10-In-comes-Euler-1-320.jpg)
![© ABCC Australia 2015 new-physics.com
To be continued on PM [011]
ABCC](https://image.slidesharecdn.com/pmb10incomeseuler-151020180420-lva1-app6891/85/PM-B10-In-comes-Euler-11-320.jpg)
PM [B10] In comes Euler
- 1. © ABCC Australia 2015 new-physics.com IN CAME EULER PM [B010]
- 2. © ABCC Australia 2015 new-physics.com Leonhard Euler The French mathematician Leonhard Euler (1707-1783) was fascinated by the similarity of these last three formulas. He ingeniously solved the problem by introducing the imaginary number 𝑖𝑖 in the equations. He had worked it out in a way different from what we are doing now. It was more analytic and harder to work out. Neither did he leave an equation exactly the same as we have now. But the credit of finding the solution was accredited to him.
- 3. © ABCC Australia 2015 new-physics.com Expansion of 𝑒𝑒𝑖𝑖𝑥𝑥 By the Maclaurin method, the vector 𝑒𝑒𝑖𝑖𝑥𝑥 can also be expanded into a similar infinite series: 𝑒𝑒 𝑥𝑥 = 1 + 𝑥𝑥 + 𝑥𝑥2 2! + 𝑥𝑥3 3! + 𝑥𝑥4 4! + 𝑥𝑥5 5! + 𝑥𝑥6 6! + 𝑥𝑥7 7! + · · · 𝑒𝑒𝑖𝑖𝑥𝑥 = 1 + 𝑖𝑖𝑖𝑖 − 𝑥𝑥2 2! − 𝑖𝑖𝑥𝑥3 3! − 𝑥𝑥4 4! + 𝑖𝑖𝑥𝑥5 5! − 𝑥𝑥6 6! − 𝑖𝑖 𝑥𝑥7 7! + · · · 𝑒𝑒 𝑥𝑥 → 𝑒𝑒𝑖𝑖𝑥𝑥
- 4. © ABCC Australia 2015 new-physics.com Shuffling the Terms Rearranging the terms into two groups: 𝑒𝑒𝑖𝑖𝑖𝑖 = 1 + 𝑖𝑖𝑖𝑖 − 𝑥𝑥2 2! − 𝑖𝑖𝑥𝑥3 3! − 𝑥𝑥4 4! + 𝑖𝑖𝑥𝑥5 5! − 𝑥𝑥6 6! − 𝑖𝑖 𝑥𝑥7 7! + · · · 𝑒𝑒𝑖𝑖𝑖𝑖 = 1 − 𝑥𝑥2 2! − 𝑥𝑥4 4! − 𝑥𝑥6 6! + · · · + 𝑖𝑖 𝑥𝑥 − 𝑥𝑥3 3! + 𝑥𝑥5 5! − 𝑥𝑥7 7! + · · ·
- 5. © ABCC Australia 2015 new-physics.com The 𝑖𝑖 does it Compare this: 𝑒𝑒𝑖𝑖 𝑖𝑖 = 1 − 𝑥𝑥2 2! − 𝑥𝑥4 4! − 𝑥𝑥6 6! + · · · + 𝑖𝑖 𝑥𝑥 − 𝑥𝑥3 3! + 𝑥𝑥5 5! − 𝑥𝑥7 7! + · · · With the cosine and sine equations again: 𝑐𝑐𝑐𝑐𝑐𝑐 𝑥𝑥 = 1 − 𝑥𝑥2 2! + 𝑥𝑥4 4! − 𝑥𝑥6 6! + · · · 𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥 = 𝑖𝑖 𝑥𝑥 − 𝑥𝑥3 3! + 𝑥𝑥5 5! − 𝑥𝑥7 7! + · · · It is not hard to see that: 𝑒𝑒𝑖𝑖𝑖𝑖 = 𝑐𝑐𝑐𝑐𝑐𝑐 𝑥𝑥 + 𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠 𝑥𝑥
- 6. © ABCC Australia 2015 new-physics.com 𝒙𝒙 → 𝜽𝜽 Now, it’s time for the medusa to come in. She connects the linear world with the circular universe but in a slightly different way from the centaur. She links up the arc with the radius: 𝜃𝜃 = 𝑠𝑠 𝑟𝑟 She is a hybrid and easily slip into the equation which becomes: 𝑒𝑒𝑖𝑖 𝜃𝜃 = 𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 + 𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝜃𝜃
- 7. © ABCC Australia 2015 new-physics.com Where is the Centaur? But this is not the famous Euler equation - it is only the origin of the Euler equation. When the vector rotates to the other side of the real axis, that is, to where −1 is, the angle becomes 𝜋𝜋 (in radian measure). 𝑒𝑒𝑖𝑖 𝜋𝜋 = 𝑐𝑐𝑐𝑐𝑐𝑐 𝜋𝜋 + 𝑖𝑖 sin 𝜋𝜋 = −1 + 0 = −1 𝑒𝑒𝑖𝑖 𝜋𝜋 + 1 = 0 Reputed to be the most ‘beautiful’ equation in science and mathematics. 𝑂𝑂 𝐴𝐴 𝜃𝜃 = 𝜋𝜋 𝜋𝜋
- 8. © ABCC Australia 2015 new-physics.com The Beauty of Equation Now, maybe you’ve never thought of math equations as “beautiful,” but look at that result: It combines the five most fundamental numbers in math − 0, 1, e, 𝑖𝑖, and 𝜋𝜋 − 𝑖𝑖𝑖𝑖 a relation of irreducible simplicity. Lee Simmons@actual_self 𝜋𝜋 𝑖𝑖 𝑒𝑒 𝜃𝜃
- 9. © ABCC Australia 2015 new-physics.com The beautiful Equation Even more astonishing if you slog through the proof, which involves infinite sums, factorials, and fractions nested within fractions within fractions like matryoshka dolls. And remember, e and 𝜋𝜋 are infinitely long decimals with seemingly nothing in common; they’re the ultimate jigsaw puzzle pieces. Yet they fit together perfectly—not to a few places, or a hundred, or a million, but all the way to forever. Lee Simmons@actual_self 𝑒𝑒𝑖𝑖 𝜋𝜋
- 10. © ABCC Australia 2015 new-physics.com The Most Beautiful Equation But the weirdest thing about Euler’s formula—given that it relies on imaginary numbers—is that it’s so immensely useful in the real world. By translating one type of motion into another, it lets engineers convert messy trigonometric problems into more tractable algebra—like a wormhole between separate branches of math. It’s the secret sauce in Fourier transforms used to digitize music, and it tames all manner of wavy things in quantum mechanics, electronics, and signal processing; without it, computers might not exist. Lee Simmons@actual_self 𝑒𝑒𝑖𝑖 𝜃𝜃
- 11. © ABCC Australia 2015 new-physics.com To be continued on PM [011] ABCC