How to Use a Chinese Abacus
Date: 06/24/2004 at 21:16:05 From: Carlos Subject: how to use Chinese abacus Can you tell me how to use a Chinese abacus? Thanks!
Date: 06/26/2004 at 16:43:10 From: Doctor Katy Subject: Re: how to use Chinese abacus Hi Carlos! Thanks for writing to Dr. Math. The first step in knowing how to use an abacus is knowing how to "read" an abacus. I hope that you have an abacus yourself, because trying examples and playing around with it a little will make learning how to use it much, much easier. The abacus consists of 13 columns, each one divided into an upper deck and a lower deck; the lower deck consists of 5 beads per column, the upper deck has 2 beads per column. I included a little diagram, I hope you can make sense of it: each "o" represents a bead. Note there is always space for you to move some of the beads away from the others. Diagram of a typical abacus --------------- - | | | |ooooooooooooo| | upper deck |ooooooooooooo| - |-------------| | | - |ooooooooooooo| | |ooooooooooooo| | |ooooooooooooo| | lower deck |ooooooooooooo| | |ooooooooooooo| - --------------- ^^^^^^^^^^^^^ the 13 columns Each column represents a digit, so with 13 columns you can do calculations with numbers up to 10 trillion! The way to express a number is to move up the beads of each column to represent a digit, starting from the right. The value of a bead in the lower deck is 1, the value of a bead in the upper deck is 5. For example, to express the digit 7, you would move up two beads in the lower deck and one bead in the upper deck, so that (1+1) + (5) = 7. Make sense? Knowing that, this is what the number 237 would look like: --------------- | o| |oooooooooooo | |ooooooooooooo| |-------------- | ooo| |ooooooooooooo| |oooooooooo o | |ooooooooooo o| |ooooooooooooo| |ooooooooooooo| --------------- 237 You might notice that each column can take on a value from 0-15 and think it's silly because in our number system, each digit only takes on values from 0-9. However, this becomes handy when you need to carry digits that exceed 10, just like you do when doing addition with pen and paper. Also, it allows you to do calculations with other number systems that are not 10-based, if you want to. How about another example? --------------- | o| |oooooooooooo | |ooooooooooooo| |-------------- | ooo| |ooooooooooooo| |ooooooooooo o| |ooooooooooooo| |oooooooooooo | |oooooooooo oo| --------------- 529 Notice that the digit 5 in the third column can alternatively be expressed by moving up one bead in the upper deck instead of the five beads in the lower deck. If all this starts making sense, I think you can already see how using an abacus can be useful for simple arithemtic. However, the abacus is only useful if one can "read" it quickly enough and do the simple additions in your head (namely 1 upper bead and 4 lower beads = 1*5 + 4*1 = 9), but that really isn't hard to learn. Now, to do something like 625 + 536, first put down the number 625. --------------- | o o| |oooooooooo o | |ooooooooooooo| |-------------- | oo | |oooooooooo oo| |ooooooooooo o| |ooooooooooooo| |ooooooooooooo| |ooooooooooooo| --------------- 625 Then you proceed to add each digit, starting from the right. To add the 6 (from the number 536), simply add 1 bead to the upper deck and 1 bead to the lower deck (since 1*5 + 1*1 = 6). Next, to add the 3 to the second column, just move up 3 beads from the lower deck (1*3 = 3). To add the 5, add one bead from the upper deck. Your final result will look like this: --------------- | o o| <-- Notice that whenever you have 2 beads up in the |ooooooooooooo| upper deck, you can move them down and add one |oooooooooo o | from the lower deck in the next column. If you |-------------- don't, you will have numbers greater than 10 in | ooo| a column, which is confusing. |oooooooooo o | |ooooooooooooo| |ooooooooooooo| |ooooooooooooo| |ooooooooooo o| --------------- This simplifies to: --------------- | o | |ooooooooooo o| |ooooooooooooo| |-------------- | oooo| |ooooooooo | |ooooooooooooo| |ooooooooooooo| |ooooooooooooo| |ooooooooooooo| --------------- 1161 It looks tedious, but with practice it becomes much easier (1.3 billion Chinese can't be wrong!). So what the abacus does for you is that it "holds" the first number for you while you're adding the second one digit by digit. When you want to do addition or subtraction problems, all you're doing is shifting beads up and down each column (i.e. digit) at a time, so that even 23509725 - 9438558 isn't too difficult to do. There are lots more "tricks" that one should know to be able to do more addition and subtraction problems, but they are hard to explain without showing. But just for your information, calculations such as multiplication, division, and even taking the square/cube root are all possible, but much more difficult. For further information on this and many other things, you can visit a website dedicated entirely to the abacus: Abacus - The Art of Calculating With Beads http://www.ee.ryerson.ca:8080/~elf/abacus/ Also, you can look at another response in our archive given by a different math doctor. Using an Abacus http://mathforum.org/library/drmath/view/59147.html Feel free to write back if you have any further questions or need clarifications on anything. - Doctor Katy, The Math Forum http://mathforum.org/dr.math/
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