Thursday, April 2, 2015

Binary Logic, puzzles and Boolean Algebra

Today, I give you a nice, simple Sudoku like puzzle to start with, called binary puzzle. It comes in 3 sizes: 6x6, 8x8, and 10x10. Check it out here and here are the rules.

Simple, fun, something to do and now for an actual discussion on Binary and Boolean logic. Keeping these posts short and sweet is nice, sort of. This will help me practice for some other short posts I need to write on other topics.

I used to enjoy converting numbers into binary in high school and working with them because it was so damn easy. And it helped me understand how places work. When working with any counting system, the first number on the right is always the ones place, or x0 where x is the base. 20 = 1. 100 = 1. 10000000000000666000000000000010 = 1. Why? Well, Dr. Math has the answer. For those looking to just read this before moving on to the next blog, here's his proof:
While the above argument might help convince your intuitive side that any number to the zero power is 1, the following argument is a little more rigorous.
This proof uses the laws of exponents. One of the laws of exponents is:
   n^x
   --- = n^(x-y)
   n^y   
for all n, x, and y. So for example,
   3^4
   --- = 3^(4-2) = 3^2
   3^2


   3^4 
   --- = 3^(4-3) = 3^1
   3^3
   
Now suppose we have the fraction:
    3^4
    ---
    3^4
This fraction equals 1, because the numerator and the denominator are the same. If we apply the law of exponents, we get:
       3^4
   1 = --- = 3^(4-4) = 3^0 
       3^4
So 3^0 = 1. We can plug in any in number in the place of three, and that number raised to the zero power will still be 1. In fact, the whole proof works if we just plug in x for 3:
                  x^4 
  x^0 = x^(4-4) = --- = 1
                  x^4   
Anyway, back to binary. So one's place is always 1 * whatever the number is. Next to that is x1, so in base 10 it's the ten's place, base 16 it's the sixteenth, and the Mayans called it the 20's place. Binary was simple to convert for this reason: 1001001 would be 1 * 26 + 0 * 25 + 0 * 24 + 1 * 23 + 0 * 22 + 0 * 21 + 0 * 20 or 64 + 8 + 1 or 73 in base 10. I can't believe I just wrote that out. Just showing my work like a good student.

Boolean algebra is good stuff too. What to say about that is simple, since I already dug myself this hole yesterday? Well, here are details about it all. Boolean and binary are not the same, but Boolean can help you build a binary calculator with LEDs. That's all I have today. Come back tomorrow when I try something a little easier. Or celebrate Good Friday, that's okay too.

Twitter: @aristotlesmstk, facebook, you know the whole usual drill.

7 comments:

  1. I really want to respond to this with something intelligent and witty and above all mathematical, but at midnight in my neck of the woods, and given my casual, head-nod fascination with numbers in general, I can't make much out of any of this. So instead, you may have this, as a comment:

    Yes. Kittens.

    https://out0fprint.wordpress.com/

    ReplyDelete
  2. Holy cow. I see lots of math. I'm thinking about that big whiteboard Sheldon and Leonard have in their apartment on The Big Bang Theory.

    ReplyDelete
  3. Boolean searches - where would we be without them?

    ReplyDelete
  4. Boolean searches - where would we be without them?

    ReplyDelete