Contents

K&W Abacus Help Contents

    The K&W Abacus (© 1997 Know Where Software) is a fully functional abacus that illustrates many important properties of numbers. It will never (at least in western societies) replace a hand held calculator, but it is a valuable learning tool and a lot of fun. Although some may argue that the abacus is less a computer and more a computational aid, the abacus has a long and venerable history and is still used widely in China and Japan. Read Abacus Basics for details.

   The K&W Abacus is easy to use and intuitive. If you need help, however, click on a help topic below. If you don't know how to use Help, click on Help in the Menu bar above and then How to Use Help.

K&W Abacus

    Quick Tour

    Abacus Basics

    History

    Calculator

    File

    Exit

    Set up

Operations

    Addition

    Subtraction

    Multiplication

    Division


Quick Tour

Quick Tour of The K&W Abacus

    The K&W Abacus is patterned after the Chinese abacus and is composed of a frame, a crosspiece, a series of uprights and beads. Just as with written numbers, each successive upright from right to left represents an increase by a factor of ten. For example, the right most upright could denote units, the next to the left 10s, then 100s and so on. Or in financial calculations, different values might be assigned such as cents, tens of cents, then dollars, tens of dollars, etc.

    When a bead is moved towards the crosspiece, it is said to be put in and counts. When moved away, it is taken off and no longer counts. To move a bead -- that is, to put in or take off a bead -- simply click it. If it has been put in, it will be taken off and vice versa. Upper beads are factors of 5 times the crosspiece value; lower beads are factors of one. Thus, the number 61, has one 5-bead and 1 1-bead put in from the second upright on the right, and 1 1-bead from the right most upright.

    For details on how to add, subtract, multiply and divide numbers, see those specific topics under Help Contents or Search help.

    The menu bar lists 2 items: File and Help. Each contains a sub-menu. You select items by clicking with the left mouse button or holding down the Alt key and typing the underlined letter. To exit, click File then Exit. Clicking on Calculator will bring up a calculator. Reset to 0 will reset the abacus to zero.

    If the abacus is new to you, read Abacus Basics and History under Help Contents.


Help

Getting Help

    Help contains K&W Abacus Help. Click on Help or press the <F1>. If you are unfamiliar with Help, click on How to Use Help.

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    Clicking on Help will drop down a sub-menu containing Help, How to Use Help, version and information About The K&W Abacus.


File

File: Exiting the K&W Abacus

    Click on File then Exit to exit.


Installation

Program Set Up

    Installation of the K&W Abacus is easy. There are several ways to do this:

    1)From the original disk, simply place it in the appropriate drive. Windows 3.1 users: From Windows' File or Program Manager, click on File then Run and type a:setup or b:setup. Windows95 users: Click on Start then Run and type a:setup or b:setup. Setup will take over and install the program.

    2) If you have already used setup or if the files were sent to you without a setup.exe file, simply copy the two files abacus.exe and abacus.hlp to a directory of your choosing. Normally this would be c:\abacus. The Microsoft run time file VBRUN300.DLL is also needed to run the K&W Abacus and it should be copied to the windows/system sub-directory. If you obtained K&W Abacus without VBRUN300.DLL, you can download it from any on line service such as America On Line. Otherwise, it is available free from Microsoft (microsoft.com) on the World Wide Web.

   3) The CD version of the K&W Abacus can be run it directly from the CD or you may copy it to your hard drive. Follow the directions in 2) above. Usually running from the hard drive is faster.


Calculator

        The Calculator allows you to perform simple calculations -- many of which cannot be done with an abacus. You may type in the numbers or use the numbered buttons provided. Enter operations using the keys provided. A unary operation is completed by clicking a unary operation button (e.g., cos, sqr, log, etc.). Binary operations (+, -, x and /) are completed with the equals sign (=).


