Monday, January 30, 2017

The Beauty of Binary

Boolean algebra was invented by George Boole in the mid-1800s, long before binary numbers had any practical purpose. While binary, which is base 2, is not the simplest numeral system for humans, it's ideal for computers. It's a simple way to store and transfer information. As a matter of fact, you can think of DNA as binary since it only has two combinations (CG or AT) that store all the genetic information of our makeup.

(The simplest and oldest numeral system for humans is unary, which is base 1. Think: tallying numbers with four ones, 1111, while the fifth tally is a diagonal line striking through the four tallies to make one group of five. Actually, traditional tallying seems more like a cross between base 1 and base 5, but I digress.)

There is an elegant simplicity in binary in that each digit is either a one or a zero. On or off. No room for any gray area, even though fractions and negative numbers can still be represented in binary. Additionally, some numbers that can't truly be expressed in one base, for example, 1/3 in base 10, can be simply written in, say, base 3 as 0.13.

Since computers use binary, some integers operations are child's play to a computer, especially bit shifting. As humans, we can't easily figure out multiplication of large numbers in our head. For example, what is 123 x 45? That will require a pencil and paper or calculator. But, we can easily figure out the answer to 12345 x 100, even though the latter deals with much larger numbers because we simply shift the digits three places. But, for humans, this calculation only works for powers of 10, since we think in base 10. Computers, however, get this luxury when they're multiplying by integers that are a multiple of the base. Multiplying a base 2 number by 2, 4, 8 is as simple as shifting the bits by one, two, three, or four places. For a computer, like a human, this is a much simpler task than working through the traditional arithmetic.

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