Now it must be halved to its original 3 and we have the answer to the problem on rods B & C. In Step 3 we doubled the 3 on rod B to equal 6. Step 4: Now mentally square the 4 on rod C and subtract the product from the remainder on the right. Multiply 6 x 4 to equal 24 and subtract 24 from 25. 6 goes into 25 four times with a remainder. Step 3: Double the 3 on rod B to equal 6 and divide 6 into 25 on rods G & H. This forms the first part of the root number. Set the square root of 9 (namely 3) somewhere on the left-hand side of the abacus, in this case on rod B. (In this case the 11 on rods F & G.) Find the largest perfect square that is less than or equal to 11. Step 2: Now look at the first group of numbers. Because there are two groups (11 & 56) the answer will have two whole numbers possibly followed by a decimal in the answer. If there's room, pair it off 11 56 or do it mentally. Step 1: Set 1156 on the right hand side of your abacus. Doing the following examples below on your abacus should clarify things. What remains will be the square root answer.Ĭonfused? I think I would be. When finished go back and halve the numbers that you doubled in the root number on the left.In the event the square number is not a perfect square, when the decimal is reached further groups of 00 may be added and the process continued. Continue the process until the square number on the right is used up.Mentally square the quotient and subtract it from the square number on the right hand side.Place the quotient along-side the first root number. Double the root number and divide it into the next group in the square number on the right hand side.This number on the left is the "root number" and will eventually form our answer. Dealing again with the perfect square, take its square root and place it on the left hand side of the abacus.Subtract the perfect square from the group. Look at the first group of paired numbers (or number) and determine the largest perfect square that is less than or equal to the group.Group the square number into pairs as described above.This number on the right is the "square number". Begin by placing the number from which the square root is to be extracted onto the right hand side of the abacus.It might just be better to jump ahead and learn from the examples below. For this reason trying to explain how it all works may be an exercise in confusion. Kojima does the same in his book.Īlthough the technique involves only simple mathematics and is therefore quite easy to do, the extraction process is fairly involved and labor intensive. Because of the constraints of using a small cyber abacus, we will do the separation and pairing mentally. 236 starting at the decimal point and moving left and right the numbers would be separated 1' 73' 62'. In preparing to find the square root of a number, whether it be on paper or on the abacus, the number is normally separated into pairs of digits. To square (or squaring) a number refers to multiplying a number by itself as in the example, 3 x 3 = 9. It's always set on the left-hand side of the abacus. The root number refers to the number that eventually forms our answer. It's always set on the right-hand side of the abacus. The square number refers to the number from which the square root is to be extracted. In order to illustrate the technique, I'll use the terminology square number, root number and square. It certainly makes it less confusing because I don't understand algebraic formulas anyway. Rather than get muddled up with a bunch of algebraic formulas, I'll stick to how it's done on the abacus. I'll use the one that Takashi Kojima seems to favor in his book, Advanced Abacus - Theory and Practice. There are different techniques that can be used to extract square roots using an abacus. In fact not only can it be done, using an abacus makes it easy. I guess I just wanted to learn the technique because I think it's so amazing that it can be done on the abacus. When it comes to extracting square roots, it's true I can't remember the last time I needed to extract the square root of a large number and apply it to anything practical. I recreate it here sinceĭave's pages have been offline as of May, 2006. This tutorial dates back to October 2003 when I wrote it as a chapter for Dave Bernazzani's Square Roots as solved by Takashi Kojima ABACUS: MYSTERY OF THE BEAD
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