BASICS

Abacus Basics

    The abacus can take on several forms. The K&W Abacus is patterned after the Chinese version and is composed of a frame, crosspiece, uprights and beads. The crosspiece divides the frame into upper and lower parts. The nine uprights contain seven beads each; five in the lower and two in the upper part. Moving from right to left, each successive upright represents a power of ten larger than the preceding upright. Often, the upright at the extreme right is used to represent units; the next upright 10s, then 100s and so on. At the extreme left are 100 millions. This need not necessarily always be the case. For example, in financial computations the user might decide that the right most upright stand for .01 (cents), the next one .1 (tens of cents), then units (or dollars), 10s (tens of dollars) up to 1,000,000 (millions of dollars) on the far left. In some operations, uprights may be assigned temporary values.

   When a bead is moved to the crosspiece it is counted or put in. When moved away from the crosspiece it no longer counts or is taken off. Thus the number zero has all beads taken off or moved away from the crosspiece. Putting in and taking off on the K&W Abacus requires that  you simply click a bead. Any beads between it and the crosspiece or any beads between it and the frame will be automatically moved depending on whether it has already been put in or taken off. Each move is accompanied with a click from your computer speaker.

    Beads in the lower part depict a multiplier of 1 times the value assigned to that upright. The upper beads represent a multiplier of five. We will call the beads on the upright assigned the units value 1-beads and 5-beads; on the upright immediately to the left , 10-beads and 50-beads. Similarly for beads on the other uprights. Sometimes we may say, for example, put in 1 1-bead from the 100s upright meaning put in 1 100-bead or put in or take off a bead from upright #2 or #3 or whatever. If it is clear which upright is involved, we may just say put in or take off some 1 or 5-beads.

   As examples and assuming the right most upright (#1) has been assigned a units value, the number 2 has the first two lower 1-beads on the right most upright put in. Seven would have, in addition, the first 5-bead put in. If the same beads had been put in from the second upright, they would depict the numbers 20 and 70 and, on the third upright, 200 and 700, respectively.

    The abacus makes it particularly easy to carry and borrow numbers. As with addition using pencil and paper, we carry numbers by removing 10 from one upright or column (either two 5s or one 5 and five 1s) and then putting in a 1-bead from the next upright to the left. Similarly during subtraction, a ten can be borrowed from the left by taking off or removing a 1-bead and putting in a ten on the upright immediately to the right (two 5s or one 5 and five 1s).

    Unlike many computations with pencil and paper where we start with the right most column (long division being an exception), abacus bead movement usually starts on the left. This is not an absolute necessity but it does make many operations more natural. For particulars on addition, subtraction, multiplication and division see the individual topic under Help Contents or Search help.


Addition

    The simplest operation to perform on the abacus is addition. A trivial  example is to add 1 and 4. First, put in 1 1-bead and then 4 1-beads. The answer is, of course, 5. More complicated is to add 2 and 4. First, put in 2 1-beads then, thinking of 4 as (5 - 1), put in 1 5-bead and take off 1 1-bead. The answer is 6 since 1 5-bead and 1 1-bead are left on the crosspiece.

   Now try 9 + 8. Put in 4 1-beads and 1 5-bead to represent 9. Thinking of 8 as (10 -2), put in 1 10-bead and take off 2 1-beads. Next to the crosspiece are left 1 10-bead, 1 5-bead and 2 1-beads. Thus the answer is 17.

    One of the tricks required to use an abacus is to mentally decompose numbers as we did above. Thus, we can think of 4 as simply 4 or, perhaps, (5-1) or (10-6). Once mastered, this technique proves useful not just for the abacus, but also in doing mental arithmetic.

    Generally, abacus bead movement starts on the left and moves right -- just the opposite of what we normally do with pencil and paper. We could start on the right, but it is much simpler to start from the left mainly because that is the way we pronounce or verbalize numbers. For example, the number 375 is pronounced three hundred seventy-five. To move from right to left, we would have to go through the extra effort of remembering numbers backwards or writing them down and then reading them backwards.

