I've always been fascinated by numbers. I speculate that counting evolved with ancestors assembling in groups: for survival, procreation, and community. As primitive hunter/gatherers, they may have assigned individual stature within the group to the number of spears, clubs, or whatever each possessed. I suspect the evaluation was whatever indicators meant "more", "less", or "same". I dare not stretch my ignorance any further.
Let's move on to some BCE civilizations. First, the primary reason that civilizations developed number systems appears to be counting money — no surprise there. Before the Romans, the Greeks had developed a number system using the first nine letters of their alphabet. If you want the details here's the link to Mac Tutor. Most of us, I suspect are more familiar with Roman Numerals: clock faces, front matter in books, etc. using letters from the Roman alphabet. I, V, X, L, C, D, and M. Further information, if you want some (here.)
The number system we all know, is Hindu-Arabic which paved the way for more than just counting money. Like the other systems above, it is a base-ten, or Decimal, system. A quick look at your hands or feet should make the reason for it obvious. If you're Mayan, you may be used to a base-twenty, or Vegesimal, system. Let's stick with Decimal for now.
A long time ago I ran for the school board in the Village of Millbrook. In the process of familiarizing myself with elementary, middle, and high school, I queried teachers at each level about numbers. When I asked in the elementary school, how they taught the decimal numbering system, the answer was the digits 1,2,3,4,5,6,7,8,9,10. When I mentioned that 10 was not one digit, it was two, some looked at me like I had that many heads. Not until I got to speak with high school math teachers did I hear the laments about having kids who didn't know the decimal number system. They did know counting 1 - 10+ and the arithmetic operations, but had to learn the importance of 0 as the first digit in our cardinal system, in order to reach into higher math.
I lost the election.
If you've stuck with me this far, let's trek into some of my favorite things about our number system.
First are the digits 0 and 1 and the arithmetic operations add, subtract, multiply, and divide. first the number 1 — also called 'unity' in mathematics. if you add 1 to a number it increases the number, if you subtract it the number is decreased. If you multiply, or divide, the number by 1 you get the number you started with! Cool huh? How about 0? Well if you add or subtract 0, you get the number you started with. If you multiply by 0 you get 0. If you divide by 0 you get...undefined! Explained here.
Another favorite of mine is INFINITY (∞). Let''s begin with the positive integers (counting numbers, for those of you who hated math). how many are there? Well there are an infinite number which means you cannot reach the largest integer because there is always one more. Okay now let's look at fractions. How about fractions between 1 and 2? How many are there? You guessed it, an infinite number of fractions between 1 and 2. Likewise, between any two integers there are an infinite number of fraction. But wait! If there are an infinite number of integers and an infinite number of fractions between each set of integers, that must mean there are two infinities. Yes there are two types of infinities: countable and uncountable. If you want to go down that rabbit hole, Have at it!
Until next time,
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Comments are always from "anonymous". Often I can identify the author by the content of the comment, but that much cogitation makes my 80 year-old brain tired. Please help out an old man and identify yourself within the text of the comment. Thanks for the comments whether or not you ID yourself. Tom