Use an Abacus

The abacus is a deceptively simple calculating tool still used all over the world. It's a useful learning device for the visually impaired, as well as for anyone who wants to learn the roots of the modern calculator. Read these simple steps and get calculating.

Steps

Counting and Basics

  1. Orient the abacus correctly. A basic abacus consists of two rows of beads arranged in a variable number of columns. Each column in the top row should have one or two beads per row, while each column in the bottom row should have four. When you start, all of the beads should be up in the top row, and down in the bottom row. The beads in the top row represent the number value 5 and each bead in the bottom row represents the number value 1.
    • Once you become more familiar with the function of the abacus, you can assign different values to the beads in the bottom row to perform more complex operations. The beads in the top row, however, need to be 5 times the value of each bead in the bottom row for the abacus method to work.
  2. Assign each column a place value. As on a modern calculator, each column of beads represents a "place" value from which you build a numeral. So, the farthest column on the right would be the "ones" place (1-9), the second farthest the "tens" place (10-99), the third farthest the hundreds (100-999), and so on.
    • Depending on your calculation, you can also assign a decimal place that you'll need to keep track of. If you want to enter the number 12,345.67, the 7 would be in the first column, the 6 in the second, the 5 in the third, and so on. When doing those calculations, you just have to remember where the decimal place is, mark it on the abacus with a pencil, or you can skip a row and leave it "blank" if that helps you remember.
  3. Start counting. To count a digit, push one bead to the "up" position. "One" would be represented by pushing a single bead from the bottom row in the farthest column on the right to the "up" position, "two" by pushing two, etc.
  4. Complete the "4/5 exchange."[1] Since there are only four beads on the bottom row, to go from "four" to "five," you push the bead on the top row to the "down" position and push all four beads from the bottom row down. The abacus at this position is correctly read "five." To count "six," push one bead from the bottom row up, so the bead in the top row is down (representing a value of 5) and one bead from the bottom row is up.
    • The process is essentially the same across the abacus. Go from "nine," in which all the beads in the ones place are pushed up and the bead in the top row is pushed down, to "ten," in which a single bead from the bottom row of the tens place is pushed up.
    • So, to illustrate: 12345 would be represented with the top bead down in the ones place, four beads from the bottom row of the tens place pushed up, three beads up in the bottom row of the hundreds place, two beads up in the bottom row of the thousands, and a single bead from the bottom row of the ten-thousand place.
    • It's easy to forget to push the beads in the bottom row down when exchanging a place, making the board will show the wrong value. It's easy enough to keep track of when you're counting, but when you get into complicated arithmetic it becomes more difficult.

Adding and Subtracting

  1. Input your first number. Say you've got to add 1234 and 5678. Enter 1234 on the abacus by pushing up four beads in the ones place, three in the tens place, two in the hundreds place, and one in the thousands place.
  2. Start adding from the left. Unlike traditional arithmetic, in which you'd start from the ones column and move left, the abacus works from left to right. So, the first numbers you'll add are the 1 and the 5 from the thousands place, in this case moving the single bead from the top row of that column down to add the 5, and leaving the lower bead up for a total of 6. Likewise, you'll move the top bead in the hundreds place down and three more beads from the bottom up to get an 8 in the hundreds place.
  3. Complete an exchange. Here's where things get creative. Since adding the two numbers in the tens place will result in 10, you'll carry over a 1 to the hundred place, making it a 9 in that column. Next, put all the beads down in the tens place, leaving it zero.
    • In the ones column, you'll do essentially the same thing. 8 + 4 = 12, so you'll carry the one over to the tens place, making it 1, leaving you with 2 in the ones place.
  4. Count up your beads. You're left with a 6 in the thousands column, a 9 in the hundreds, a 1 in the tens, and a 2 in the ones: 1,234 + 5,678 = 6,912.
  5. To subtract, do essentially the exact same process in reverse. Borrow digits from the previous column instead of carrying them over. Say you're subtracting 867 from 932. After entering 932 into the abacus (the upper bead in the up position and all four lower beads up in the hundreds column, three lower beads up in the tens column, and two lower beads up in the ones column), start subtracting column-by-column starting on your left.
    • 8 from 9 is one, so you'll leave a single bead up in the hundreds place. In the tens place, you can't subtract 6 from 3, so you'll borrow the 1 in the hundreds place (leaving it zero) and subtract 6 from 13, making it 7 in the tens place (the upper bead up and two lower beads). Do the same thing in the ones place, "borrowing" a bead from the tens place (making it 6) to subtract 7 from 12 instead of 2. There should be a 5 in the ones column: 932 - 867 = 65.

