Evaluation of Powers https://janmr.com/posts/evaluation-of-powers/ #programming #algorithm #cpp #exponentiation #optimality

• \( x^0 = 1 \)
• \( x^{2k} = (x^k)^2 \)
• \( x^{2k+1} = x \cdot x^{2k} \)

Never thought exponents as time:

Understanding Exponents (Why does 0^0 = 1?)

https://betterexplained.com/articles/understanding-exponents-why-does-00-1

#math #arithmetic #exponentiation #mathematics #betterexplained

Attn Calc II students: This is a fun little integral. \( \int 2^{\ln(x)}dx \) A substitition \( u=\ln(x) \) won't work since there is no \( \frac{1}{x} \) to make the match. #integration #exponentiation

Think about before reading on.

Nifty trick! Since \( 2=e^{\ln(2)} \), then \( 2^{\ln(x)} = (e^{\ln(2)})^{\ln(x)}=(e^{\ln(x)})^{\ln(2)}=x^{\ln(2)} \)

Now the integral becomes \( \int x^{\ln(2)}dx = \frac{1}{1+ln(2)} x^{1+\ln(2)} + C \)

Swapping from an constant-to-variable to a variable-to-constant made a world of difference. Sweet!

#functionalprogramming

Many people's vote for most #beautiful construct in #math is \[ e^{i\pi}+1=0. \]

Yeah, maybe. But I think a close contender (if you include the #CS realm) is \[ \lambda b.\lambda e.eb . \]

This serves as the complete #Church encoding of #exponentiation in #LambdaCalculus, driving home subtler points about #function #mapping and ordered pairs and the primacy of exponentiation over add/mult, both of which have uglier #LC representations.

https://en.wikipedia.org/wiki/Lambda_calculus#Arithmetic_in_lambda_calculus

#math #education #school #arithmetic

At risk of pointing out the obvious, here's something that didn't occur to me in these plain terms until last week:

Everyone from little kids on up understands base-10 numbers to one degree or another, and yet decoding place-value #numbers requires using #addition, #multiplication, and #exponentiation in concert.