Things I learned this week!
Once a week I'll be posting 5 interesting things I've learned followed by my favourite equation(s) of the week!
1. "Nanoputian are a series of organic molecules whose structural formulae appear human" -
http://en.wikipedia.org/wiki/Nanoputian
2. People wake board in cranberry fields! - http://www.youtube.com/watch?v=qGf6earGAOc
3. The term "mother load" is actually "mother lode" and is a term commonly used in mineral mining. - http://en.wikipedia.org/wiki/Mother_lode
4. The British Flag is actually a combination of the English, Scottish, and Irish Flags. - http://upload.wikimedia.org/wikipedi...Union_Jack.png
5. U.S army flag patches are always reversed when worn on the right shoulder. The blue field of stars should always be in the highest position of honour. Imagine the flag on a pole rather then a patch, therefore if something is moving the stripes will trail behind with the stars in the most honourable position - front, or in respect to the badge the top right. - http://www.marlowwhite.com/faq-why-i...-reversed.html
All of these are attributed to reddit.com/r/todayilearned
Equation of the week:
Feynman called it our "jewel" and "one of the most remarkable, almost astounding, formulas in all of mathematics."
[tex]e^{ix} = cosx + isinx[/tex]
Derivation:
A simple derivation comes from calculating the Maclaurin Series of [tex]e^{ix}, cosx, isinx[/tex].
[tex]M = \sum_{n = 0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n[/tex]
[tex]M_{cosx} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ... [/tex]
[tex]M_{isinx} = ix - \frac{ix^3}{3!} + \frac{ix^5}{5!} - \frac{ix^7}{7!} + ...[/tex]
[tex]M_{e^{ix}} = 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \frac{x^6}{6!} - \frac{ix^7}{7!} + ...[/tex]
It is clear then that adding
[tex](M_{cosx} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...) + (M_{isinx} = ix - \frac{ix^3}{3!} + \frac{ix^5}{5!} - \frac{ix^7}{7!} + ...)[/tex]
[tex]=[/tex]
[tex]M_{e^{ix}} = 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \frac{x^6}{6!} - \frac{ix^7}{7!} + ...[/tex]
I hope you guys enjoyed!
1. "Nanoputian are a series of organic molecules whose structural formulae appear human" -
http://en.wikipedia.org/wiki/Nanoputian
2. People wake board in cranberry fields! - http://www.youtube.com/watch?v=qGf6earGAOc
3. The term "mother load" is actually "mother lode" and is a term commonly used in mineral mining. - http://en.wikipedia.org/wiki/Mother_lode
4. The British Flag is actually a combination of the English, Scottish, and Irish Flags. - http://upload.wikimedia.org/wikipedi...Union_Jack.png
5. U.S army flag patches are always reversed when worn on the right shoulder. The blue field of stars should always be in the highest position of honour. Imagine the flag on a pole rather then a patch, therefore if something is moving the stripes will trail behind with the stars in the most honourable position - front, or in respect to the badge the top right. - http://www.marlowwhite.com/faq-why-i...-reversed.html
All of these are attributed to reddit.com/r/todayilearned
Equation of the week:
Feynman called it our "jewel" and "one of the most remarkable, almost astounding, formulas in all of mathematics."
[tex]e^{ix} = cosx + isinx[/tex]
Derivation:
A simple derivation comes from calculating the Maclaurin Series of [tex]e^{ix}, cosx, isinx[/tex].
[tex]M = \sum_{n = 0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n[/tex]
[tex]M_{cosx} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ... [/tex]
[tex]M_{isinx} = ix - \frac{ix^3}{3!} + \frac{ix^5}{5!} - \frac{ix^7}{7!} + ...[/tex]
[tex]M_{e^{ix}} = 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \frac{x^6}{6!} - \frac{ix^7}{7!} + ...[/tex]
It is clear then that adding
[tex](M_{cosx} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...) + (M_{isinx} = ix - \frac{ix^3}{3!} + \frac{ix^5}{5!} - \frac{ix^7}{7!} + ...)[/tex]
[tex]=[/tex]
[tex]M_{e^{ix}} = 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \frac{x^6}{6!} - \frac{ix^7}{7!} + ...[/tex]
I hope you guys enjoyed!
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