    To illustrate, consider the sum of 375 and 148. Put in, from left to right , 3 100-beads, 2 10-beads, 1 50-bead and 5 1-beads. That represents 375. Basically, we are now going to add, in this order, 100 then 40 then 8 for a total of 148. Start by putting in another 100-bead (or 1 1-bead from the third upright). Next, think of the 4 in 148 as 10 -6. (Actually, because these beads are on the second upright, it is 100-60.) Hence, put in another 100-bead (add 100) and take off 1 10-bead and 1 50-bead (subtract 60). Now for the 8. Since 8 is the same as 10 -2, put in 1 10-bead and take off 2 1-beads. The answer is 523. Note that the order in which we carried out the addition is the same as the way we pronounce the numbers. That is, we added 100 first, then 40 and finally 8 -- one hundred and forty-eight. It would have made no difference if we had proceeded from right to left. That is, we still would have arrived at 523 if we had added 8 then 40 then 100. It is just more natural to go from left to right.

     Handling decimals is done in the same manner as on paper. That is, you add, for example, 100s to 100s, 10s to 10s, 1s to 1s, tenths to tenths and so -- keeping track of the value each upright represents. It is columnar arithmetic.

    Normally, when performing additions, the second 5-bead (i.e., the upper most of the two) is not touched. It is used, however, in subtraction to borrow.


Subtraction

    Subtraction is almost as easy as addition, but you should probably master or at least become familiar with addition first before trying subtraction.

    Addition requires carrying over numbers to the upright or column immediately to the left. Subtraction involves the opposite, that is, borrowing from the column to the left. Whereas in addition, the upper 5-bead is not touched, in subtraction it is used to borrow. Again, bead movement is from left to right.

    Consider 10 - 4. As usual put in 1 10-bead to represent 10. Next, borrow that 10 by putting in 1 5 and 5 1-beads. Take off 4 1- beads and the result is 6. Now try 26 - 7. Put in 26 using 2 1-beads, 1 5-bead and 1 1-bead. Subtracting 7 is the same as subtracting 10 and adding 3. That is, 26 -7 = 26-10+3. Thus take off 1 10-bead and add 3 1-beads. The answer is 19.

   When summing numbers, we may think of adding 7 as adding (10-3), i.e., as adding 10 to one upright then subtracting 3 from the upright to the right. In subtraction, we may think of subtracting 7 as subtracting 10 and adding 3 -- just the opposite. It all depends on how many available beads there are on a particular upright.

    Lets try 375 - 148. Put in 375. Take off 1 100-bead. Next, we note that subtracting 4 is the same as subtracting 5 and adding 1. Thus, take off 1 50-bead and add 1 10-bead. Now we have to borrow 1 10-bead. Do this by taking off 1 10-bead and putting in both 5-beads. To subtract the 8, take off 1 5-bead and 3 1-beads. The answer is 227.

    Numbers that contain decimal values (i.e., tenths, hundredths, etc) are treated just like in addition.


Multiplication

    The abacus is less well suited for multiplication and division than addition and subtraction. However, once you understand the technique, you will see that it is not all that different from multiplication with pencil and paper. You must know the multiplication tables up to 10 x 10 and you should be familiar with addition. There are several techniques used to perform multiplication. Following is one that should be the most comfortable with most of us since it resembles doing multiplication with pencil and paper.

    We start with whole numbers (i.e., no decimal part). Consider 67 x 3. We will temporarily decompose the numbers into a more convenient form to illustrate the process. That is, think of 67 x 3 as (60 + 7) x 3 and finally as (60 x 3) + (7 x 3). We proceed from here and put in 180 ( the result of 60 x 3) then add 21 (the result of 7 x 3). When we do this, we get 1 100-bead, 10 10-beads and 1 1-bead. Carrying the 10 10-beads over into the 100s upright, we arrive at 2 100-beads and 1 1-bead or 201. This method works, but it is neither the best nor the fastest.

