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Calculate Square Root
. . .
without using a square root calculator

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Calculate Square Root -
without using a square root calculator . . .
Introduction - How to Calculate Square Root by Hand:

There are many methods that can be used to calculate the square root of a number without a calculator. Four of the most prominent methods are discussed below.


  • Direct Calculation (The Chinese Method) - Probably the most popular method of computing square roots without a calculator. This is a precise, digit by digit calculation similar to long division. It is often found in textbooks.

    This technique can be modified to calculate cube roots and other roots.

    Skip to example - click here



Direct Calculation - free ON-LINE Calculator (The Chinese Method) - Shows step by step calculations using the Direct Calculation Method to compute the square root of any number. You can practice using the Direct Calculation Method, and check each step using this on-line calculator:

http://barnyard.syr.edu/longroot.html


  • Newton's Method (Newton's Method is similar to the Babylonian method, but incorporates mathematical insights gained from calculus.) - It is also a popular method of computing square roots without a calculator. Newton's Method is a fast approximating sequence which converges quickly to a high degree of accuracy . The procedure is straightforward and is easy for anyone to use.

    Skip to example - click here

    However, Newton's Method goes far beyond calculating square roots. It is a numerical method of solving all sorts of complex equations. For example, it can also be used to find numerical solutions to equations containing higher order roots such as cube roots.

    On-line calculators which use Newton's Method can be found at:

    http://www.math.sc.edu/cgi-bin/sumcgi/Newton.pl

    http://www.shodor.org/UNChem/math/newton/nmethod.html




  • Guess & Check Method The Guess & Check Method to Calculate Square Roots is easy for anyone to use. If you have access to a simple calculator (without a square root key), this method can be very fast, although its accuracy is limited.

    This method can also be used to calculate higher order roots such as cube roots.

    Skip to example - click here



  • Geometric Diagram The ancient Greeks developed a method for calculating the square root of a number using a geometric construction of circles and triangles. To use this method, you need only a compass and a straight edge.

    The geometric construction used to Calculate Square Roots actually calculates the geometric mean of two numbers. So, setting one of the numbers to 1, the geometric mean is the square root of the other number. The final answer (the square root) is the length of one side of a right triangle. Examples of how to use this method will not be covered here, but you can read about how this is done by clicking on the following links:

    http://www.unl.edu/tcweb/fowler/2006MathPapers/WhatisaSquareRootRudy.pdf

    http://www.egge.net/~savory/maths9.htm

    http://en.wikipedia.org/wiki/Square_root#Geometric_construction_of_the_square_root


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The Properties of Radicals & Roots - Real Numbers - click description
Simplifying Radicals - Math - Radicals - adding and subtracting
Math - Multiplying Radicals
Math - Dividing Radicals
Math - Rationalizing the Denominator
Math - Fractional Exponents & Radicals
Math - Simplifying Radicals
Math - Calculate Square Root Without Using a Calculator
Math - Calculate Roots Using Equations
Math - Free Videos on Radicals & Roots + Free Resources
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Calculate Square Root . . . without using a square root calculator



Example of Direct Calculation
Precise, digit by digit method used to Calculate Square Roots (The Chinese Method).


Video demonstrating this method - click here


Math - Calculate Square Root without a calculator - example 1 - Precise, digit by digit calculation Calculate Square Root: Math - Simplifying Radical Expressions - Initial Problem - example 1

Step 1:

Beginning at the decimal point, divide the radicand into groups of two digits in both directions. The decimal can be extended as far as you wish.

Math - Calculate Square Root without a calculator - example 1 - Divide the radicand into groups of two digits - Step 1


Step 2:

Beginning on the left, select the first group of one or two digits.

In this example, the first group of digits in the radicand is simply the number 2.

Math - Calculate Square Root without a calculator - example 1 - First Group of Digits - Step 2


Step 3:

If possible, find the perfect square root (expressed as an integer) of the first group of digits in the radicand.

Or, If the first group of digits does not have a perfect square root (expressed as an integer), find the perfect square root of the number closest to, but less than, the first group of digits.

