sums will set you free
how to teach your child numbers arithmetic mathematics
understanding, calculating, and changing bases
The problem with teaching arithmetic in different bases is that it is so easy that people think that there must be more to it than this - but there isn’t.
You will learn far more by playing around, and becoming familiar with the Counter than you will by me listing all sorts of examples.
Count up in ones and the tenth one clicks over into the tens column, then the tenth ten clicks over into the hundreds column, and so on. Try it out on the counter below.
The large counter number counts up (increasing): 1,2,3,4, etc to 9, on the tenth click the number clicks over to a 1 in the tens column and a 0 in the units column.
If you are stepping (counting) up in ones from zero using a non-base-ten base, the decimal value will show as the red number of manual steps.
I will start by giving two simple examples. [See here for a full description of abelard’s educational counter.]
Now try setting the counter to base 3, and then to base 5.
The large counter number counts up (increasing): 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110 etc.
The large counter number counts up (increasing): 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, etc.
Now choose any base of your own and try again. (Note that the Counter goes up to base 32.)
(Or trees, or bollards, or ...)
There are ten numbers/digits in the decimal (base 10)
Likewise, there are two digits/numbers/objects necessary and used in the base-2 number system, and 16 digits/numbers/objects when counting in base-16, and so on. Each number system, therefore, uses zero (0) and n-1 other digits (where n is the number of the system concerned - for base-2, n=2; for base-16, n=16).
As we run out of standard base-10 digits, we switch to using letters, starting with a, b, c etc.
Thus, base-12 counting goes like this:
Bases other than ten can be found in everyday life. A clock counts in sixties (seconds and minutes) and in twelves (hours), or in sixties and in twenty-fours.
Before many weights and measures became decimalised, other bases were commonly used :
The ancient Babylonians used a sexagesimal system of counting, that is a base-60 number system. This system still survives in the form of degrees - a circle is divided into 360° , and in minutes and seconds used for measuring time.
The Pre-Columbian Mayans used a vigesimal, base-20, counting system, so does the West when counting in scores, such as the number of hundredweights in a ton.
Paper is counted in quires of twenty-five sheets.
Computers use a binary, base-2, counting system at their most basic level, and sometimes programmers use hexadecimal (base-16).
link logic for computers
A computer has a memory. In concept, this is no different from writing words in a book, or drawing pictures on a wall. Just as you need a pencil to write on paper, or maybe chalk to write on a wall, in a computer you write with electrons on magnets, very small magnets.
So why bother with a computer at all? Essentially, it is because you can write very more quickly in very much less space. For example, a small library of books can already be held in some modern telephones, or in specialist portable readers.
Much easier than carrying a few shelves of books around with you, and far easier to search for a word, title, sentence, or name.
Although earlier attempts at computers were made with cogwheels and springs, and even with hydraulic systems, it has been found that the natural way and most efficient way with electronics is to use various forms of on/off switches. So to describe these switches, we only need two ‘numbers’, one each for on and for off - one and zero - 1 or 0. This system is usually referred to as the binary system, that is counting in base two . Try setting base two on the counter above, and see what happens when you play with it.
As you will see, if you do ten steps when the counter is set to base 2, you will reach the binary number 1011. This the number 10 represented in binary [base 2]. You will also see that to represent 10 in usual [decimal or base 10] counting takes two digits, where as in binary it takes four digits [bits]. Likewise, representing letters and other characters usually uses 8 binary ‘digits’ [bytes].
By extension, you may represent any idea at all with a sufficient number of binary bits. You can represent a picture complete with colours, the words in a book, the plans for a space craft, or a telephone directory.
But a computer also has to do other things. The computer has to know where it is and what it has to do with the various types of data that are written on its switches. Everything in the computer has an address - for example, the fifth number or letter, or the 5 billionth. Therefore, the computer has to know whether what it is looking at is an address to find the numbers or words in a book, or whether it is actually looking at those numbers or words (commonly called data).
Not only does the computer need to know whether it is looking at data or at an address (another form of data), it also needs to know what to do with those addresses and data. The instructions telling the computer what to do are also encoded in binary - yes, another form of data.
These forms of data will all look the same to you, or to the computer - just strings of ones and noughts Thus, in writing for the computer, it is very important that the computer can distinguish between the different sorts of data: the words in your book, where those words are, and what to do with those words.
This is the sort of reason that the systems software in your computer can often go berserk. For example, when the computer thinks that it is reading instructions, and suddenly jumps into the middle of your book and tries to use the words in your book as instructions to run the computer. But you are not going to have to worry about this as long as you are using the computer for your homework, or for a letter to Aunt Polly.
You will only need to start worrying about this when you go to work for Microsoft or Google or Apple, and go home after work worrying about the trillions of users who are cursing you out because their computer stalled, or that the computer ordered you a thousand Mercedes when you only wanted one.
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