pi in base 12 starts out 3.1848. Is this more or less accurate than the base 10 value 3.1416?

π ≈ 3.141592653589793...

The number 3.1848 (base 12) is equal to:

3 + (1 * 1/12) + (8 * 1/144) + (4 * 1/1728) + (8 * 1/20736)

≈ 3.141589506172839506172839506...

That's more accurate than 3.1416.

Answer:

More accurate

3.1416_10 = 3.1416_10

3.1848_12 = 3 *12^0 + 1/12^1 + 8/12^2 + 4/12^3 + 8/12^4 = [left for you]_10

So which is closer to π ~ 3.1415926535897932384626_10 ?

More accurate in this sense: The maximum error base 10 is 5/(10^5). The maximum error base 12 is 6/(12^5), a smaller number. To say which is actually closer you would have to compute the difference between the 2 values and the actual value to additional places. My very large bet is that the base 12 number is closer.

3.1848 in base-12 is equal to...

3(12^0) + 1(12^-1) + 8(12^-2) + 4(12^-3) + 8(12^-4)

= 3.14158950617

Pi to 20 digits is 3.14159265358979323846.

3.1416 is 0.00023384349% off from this value.

3.14158950617 is 0.00010018548% off from this value.

0.00010018548% < 0.00023384349%

So that number in base-12 is more accurate.

It's pretty easy to see why as well, because with higher bases, you can store more information with the same amount of digits. Let's look at the amount of different numbers we can store with only 5 digits at different bases:

base-10: 10^5 = 100,000

base-12: 12^5 = 248,832

So with only 5 digits, base-12 stores almost two-and-a-half times the amount of information as base-10.

more accurate because higher base has a higher value (magnitude) in the same number of digits.

to check:

3.18479 ➜ 3.1415774

3.18480 ➜ 3.1415895

3.18481 ➜ 3.1415935

So an error of ±0.00001 base 12 ➜ 3.1415935–3.1415774 = 0.000016

a higher error

It's not, pi is 3,1415926, so base 12 is wrong.