Tools for Thought
It may seem unimportant, but when I started out […] People drew pictures of surfaces one way and I started drawing them a different way. There's something significant about how the representation in your head profoundly changes how you think. It's very hard to do a brain dump. … Words are one thing, […] There are many precise mathematical words that could be used, but they don't automatically convey a feeling for it. I probably can't convey a feeling for it either, but I want to try." - William Thurston, The Mystery of 3-Manifolds 𝑓 ( 𝑥 ) = 2 𝑥 f(x)=2 x 2 ⋅ 2 𝑥 = 2 1 + 𝑥 2⋅2 x =2 1+x 2 𝑎 ⋅ 2 𝑏 = 2 𝑎 + 𝑏 2 a ⋅2 b =2 a+b ( 2 𝑥 ) 2 = 2 2 𝑥 (2 x ) 2 =2 2x ( 2 𝑎 ) 𝑏 = 2 𝑎 ⋅ 𝑏 (2 a ) b =2 a⋅b 𝑦 = 2 𝑥 ⇔ 𝑥 = 𝑙 𝑜 𝑔 2 𝑦 y=2 x ⇔x=log 2 y 𝑓 ( 𝑥 ) = 𝑥 2 f(x)=x 2 𝑥 2 + 2 𝑥 = ( 𝑥 + 1 ) 2 − 1 x 2 +2x=(x+1) 2 −1 𝑥 2 + 𝑏 𝑎 𝑥 + 𝑐 𝑎 = ( 𝑥 + 𝑏 2 𝑎 ) 2 + 𝐶 x 2 + a b x+ a c =(x+ 2a b ) 2 +C