# $\left(1+ \frac {1}{n}\right)^n$

*This doesn’t appear to work on all models of calculator. Let me know whether yours handles it properly…*

“I saw this thing about Euler’s identity,” said the student, and the words “ut” and “oh” forced themselves, unbidden, into my head.

You’ve maybe seen it: it’s $e^{\pi i} + 1 \equiv 0$, or - if you prefer $e^{\pi i} \equiv -1$. I can go either way. It’s frequently voted the most beautiful equation in all of maths (even though it’s an identity rather than an equation).

I was thinking ut-oh because, in honesty, I didn’t fancy explaining the idea of an imaginary power to someone who has only just about grasped fractional ones. But he surprised me with his follow-up: “what’s $e$?”

### Well, that’s a freebie

“It’s about $2.718281828$,” I said, in forming a mnemonic to remember a constant in analysis.

“I know *that*,” said the student, as if he knew that. “But why is it important?”

My rabbit-hole alarm went off. This was an opportunity for a 10 minute lecture discussing the intricacies of calculus in the hope of inspiring the student to see that there’s way more to maths than what’s in the GCSE, but I picked a different route, since we’d just been talking about compound interest.

“Let’s say you’ve got £100 invested at 100% interest in some weird Ponzi-scheme bank account,” I said, brushing off his “what’s a Ponzi-scheme?” as irrelevant. “How much would you have after a year?”

“£200!” he said, brightly.

“Good. How about if you had interest payments every six months?”

“So 50% after 6 months, and 50% compounded after another six?”

“Yup.” I rolled my eyes as he reached for a calculator, but he was going to need it shortly anyway.

“£225,” he said.

### “What about every month?”

“Um…” tappity tap, “… so that’s 8.3% every month? Do I have to add that on each time? No, wait, we talked about this - you can just make it $1.083^{12}$, can’t you?”

“Use your answer button rather than rounding it, but yes.”

“£261.30,” he said, dollar signs starting to form in his eyes. “The more often you compound it, the more you get!”

I nodded. “How about daily?”

“Mutter mutter 0.274 mutter mutter “£271.46 - it’s not gone up by much, has it?”

### A brief aside about calculator use

If I was doing this - for example, to research a blog post - I’d be working out the number of interest periods (say, $365 \times 24$, ignoring that there are more than 365 days in a year, and less than 24 hours in a day) and then typing in $((Ans + 1) ÷ Ans)^{Ans}$ - and making copious use of the scroll-back function on my trusty Casio.

### Back to the action

“Not really, no. It’s almost as if there’s a hard limit!”

“How about every hour?”

“Go for it.”

“£271.81… that’s hardly any change. And isn’t it…?”

Luckily, I’d written $e$ on the board. “It’s almost $e$ times what you started with.”

“Minutes!” he insisted. “£271.8279! Within a fraction of a penny! We must be nearly there! Seconds! £271.8281615!”

“That’s pretty close.”

“Milliseconds!” he nearly yelled, triumphantly. “Two hundred and seventy one pounds… 74? That’s gone down.” He looked deflated.

I raised an eyebrow. “That’s not meant to happen,” I said. “I fear, my friend, you’ve got a defective calculator.”

### How about you?

Does your calculator break like this? Why? ((Yes, I have an idea why.))