    A skilled abacus user would proceed as follows. The fact there is one number to the right of the 6 in 67 and none to the right of the 7 means that you temporarily assign the units upright not to the right most upright, but the one immediately to the left (#2). If there were a total of 2 numbers to the right, say as in 612 x 3 or 67 x 31, use the third upright from the right. So you put in 18 (6 x 3) using the second upright as units. Then add 21 (7 x 3) with the far right upright as units since there are no numbers to the right of the 7 or the 3. In far fewer words, put in 6 x 3 using the second upright then add 7 x 3 using the first. This is completely equivalent to moving columns to left and adding when we do multiplication with pencil and paper:

                                                                       67

                                                                      x 3

                                                                       21

                                                                  +18           (note how the column has been shifted to the left)

                                                                     201

    Note when we add the shifted 18 to 21, it is as if we added an unshifted 180 to 21. With the abacus, the temporary assigment of the second upright to unit values is the same as shifting a column left on paper.

    What about, say, 67 x 31? We could decompose this to (60 + 7) x (30 + 1) = 60 x 30 + 7 x 30 + 60 x 1 + 7 x 1 and proceed. Equivalently, we can multiply 6 x 3 using the 3rd upright temporarily as units because there are a total of two numbers on the right of the multiplier (the 1 in the 31) and multiplicand (the 7 in the 67). So put in 1 and 8 on uprights # 4 and #3, respectively. This is the same as putting in 1800, the product of 60 and 30. Next add (7 x 3) using the second upright as units (there is a 1 to the right of 31 and nothing to the right of the 7 in 67). Note that there are now 10 beads on upright #3. Take these off and carry them over by putting in 1 1000-bead (#4 upright). Next add 6 x 1, again using the second upright as units (there is a 7 to the right of the 6 in 67) and finally add 7 x 1 using the normal units upright. The answer is 2077.

    Note that we proceeded from left to right, the same direction as the way we read the numbers.

    Once you master whole numbers, handling decimal values is quite simple. First, ignore the decimals and proceed as if you were multiplying whole numbers. Lastly, sum all digits to the right of both decimal points. Do not count trailing zeros. Next, count off from the extreme right of our as of yet whole number answer the same number of digits as the sum. Place the decimal point just to the left of this digit. Thus we treat the product of 6.7 x 3.1 just like the example above (67 x 31) and arrive at the whole number 2077. Next we move the decimal over two places and get 20.77 -- the final answer. This technique is identical to that with pencil and paper.

    Abacus users often do the equivalent of writing numbers down before proceeding on with the multiplication. That is, the user enters them on the abacus as a reminder. Usually, one of the numbers is entered on the extreme left; the other somewhere near the middle of the abacus. When doing this, units uprights are temporarily assigned as required. The right part of the abacus is then used to carry out the multiplication.


Division

    On paper and on the abacus, division is the most difficult of the simple arithmetical operations. However, the two procedures are quite similar. The following procedure is one of several that can be used and equivalent to that done on paper.

    Lets look at 762 ( the dividend) divided by 3 (the divisor). We will first do this on paper so we can see the parallels between the two techniques. We get

    Note that we divided 3 into the 7 to get 2. Then multiplied 2 x 3 to yield 6 and subtracted this from 7 to get 1. Next we dropped down the 6 in 762 to get 16. Continuing, we divide 3 into 16 to get 5 with a remainder of 1. Dropping down the 2 gives us 12. Finally, dividing 3 in to 12 yields 4 with no remainder. The final answer or quotient 254. On the abacus we will proceed in exactly the same way.

    First enter the divisor on the left side with the required units upright being given a temporary assignment. Next, enter the dividend on the right. Hence, put in 3 to the left and 762 on the right.

    Mentally divide 3 into 7. The nearest whole number that can do this without the two products exceeding 7 is 2. Put in a 2 somewhere in the middle of the abacus, say the upright #7. Now multiply 2 times the 3 and subtract the result from the 7. The result is 1 on #3. Note this is exactly equivalent to what we did above on paper. To the right of the 1 on the abacus is a 6. The two numbers taken together are 16. (This is the same as dropping down the 6.) Divide 3 in 16 to get 5. Put 5 in on #6. Multiply 3 times 5 and subtract the result from 16. We get 1 on #2. It along with the remaining 2 on the first upright makes 12. Divide 3 into 12 to get 4 and put this in on #5. Subtract 3 x 4 from 12 to get zero and we are finished. The quotient is 254 with the fifth upright being used as units.

