a reply to: Peeple
Well, the whole point of using modulus is to not have to deal with them as fractions. But there's several different ways to handle such equations.
What I gave was a rather lengthy explanation to demonstrate how modulus doesn't have to be a lengthy process, lol. You use what works for you.
I think where a lot of people get confused is they try to work the problem in base11. That's possible, but it can also be a bit laborious and requires
a good understanding of different bases.
Your statement above about imagining mod11 as a clock with 11 hours is spot on. In reality, we use a base12/base60 system for measuring time anyway,
so it's a good way to point out how the modulus works. You could think of mod11 as such a clock and move one "hour" for every count. When you're
finished, whatever hour you are pointing to is the modulus. How many times that clock went around before you finished is the complete number.
You can also just divide and subtract, like this. Divide the first result (16) by 11 to get 1.4545. Now take the whole part of that, the 1, and
multiply it by the 11 to give 11. Subtract that 11 from the original 16 and you'll get 5. Same with the 27... 27/11 = 2.4545. The whole part of 2.4545
is 2, so 2*11 = 22. 27-22 = 5, which is the modulus using base11.
Once you realize how our base10 system works, other bases become a lot easier to understand. Unfortunately, this is not normally taught in schools
outside "advanced" math courses which no one seems to want to take. In truth, math is the easiest subject there is in school, but it is taught in a
haphazard fashion instead of a logical progressive fashion.
Take the number (in base10) 45,692 (forty five thousand six hundred ninety two). How that number comes about using the various digits is as follows:
imagine five columns, containing, in order, 4, 5, 6, 9, and 2 reading left to right. Now imagine each column having a number itself... the 4 is in the
5th column, the 5 is in the fourth column, the 6 is in the third column, the 9 is in the second column, and the 2 is in the first column. Yes, they
are in reverse order, because the first column is to the left of the decimal point, which in this case comes after the 2.
The 4 is in the fifth column, so it means 4*(10^4). The 5 is in the fourth column, so it means 5*(10^3). The 6 is in the third column, so it means
6*(10^2). The 9 is in the second column, so it means 9*(10^1), and the 2 is in the first column so it means 2*(10^0). Notice that we drop the column
number by 1 in every case; that's because we start counting powers from 0, not from 1 like the columns. Now put it all together and we get
(4*10,000)+(5*1000)+(6*100)+(9*10)+(2*1) = 45,692.
Schools teach about the value of the columns, but they don't make that connection that they are just powers of the base (10) in sequence.
Now apply that to base 11. The columns are still numbered the same way, but they represent powers of 11 instead of powers of ten. If we try to convert
45,692 base11 to base 10, we get (4*11^4)+(5*11^3)+(6*11^2)+(9*11^1)+(2*11^0) = (4*14,641)+(5*1331)+(6*121)+(9*11)+(2*1) = 58,564+6655+726+99+2 =
Now you know why we don't use base 11 very much! LOL!
You can produce numbers in any base this way... it is possible to have base 5,832,853 if you want it. Just don't ask me to do the conversions!