Hello everyone. This is going to be one of my first technical diaries I plan on posting for kicks and stuff. Today's topic is going to be about the seemingly simple math problem that we grew up with, and looking at it in a different light. Namely, realizing that there is more to mathematics than most people understand. The title of this journal insinuated that 1 + 1 does not equal 2, but rather 10 AND 1 (technically, in my first example 2 = 10). I stand firm in that this is, indeed, correct. The issue arises not in faulty math, but lack of context, and by showing this context I plan on giving insight on two ideas that have farther reaching intricacies: non-decimal-based number systems, and the Boolean Algebra.
The number system we're all used to is the base-10. Why not, we have 10 fingers and toes so it seems to be nice for us. The fact of the matter is, other cultures have used other bases for their numeric systems, and ultimately there is nothing special about base-10 at all. In fact, you can make any integer outside of 0 and 1 as your numeric base. But our attention here will be relegated to base-2 for the most part with some delving into the applications.
Base-2--also known as binary--is very simple in that it consists of only 0's and 1's in every digit. For every day life it is highly impractical, in that it takes 9 digits just to write the 10-number 1000 in base-2, 111101000. [For brevity and inability to use subscripts akin to convention, I will denote n-number to say that the next entry will be a number of base-n]. The application is mostly in digital systems (Computers), where the easiest way to represent values is either a high voltage or a low voltage, representing a 1 or 0 respectively; a binary system. In fact, binary is the only thing computers work with, so to understand computers one must understand binary. Now I'm not expecting people to take it that far here, but let's dispel some of the mystery surrounding base-2 calculations.
First, let's show off some numbers in binary using the dec-bin convention:
As we can see, the methods of counting in the digits are very much like in base-10. Take notice of how inefficient the space is when we approach larger numbers. In case there is some confusion on how to read the numbers, let's look at the scientific notation breakdown with first an example of how it's done for a decimal number, then looking at the number 43 in both decimal and its binary notation.
Before we do some calculations, a strong note is in order: all of the arithmetic methods you used with base-10 all of these years work in much the same way in base-2, the only issue is that you make all of your carries and such using 2 as your base. We will not cover negative numbers for the sole reason that, ironically, there is multiple ways to implement them, and I doubt many of you want me to show 2's-compliment and how they're found. So without further adieu:
Addition and subtraction are simple, right? Well how about multiplication and division? Well multiplication is almost simpler (more work but less memorization), division can get ugly especially with decimal-representation of remainders but again less mental calculation (we will use the quotient R remainder convention for simplification). For each of them, I will do a couple examples using the by-hand methods of finding the products and quotients, then omit them in other examples.
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So now we've seen how to calculate using base-2. Simple right? Now we understand why I said 1+1=10, so how about the other problem I posited, that 1+1=1? Well that'll be a story for next time, since that'll take a lot more time to discuss (since we'll be talking about a whole lot about logic and some abstract algebra) and I'd rather make the brain hurt for only one reason a day. So tune in next time.