Micro-challenge: Special Event

[johnb]

(12-23-2022 06:45 PM)rprosperi Wrote:  Thanks for replying Albert, and while this answer no doubt is useful to math-heads, I was hoping for an English explanation of the meaning of a negative base. Equations can show and even prove many things, tragically only a small percent of them are meaningful to the rest of is.

Okay, here you go! Merry Christmas, Bob!

Being super pedantic here, so that everything is clear.

We already know how to handle positive bases. Base 10 means that there the 1's place (10^0), the 10's place (10^1), 100's place (10^2), and so on. Same for base 2: 2^0, 2^1, and so on.

[At this point, the reader is supposed to say, "DUH, John. So what?"]

So let's take base -8.

(-8)^0 = (+)1's place.
(-8)^1 = -8's place.
(-8)^2 = (+)64's place.
(-8)^3 = -512's place.

Thinking about it a bit, it makes counting a BEAR.

0,1,2,3,4,5,6,7, then what? -10? Yup. Because the leading 1 here is in the -8's place, so you have to negate the entire string of digits.

Encoding and decoding by hand is a bear, too. (So glad we have calculators!)

What is 411 (base 10) in base -8?