ADDING MACHINE

We are used to writing numbers in base 10, because this system was what prevailed in our culture (and because there are ten fingers). When we see 109 we think of “1 hundred plus 0 tens plus 9 ones”.

From a certain point of view, it would be more economical to use base 2, since only two digits are needed: 0 and 1. In this case, the positional notation is based on powers of 2. For example, the representation of 109 in base 2 is 1101101: 109 = 1×64 + 1×32 + 0×16 + 1×8 + 1×4 + 0×2 + 1×1.

Adding two numbers in base 2 is similar to adding in base 10: the addition is done from right to left, noting that in each column 0+0 is 0, 1+0 is 1, 0+1 is 1, 1 +1 results in 0 and “goes 1”, 1+1+1 results in 1 and “goes 1”.

This machine performs exactly that algorithm.

Image: Rodrigo Tetsuo Argenton.

HOW TO ENTER A NUMBER?
Let's say you want to put 23. The biggest power of 2 that fits in 23 is 16: put a marble in 16. For 23, there's 7: it doesn't fit 8, but it fits the 4. Put a marble in 4. There are 3 left: it fits the 2. Put a marble in 2. Missing 1, put a marble in 1.

HOW TO READ THE RESULT?
Add the powers of 2 of the positions where a ball was.

If you want to see a little more about this machine, check out the video below! Seize the opportunity to check out the other cool videos on the Matemateca's channel!

                 

USEFUL LINKS

Although we are used to the decimal system, there are many other different systems, such as base 2. This UFSC's (Santa Catarina's Federal University) article talks about number systems, showing the connection between binary and hexadecimal (base 16) systems with the ASCII code of basic computer characters.

As this machine works on base 2, this Wikipedia page is great to increase understanding of the binary system, since, in addition to explaining its operation, it also brings decimals encoded in binary and a little bit about binary scientific notation, including showing part of an open code that supports this functionality.

And if you associated a binary system with the binary code that underlies the language of computers, this TecMundo page explains this relationship, showing how it works and how data is stored in computer memory. Furthermore, it shows how the evolution of machines has made it much easier to "talk" to the computer, since the conversion is done automatically.

Finally, if you are interested in the subject and would like to start learning how to add numbers in the binary system (as our machine does), this Calcular e Converter calculator lets you do just that. In addition to explaining in detail how the sum should be performed and giving some examples, it allows you to perform it on the website, to check if you were right!