Blog Posts
If the Sum of the Digits of a Number is Divisible by Three, the Number Is Divisible by Three [August 22, 2019]
Miscellaneous
Phi (\(\phi\)) is the golden ratio.
\({\Large \phi = \frac{ 1 + \sqrt{5} }{2}}\)
Look into Toads and Frogs game. See OEIS “A005563 a(n) = n*(n+2) = (n+1)^2 – 1″ or OEIS search for “frog”.
x^2+(5y/4-sqrt(abs(x)))^2=1
x^2+(5y/4-sqrt(abs(x)))^2=1
Numbers
\(3^3 + 4^4 + 3^3 + 5^5 = 3435\) — The only other number with this property is 1.
Notation
\({}^n C {}_r\) — N Choose R
\({}^n C {}_r\) (“n choose r”) is an alternate notation for the binomial symbol:
$${}^n C {}_r = {n \choose r} = \frac{n!}{r!(n−r)!}$$
\({}^n C _r\) is an alternate notation for the binomial symbol \({n \choose r} = \frac{n!}{r!(n−r)!}\)
Inline style: \({}^{10}C_{3} = {10 \choose 3} = \frac{10!}{3!(10−3)!} = \frac{10!}{3! 7!} = \frac{3628800}{6×5040} = \frac{3628800}{30240} = 120\)
\({}^{10}C_{3} = {10 \choose 3} = \frac{10!}{3!(10−3)!} = \frac{10!}{3! 7!} = \frac{3628800}{6×5040} = \frac{3628800}{30240} = 120\)
Display style: $${}^{10}C_{3} = {10 \choose 3} = \frac{10!}{3!(10−3)!} = \frac{10!}{3! 7!} = \frac{3628800}{6×5040} = \frac{3628800}{30240} = 120$$
$${}^{10}C_{3} = {10 \choose 3} = \frac{10!}{3!(10−3)!} = \frac{10!}{3! 7!} = \frac{3628800}{6×5040} = \frac{3628800}{30240} = 120$$
\(\sum_{n=1}^{\infty}\) — Summation
Inline style: \(\sum_{n=1}^{\infty} 2^{-n} = 1\)
\(\sum_{n=1}^{\infty} 2^{-n} = 1\)
Display style: $$\sum_{n=1}^{\infty} 2^{-n} = 1$$
$$\sum_{n=1}^{\infty} 2^{-n} = 1$$
Inline style: \(\sum\limits_{k=1}^{n} k\)
\(\sum\limits_{k=1}^{n} k\)
Display style: $$\sum\limits_{k=1}^{n} k$$
$$\sum\limits_{k=1}^{n} k$$
See Also
Xcas, the swiss knife for mathematics. Giac/Xcas is a free computer algebra system for Windows, Mac OS X and Linux/Unix.
Use Xcas online in your web browser.
References
Alekseyev, Max. “PARI/GP Scripts for Miscellaneous Math Problems“. George Washington University.
De Graeve, Renée; Parisse, Bernard. “Symbolic Algebra and Mathematics with Xcas“. University of Grenoble I.
“MathJax TeX and LaTeX Support“. MathJax Consortium.
Wikipedia contributors. “List of computer algebra systems“. Wikipedia.