# The Mathematical Ninja and the Power of Ten

“We’ve been through this a hundred times, sensei. I say something like ‘$10^{1.35}$. Hm, let me get my calculator’ and you torture me in some unspeakable way an blurt out the answer…”

“22.4”

“… thank you, especially for refraining from the torture bit.”

“You’re welcome.”

“Then, of course, you tell me how you did it.”

“$10^1$ is 10. Obviously. $10^{0.3}$ is a smidge less than 2, so we’re somewhere a bit more than 20.”

“OK.”

“$10^{0.03}$ is about $\frac{15}{14}$ and $10^{0.01}$ is about $\frac{45}{44}$.”

“Those ones I didn’t know.”

“Well, you do now.”

“So you did $20 \times \frac{225}{196} \times \frac{44}{45}$?”

“No. I said that multiplying by $\frac{15}{14}$ is the same as adding a bit more than 7%, and that $\frac{45}{44}$ is the same as adding about 2.3%. So here, I needed to add about 12% to my 20, which is 22.4.”

“Can I check the actual answer?”

“Do you feel lucky ((22.387)) ?”

* Edited 2020-09-14 to fix LaTex. Thanks to @hartkp for pointing it out!