Multiplying

  1. Transpose the problem onto the abacus.[2] Unlike adding, it helps to start at the farthest left column of the abacus when multiplying. Say you're multiplying 34 and 12. You need to assign columns to "3" "4" "X" "1" "2" "=" and leave the rest of the columns to the right of them blank for your product. For this problem, you'll need at least three.
    • The "X" and the "=" should just be spaces that you leave blank, to keep your numbers separate, so it will take a total of six columns to enter "34 x 12 =" into the abacus.
    • The abacus should have 3 beads up in the farthest column left, four up in the next farthest, a blank column, one bead up, two beads up, another blank column, and at least three columns open to record the product.
  2. Multiply by alternating columns. The order here is critical. You need to multiply the first column by the first column after the break, then the first column by the second column after the break. Next, you'll multiply the second column before the break by the first column after the break, then the second column before the break by the second column after the break. It should always be done in this order.
  3. Record the products in the correct order. First, you'll multiply 3 and 1, recording their product in the first answer column, which in this case will be the seventh column from the left, accounting for each digit and each necessary blank column. Push three beads up in that seventh column. Next, multiply the 3 and the 2, recording their product in the eighth column. Push up the upper bead and one lower bead in that column.
    • Here's where it gets tricky. When you multiply the 4 and the 1, you'll need to add that product to the eighth column, the second of the answer columns. The product of 4 and 1 is 4, and since you're adding a 4 to a 6 in that column, you'll need to carry one bead over to the first answer column, making it a 4 in the seventh column and a zero in the eighth.
    • Multiply the last two digits in the problem, 4 and 2, and record that product in the ninth column, putting an 8 in the last of the answer columns, which should now read 4, blank, and 8, making your answer 408.

Dividing

  1. To divide, leave space for the answer between the divisor and the dividend.[3] Division is a more fluid process than multiplication, and it works best when you don't leave blank spaces between the numbers involved.The left-most column on the abacus will be the divisor, the number being divided by. The next spaces to the right should be left for the answer.
    • Say you're dividing 34 by 2. You know the answer will be at least two columns, so leave two columns between 2 on the right and the 3 and the 4.
    • To summarize, to enter 34 divided by 2 on the abacus, you should have 2 in the left most column, two columns to record the answer, the 3 in the fourth column, and the 4 in the fifth.
  2. Record the quotient. Take the first number in the dividend (3) and the divisor (2) in the first answer column. 2 goes into 3 once, so record a 1 in column 2.
  3. Determine the remainder. Next, you need to multiply the quotient in column two (1) by the dividend in column one (2) to determine the remainder. This product (2) needs to be subtracted from column four. The divisor should now read 14.
  4. Repeat the process. Record the next digit of the quotient in the third column, subtracting the product from the divisor (here, eliminating it). Your board should now read 2, 1, 7, leaving your dividend and the quotient, 17.



Tips

  • If you are dealing with decimals, you can move the units column to the left.
  • If you are dealing with numbers where the units value is insignificant because of the large magnitude of the numbers, you can move the units column to the right (out of the abacus).
  • The Japanese "soroban" method works slightly differently in terms of how the individual columns are added and subtracted, and the "five" bead is counted by moving it into the down position, rather than up, but still moves left to right in essentially the same way for calculations and operates on the same basic principle.

Related Articles

  • Teach a Child Skip Counting with a iPhone Calculator

Sources and Citations

  1. ↑ http://www.tsbvi.edu/abacus/3284-using-an-abacus-and-the-counting-method-
  2. ↑ http://www.abacusmaster.com/Mypage.aspx?p=Abacus-Multiplication.htm
  3. ↑ http://scienceblogs.com/goodmath/2006/09/25/division-on-the-abacus/
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