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

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.

"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.

"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?"

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

"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!"

"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."

Does your calculator break like this? Why?1

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

1. Yes, I have an idea why. []

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