bases and number systems- 1510

I admit it. This topic confounds me. I had a really hard time with the whole base two idea. The book explained it but not much made it in my head. I'll try and untangle this issue with other sources. One source I found here discusses a "base table" when doing the conversion from a base 10 to any other base. So for instance...we could convert the base 10 number 137 to a base 6 number by making the base chart...

6^3 \   6^2\  6^1\  6^0\

216 \   36  \  6     \     1 \

So the ONLY numbers that can be part of base six are 0, 1, 2, 3, 4, 5, 6 . 

The first thing to do is figure out how many groups of 6 will fit into the number 137.  We can do this with a standard division algorithm            137/6 =  22.8    (we can ignore the .8 and just focus on the 22)

                        This means that there are 22 groups of 6 in  the number.  But since base six doesn't have a 22 we'll have to break that down a little more...

 Next we'll figure out how many times 6 goes into 22...well, that would be three, or 3 .

Okay...I just got lost myself. I'm going to explain it this way then...
How many groups of 36 can we break 137 into? why that would be three!! So I can put a three under the 36 in the base chart.  36 x 3 = 108 so we have some left over.  That would be 29 and some change left over so we have to determine how may groups of six go  in there and my brain says 4! (put that 4 under the 6^1 column) 4 x 6 equals 24 so we have some left over (5 to be exact) and we can add those to the 6^0 column... which makes our base table look something like this...

6^3 \   6^2\  6^1\  6^0\

216 \   36  \  6     \     1 \
        \   3    \  4     \    5  \

which makes our final answer 345 six

Whew...I did it! I wasn't sure I could but using a little of the ideas found at this website - namely the base chart- and some other brain figuring I came up with the solution.  How do I know it is right? I googled it, of course!

The ideas of other bases and alternate number systems was a new trick for this old dog.  I found that the system of the Egyptian numbers seemed the most straight forward, most likely due to it being a base ten system.  The downside is that the numbers can get very long.  The Mayan system was the next best but the way of reading it top down was certainly a novel idea.  The most confusing for me was the Babylonian system which works subtraction into the characters for numbers.  It also seemed to figure backwards from how my brain wanted it to go and so I was often quite muddled.  There are people who truly understand all this and even chose to blog about it...I found one such person here at hyper mathematics.
Here is a site that discusses Egyptian numbers.  Overall, I think this will always be a part of math that makes my head ache and my hands sweat but perseverance is the name of the game.  

Seeing as I love videos and we've all had such fun with the conversion of hindu-arabic numbers to Roman numerals, please enjoy one of my favorite catchy tunes by Trout Fishing in America- sing along!