    More complicated numbers can be handled precisely in the same manner.

    The placement of the decimal in division is similar to the method we use with pencil and paper. First, we eliminate any decimals in the divisor by counting how many digits it has to the right of the decimal point. Trailing zeros do not count. Say, this number is N. Next multiply both the divisor and dividend by 10N  by moving the decimal to the right in both N places adding zeros as needed. This removes the decimal in the divisor. However, the dividend may still contain a decimal point. As with pencil and paper, decimal placement in the quotient depends on the location of the new decimal in the dividend and on whether the final result is a whole number. When the dividend was put in, one of the uprights was assigned a value of tenths (1/10). During division, the decimal position in the quotient is set when you encounter the tenths upright as you move from left to right in the dividend. Successive numbers after that are 1/100s, 1/1000s and so on. Again, there is a direct parallel with pencil and paper division.


History

    The abacus (from the Latin for calculating table or board ) may have been invented in Mesopotamia by the Sumerians, although, people the world over had used pebbles or other such objects to help them count and do simple arithmetic. The abacus can take on many forms. The K&W version replicates the Chinese abacus with 9 uprights and 2 5-beads and 5 1-beads on each upright. The Japanese version contains up to 21 uprights and has only 1 5-bead and 4 1-beads on each. The abacus techniques used by the Japanese are accordingly somewhat different that those employed by the Chinese.

     In ancient times, even the most basic arithmetical operations were difficult without these devices because of the awkward way numbers were written. Early enumeration methods were designed more for recording or tabulating numbers than doing arithmetic. One drawback was that the numerical value a symbol represented was often independent on its position within the number. For example, 1000, 1001, 1002 and 1333 written in the once popular Roman numerals are M, MI, MII and MCCCXXXIII, respectively. The position of the M seems completely arbitrary even though in all four instances it represents the same value -- namely, 1000. Secondly, there was no symbol for zero. In some systems, it was difficult to distinguish between 15 and 105 or 15 and 1050. Furthermore, without the place holder zero, carrying and borrowing numbers is cumbersome and messy. To add, say, XLII (42) to CIII (103), is not straightforward and multiplying these numbers is next to impossible. Indeed, special tables were designed and complex procedures followed to do all but the simplest of problems. The use of an abacus overcame many of these difficulties and sped up arithmetical calculations. It is no wonder that for centuries it was used extensively from Europe to China.

    The abacus has seen many forms. At first, it was simply a board with marked columns on which pebbles, metal disks or some other small objects were moved. Later, the board was grooved and eventually fitted with runners onto which beads were strung. Some variations had ten or more beads per runner. The modern version (if by modern we mean something that is many of hundreds of years old) is divided into upper and lower parts with the upper beads having a value of 5; lower beads a value of one. The number of uprights can also vary. The Chinese version is more compact with 9, but the Japanese version with up to 21 uprights, makes it more convenient to temporarily store numbers.

    In 1202 the Italian mathematician Leonardo Fibonacci introduced the Hindu-Arabic method of writing numbers to Europe. The Hindus invented this technique in the 8th century AD and it was later adopted by the Arabs. It was no doubt heavily influenced by the columnar plan of the abacus. The system was based on ten symbols that eventually evolved to the familiar 1 through 9 plus the all important 0. By writing numbers with this system, symbol values became dependent on columnar position mimicking abacus bead values on the various uprights. Blank or vacant columns were indicated with a zero. Borrowing and carrying numbers on paper became as easy as it was using the abacus.

    The adoption of Arabic numerals (as they were eventually called) caused the abacus to fall into disuse in many areas, particularly the West -- where it is now more an object of curiosity. However, even with the advent of the hand held electronic calculator, it is still the calculating device of choice for many in China, Japan and elsewhere.