The Binary Number System

At the core of digital electronics, the binary number system defines digital electronics itself.  As will be discussed in Digital signals in more detail, digital electronics consists of two positions or values, on and off.  When making truth tables, one must be able to count in binary and understand the meaning of the numbers to successfully complete it. It is therefore that binary is the heart of the computer and the computer's calculations.

Perhaps the easiest way to understand binary is to compare it to base ten, the number system that humans use to do calculations.  Base ten uses a series of numbers 0-9 to represent the quantity of the place value it holds.  With all bases (base two, base eight, base seven-hundred thirty-seven, ect.) the first place holder (the rightmost) is always the "one's" place.  Each place holder to the left is a multiple of the one before it.  In base ten, then next place is the "ten's" place, and the following is the "hundred's."  So each place holder to the right is the previous place holder X 10.  In base ten, if you were to write "37" that means that you have three tens, and seven ones.  You would then add thirty and seven to get thirty seven.  Base two (Binary) multiplies X 2.  So the place to the right of the "one's" place is the "two's" place and to the left of that, the "four's" place.  With base two, the only numbers available are 1 and 0.  If you were to write 100101, that means you would have 1 one, 0 twos, 1 four, 0 eights, 0 sixteens, and 1 thirty-two.  Since there are no twos, eights or sixteens we can forget about them for now, the zero is a place holder to say that we do not have any of that particular place.  If we were to add what we are left with: the one, the four, and the thirty-two, we would get 37.  This number is in base ten.  This is the way to convert binary to base ten.  To convert from base ten to binary, the opposite process should obtain the correct results.  Say that you have 23 in base ten, and you want to convert that to binary.  First it might be beneficial, but not necessary, to find out how many places you need.  To do this, find the highest place value that does not exceed the number you have.  In this case it would be the sixteens place.  We do not need any places beyond this because the next value would be thirty-two's and we have a number below it, so we have zero thirty-two's.  So we can safely say that we do have one 16, because 23 is indeed higher than sixteen.  If we were to take sixteen away from our 23, we would have 7 left.  The next lowest value is a eight.  Eight exceeds the 7 we have left, so we have 0 eight's.  The next lowest value is four.  There is one four in the 7 we have left so we can put a 1 in the "four's" place.  We now have a 3 left.  The next lowest place is the "two's."  There is one 2 in what we have left, so put a 1 in the "two's" place.  Now all we have left is 1, and the final place value is 1, so we have one 1.  It is in this way that we can write the number 23 as: 10111. We use binary everyday without thinking about it.  Flipping a switch to turn a light on or off is binary.  Turning on the TV or turning it off in involves binary. 

But why intentionally use binary?  In digital electronics, binary is used on the basis of calculating an output in a certain scenario.  This is imperative when creating a circuit, especially if you want it to work correctly.  But more importantly, it makes you look incredibly intelligent. On a more serious note, binary is not commonly used for other uses.  The amount of ones and zeros involved with writing numbers in binary, make binary very complex to decipher, and we already have all of the calculations we do on a day-to-day basis set up for base ten, so converting to binary for any reason would be costly, and extremely difficult.  Therefore, outside of digital electronics, binary is almost obsolete.  Digital electronics, however is centered around binary, and without it digital electronics would fail.  Also, fancy watches would not exist as they do today.

Here are some links to check out to learn more:
http://binarytranslator.com/what-is-binary.php

http://l3d.cs.colorado.edu/courses/CSCI1200-96/binary.html