I might be different, but 355/113 is very easy to remember and is not 7 unique segments of information. I just think "double the odds"

113355

We know we want a fraction, not a single number, so split down the middle:

113/355

And we know pi won't be less than 1, so flip it: 355/113. Knowing that the digits sequences are doubled lets you do some cheap, mental run length encoding. In a case where you need a hand calculation, spending 3 seconds to re-derive the sequence seems tolerable.

Unwillingly, I just memorized a Pi approximation.

As did I.

I came back here 6 hours after reading it to confirm that it is still stuck in my brain, I am guessing permanently. Although I already knew Pi to 6 decimal places so its not like I am gaining a lot of accuracy out of this.

Required information for your approach:

Repeat the digits

Use odd digits

Use increasing sequence

Use three digits

Split the result in half and divide

Pi is greater than 1

But since humans have associative memory, these are easier to learn than a seemingly arbitrary digits sequence.

Rational approximations are good for finding reasonable answers without a calculator. My contrived example is the area of a circle, r=7. i can do 22 * 7 in my head and get 154, which is pretty darn close to 153.86. 7 is awkward (for me anyway) but it does offer opportunities to cancel.

How fast can you do 22 * 7 in your head? I get a lot of mileage out of simply approximating pi=3, and optionally adding in a factor of 10% later. For your example, I know 7^2=49 immediately, which I round to 50, 50 * 3=150 which is fairly close to 153.938. (Use more digits of pi!) I can quickly improve my estimate by adding 4.9. (Edit: It's also fairly trivial to get even better estimates quickly from here, but at this point it's probably faster to grab a cell phone / [favorite language] repl. From 154.9, subtract the additional 3 gained from using 50 instead of 49, now 151.9, add 4.9/2~=2.4, now 154.3, subtract 4.9 * 10% ~= .5, 153.8.)

22 times a single digit is pretty easy, double the digit, shift the decimal, add the doubled number - but it's not trivial. As i said, a contrived example.

The point is, you yourself stick to integer arithmetic, then try to fix it up with a 10% modifier at the end. People have been using pi for a long time. Easy access to calculators is, what, about 50 years now? I'll happily agree that rational representation is a historical artifact. But i still believe the vast majority of people doing arithmetic pre 1960 with floating point numbers did it like you and i do. They would put off the decimal representation as long as possible.

> optionally adding in a factor of 10% later

You're better off not adding anything. If you could add 5% instead, then you'd be much better still.

Ideally you add 14%. Why do you say 5% (or nothing) is better than 10%?

He's taking the % of the wrong number: the number you get after multiplying by three, not the original. I do what you do: 3, +10% + 5% (which is easy after getting the 10%).

It is easier to add 5% first, then triple, skipping the middle step.

Not for me- I get 10% to get the 5% so I have it anyway.

However, you could work from the 3 you've already used as the tripling in your model, then take 10% of that then 1/2 of that to get there. I've just never done it that way (these are things I just do without thinking too deeply about).