Concept of Base Numbers
I didn't truly understand the concept of base numbers until I was a junior in college. Once it dawned on me a whole flood washed over me. It was the concept of base numbers. I'd started investigating what it would be like to express our number system with more or less symbols. Our current number system has ten symbols: zero through nine. What if we tried expressing it with eleven? Nine? Twenty?
The ancient sumerians used sixty because it was divisible by so many smaller numbers: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. They are the reason we have sixty seconds in a minute and sixty minutes in an hour. They are the reason we have 360 degrees in a full circle.
Anyway, I decided to try to come up with an original number system, a way of expressing numbers that no one else has ever discovered... I later found out they'd been studied for hundreds of years and I just didn't get it until just then.
I eliminated the symbol 7 first, keeping the other nine symbols. 1 was still one, 2 was still two, but 8 became the seventh symbol. They were listed as such: 1, 2, 3, 4, 5, 6, 8, 9, 10... so on. I even eliminated the 70s, going directly from 69 to 80. What did I find? 100 was the 81st number. I eliminated another symbol so that there were only eight symbols used. What did I find then? 100 became the 64th number. I eliminated yet another. 100 became the 49th number. I finally decided to make up a symbol and introduce it in with the original ten symbols so that now I had eleven symbols. I listed them off as so: 1, 2, 3, 4, 5, 6, 7, 8, 9, @, 10, 11, 12, 13... 100 became the 121st number.
What am I getting at? Patience. Anyway, I thought about why we chose ten as our base number of choice. My most reasonable guess is that we have ten fingers and that makes ten symbols really easy to remember. Every time we add a digit to our numbers, say from 10 to 100 to 1000, we are multiplying times ten every time. If we use a number system based around eleven symbols, then 10 to 100 to 1000 would be multiplying by eleven every time.
I realized that it was a logorithmic scale. We are so used to seeing our base ten logorithmic scale that it doesn't even come across as logs of base ten. All my other base systems were simply just logs of other numbers. Hell, by the time I eliminated all the symbols except for two symbols, I realized I was expressing numbers in binary digits. Six symbols is hexidecimal. Other people have studied logs and other base systems for centuries. Natural Log is the log of Euler's number, which isn't even a whole number for christ's sake (2.718).
Other Patterns and How They Relate
I'd previously discovered another pattern. When you add the digits of any number divisible by nine, that sum adds up to nine (27 is divisible by nine. 2 + 7 = 9). I wrote a bunch of these patterns down in my other posts. So many patterns end up with nine! Every ninth number is this or that, every tenth number in the sequence was nine... nine showed up everywhere. I even had a jersey made with my name and the symbol 9 printed on it.
When I was studying the base systems, I had an epiphany. I realized that it wasn't just 9. It was the base number minus one. Our base number of choice is ten, which makes 9 show up in all these patterns. When I studied the patterns using eleven digits, suddenly @ became the symbol that showed up everywhere (I designated ten as the symbol @ above because 10 actually represented the number eleven). In fact, I found only one pattern that was the base system plus one, the rest were base minus one. I will cover that exception in a moment. Play around with numbers. You'll find so many weird things that hide in plane sight.
The exception pattern was Pascal's Triangle. I am placing an image below of it, taken from www.mathisgoodforyou.com. Each number is a sum of the two numbers directly above it. I first saw Pascal's Triangle back in elementary school and didn't work with it until seventh grade or so. I didn't use it for anything practical until trigonometry in high school. Even then it hid in plane sight that Pascal's Triangle is actually a list of the eleven to various exponents. The first row is 1, which is 11^0. The second row is 11, which is 11^1. The third row is 121, which is 11^2. Fourth is 1331, which is 11^3. So on and so on. It gets a little complex when you get digits greater than ten representing a space in the triangle. In that case, you add the first digit to the number on the left. Thus 1 5 10 10 5 1 becomes 1 6 1 0 5 1, which is 11^5.
It doesn't end there. How creepy is this: when I looked into Pascal's Triangle with other base numbers, I found that Pascal's Triangle wasn't just a list of eleven's exponents. It was our base system plus one. If you express our numbers with eleven symbols, the triangle becomes a list of twelve's exponents.
My friends ask me what I was doing to be so bored that I'd do this. I stated I was stuck in a hotel room with my grandma for six hours and suddenly they all understood.
Do we ever use scales that are mixed base numbers? Absolutely. If you've only ever used Metric, then you use less mixed systems. For example, there are twelve inches in a foot, three feet in a yard, 5280 feet in a mile. There are two cups in a pint, two pints in a quart, four quarts in a gallon. There are sixty seconds in a minute, sixty minutes in an hour, twenty-four hours in a day, and 365 days in a year.
If you want to change units for whatever you're measuring, you've got to convert (say from seconds to minutes). People who use the metric system have it all down better (except for measuring time, in which case they're equals).
So if you want to measure a distance between two cities in inches, you get tens of millions of inches. If you measure it in feet, you get hundreds of thousands. If you measure the distance in miles, you get a couple dozen. If you measure the distance in light years, you get an incredibly small fraction. Your units determine your base system, and if it's not metric it usually isn't a nice round ten.