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sums will set you free

how to teach your child numbers arithmetic mathematics

understanding, calculating, and changing bases

counting in different bases
fence posts

fields and fences - one and two dimensions

bases in everyday life
binary for computers, it’s all ones and noughts

counting in different 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.

• Here’s how:
• put 0 in Reset Counter Value, and press set to set large counter to 0;
• click red reset button to set number of manual steps to 0;
• put 1 in Change Step, and press set to set counting step size to 1;
• Now click on the Manual Step button ten times. The red number counts to 10.

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.

For base 3,
• put 0 in Reset Counter Value, and press set to set large counter to 0;
• Change Base to 3;
• Now click on the Manual Step button. The red number counts up in decimal, while the large number counts up in base 3.

The large counter number counts up (increasing): 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110 etc.

For base 5,
• put 0 in Reset Counter Value, and press set to set large counter to 0;
• Change Base to 5;
• Now click on the Manual Step button. The red number counts up in decimal, while the large number counts up in base 5.

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.)

The main practical bases used in computer technology are binary (base 2) and hexadecimal (base 16), see understanding sets and set logic and writing down sets and set logic equations.

You can use the Counter, by various settings, to watch addition, subtraction, multiplication and division in action, changing the base according to your interests and needs.

fence posts

(Or trees, or bollards, or ...)

Note that eleven trees enclose ten gaps. There are twelve trees in this photo!

There are ten numbers/digits in the decimal (base 10) number system.
As you see with the trees above (or it could be fence posts), when you go from the tree labelled zero to the tree labelled one (0 to 1) you are counting one object, which is represented by the gap between trees zero and one.
When counting to ten (or any other number), you start from an implied zero (0), as you see with the trees above.
When counting up from one (1), you will have used up all the trees (or fence posts) by the time you counted the ninth object if you forget the implied zero.
For the tenth object (gap between the trees above), you have reached the click-over number, which comprises two digits that is/are 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:
(0), 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, 10, etc;
and with base-2, counting goes
(0), 1, 10, 11, 100, etc.
Try counting in base-12 or base-2 using the Counter to familiarise yourself with what happens. Then experiment with other bases in the Counter above, according to your interest.

fields and fences - one and two dimensions

Looking at the photo-illustration above, you see that with one-dimensional (linear) counting, one more fence post (here, a white block) is required than the number of spaces (here, the gap between each pair of blocks) being counted.

However, with two-dimensional, closed loop, counting (here illustrated by a square and a circle of blocks), the number of fence posts and spaces are equal.

bases in everyday life

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 :

• base twelve for the number of inches in a foot,
• base three for the number of feet in a yard,
• base sixteen for the number of ounces in a pound,
• base fourteen for the number of pounds in a stone,
• base eight for for the number of pints in a gallon,
• dozens and the gross for counting a group of items,
• base twelve for the number of pennies in a shilling, and base twenty for the number of shillings in a pound [‘old’ money].
  and many others.

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).

binary for computers, it’s all ones and noughts

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 [2]. 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.

email email_abelard [at] abelard.org

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1015 words
prints as 4 A4 pages (on my printer and set-up)