Napier's Bones
       for




Taking
Square Roots

Read Vocabulary First       You must know Napier's Bones vocabulary and how to multiply.       Read: Vocabulary

About the Square Root

 

Take a Square Root     There is an additional Napier's bone for completeing this computation -- the square root bone.     The Napier's bones do all the hard work -- you just fill in the pieces and complete the subtractions.     The square root bone has columns for the square of a digit, double the digit, and the digit.     The impact of the square of a digit, the double of the digit, and the digit is illustrated in the algebra and figure below.

The sum of the first square (x2), the product of the double of the number which produced the square and the number which produces the second square (2xy), and the second square (y2), is the square of the sums of the two numbers (x + y)2.

See the animation "Take A Square Root" or read instructions and examples below.

The Square Root Algorithm 1st: Recall the columns on the square root bone are, from left to right, the square of the digit, the double of the digit, and the digit. The square will be used to: identify the next digit of the root, decrease the radicand making a new "radicand remainer," and to act as a bone in creating multiples as is done in multiplication and division. The double will be used to: identify the first bones used on the board to create multiples, and then to modify the set of bones on the board used to make multiples and take the root. The digit will be used to: identify the row of bones and to become the next digit of the root. 2nd: From the decimal, to the left with whole numbers and also to the right with decimal numbers, group digits of the radicand in pairs. 3rd: Begin with the left-most pair (or single digit should there be an odd number of whole number digits). 4th: On the square root bone, find the largest square smaller than or equal to this pair. 5th: Write it below the left-most radicand pair and subtract to produce a "radicand remainder." 6th: On the square root bone, on the same row, use the DIGIT as first digit of the root. 7th: On the square root bone, on the same row, use the DOUBLE to place bone(s) for this number on the board.   Once the first root digit is found, if additional digits of the root are required, this process use the following rule.   If the double has one digit, make the units digit of the DOUBLE the new last digit of the root. If the doube has two digits, make the units digit of the DOUBLE the new last digit of the root. Take the existing number represented by bones on the board, add to it the tens digit of the DOUBLE and place bones on the board. 8th: Use both the "radicand remainder," from step 5, and the next pair to create the next "new radicand." 9th: Repeat the procedure until the desired accuracy is achieved or until the "radicand remainer" is 0.

Click on graphic to enlarge.

4th, 5th, 6th steps

first part of 7th step

second part of 7th step, 8th

9th: "radicand remainder" is 0

See how to take a square root

Done step-by-step.

Done with animation.

Click on graphic to enlarge.

For each digit of the root, find the square, take the digit for the root, use the double to choose bones for the next repetition.

Resources Instructions on the Operations multiplication division root extraction   Manipulatives Napier's Bones Digital Manipulatives Napier's Bones big paper version Napier's Bones small paper version       Many other manipulatives on many different topics are found at: MathTokens.Com.

References Azzolino, Agnes "algorithm" © 2005 at www.mathnstuff.com/math/spoken/here/1words/a/a11.htm (last modified 2005), visited 10 April 2009. Azzolino, Agnes "Multiplication & Division" © 2008 at www.mathnstuff.com/math/algebra/tt15.htm#divide visited 10 April 2009. Azzolino, Agnes "multiple" © 2005 at www.mathnstuff.com/math/spoken/here/1words/m/m29.htm visited 10 April 2009. Wikipedia®', Wikimedia Foundation, Inc., "Napier's bones - From Wikipedia, the free encyclopedia" GNU Free Documentation License at http://en.wikipedia.org/wiki/Napier%27s_bones (last modified 16 March 2009, at 20:42 (UTC) ), visited 10 April 2009.

© 2009, October 2010, A. Azzolino www.mathnstuff.com/math/spoken/here/2class/60/nbonesr.htm