In this example, the number 2 does not have a perfect square root (which can be expressed as an integer).

The number closest to, but less than, the number 2 which has a perfect square root is the number 1.

The perfect square root of the number 1 is also1. Write this number above the radical sign as shown below.

Math - Calculate Square Root without a calculator - example 1 - Step 3


Step 4:

Square the answer to step 3, and subtract it from the first group of numbers.

In this example, the number 1 squared is subtracted from the number 2.

Math - Calculate Square Root without a calculator - example 1 - Step 4

Step 5:

  • A - Bring down the next group of two digits.

    In this example, bring down 73.

  • B - Double the result of step 3 (shown as a slate color), and write on lower left.

    In this example, the number 1 x 2 = 2.

  • C - Add another digit to the number computed in "B" (the slate colored number) on the lower left. Choose the new digit so that the new number created by the additional digit can be divided into the number on the immediate right the same number of times as the new digit.

    In this example, the new digit is 6. The resulting number (26), can be divided into 173 a total of 6 times.

Math - Calculate Square Root without a calculator - example 1 - Step 5



Step 6:

Repeat steps A, B, and C shown in step 5.

  • A - Bring down the next group of two digits.

    In this example, bring down 00.

  • B - Double the result of step 5 (shown as a slate color), and write on lower left.

    In this example, the number 16 x 2 = 32.

  • C - Add another digit to the number computed in "B" (the slate colored number) on the lower left. Choose the new digit so that the new number created by the additional digit can be divided into the number on the immediate right the same number of times as the new digit.

    In this example, the new digit is 5. The resulting number (325), can be divided into 1700 a total of 5 times.

Math - Calculate Square Root without a calculator - example 1 - Step 6



Step 7:

Repeat steps A, B, and C shown in step 5.

  • A - Bring down the next group of two digits.

    In this example, bring down 00.

  • B - Double the result of step 6 (shown as a slate color), and write on lower left.

    In this example, the number 165 x 2 = 330.

  • C - Add another digit to the number computed in "B" (the slate colored number) on the lower left. Choose the new digit so that the new number created by the additional digit can be divided into the number on the immediate right the same number of times as the new digit.

    In this example, the new digit is 2. The resulting number (3302), can be divided into 7500 a total of 2 times.

Math - Calculate Square Root without a calculator - example 1 - Step 7



Step 8:

Repeat steps A, B, and C shown in step 5.

  • A - Bring down the next group of two digits.

    In this example, bring down 00.

  • B - Double the result of step 7 (shown as a slate color), and write on lower left.

    In this example, the number 1652 x 2 = 3304.

  • C - Add another digit to the number computed in "B" (the slate colored number) on the lower left. Choose the new digit so that the new number created by the additional digit can be divided into the number on the immediate right the same number of times as the new digit.

    In this example, the new digit is 2. The resulting number (33042), can be divided into 89600 a total of 2 times.

Math - Calculate Square Root without a calculator - example 1 - Step 8



The final answer is: Math - Calculate Square Root without a calculator - example 1 - Final Answer to 3 decimal points

(not rounded)

Cube Roots: calculating cube roots by hand is more complicated than computing square roots by hand. For information on cube roots, see the following web sites:

http://www.itl.nist.gov/div897/sqg/dads/HTML/cubeRoot.html

http://mathforum.org/library/drmath/view/52605.html

http://www.mathpath.org/Algor/cuberoot/algor.cube.root.htm




Example of Newton's Method
Newton-Raphson method: consecutive numerical approximations used to
Calculate Square Roots (approximations become more and more accurate
with each iteration)
.

The three steps used to Calculate Square Roots are:


  • A - Initial Estimate of the square root of the radicand. (The initial estimate can be any number. Subsequent steps will refine the estimate.)

  • B - Divide the radicand by the estimate in part A.

  • C - refine the estimate by computing the Average of the estimate (from part A) and the quotient (from part B).


Video demonstrating this method - click here



Math - Calculate Square Root without a calculator - example 2 - Newton-Raphson numerical process Calculate Square Root: Math - Calculate Square Root without a calculator - example 2 - Newton-Raphson numerical process - Initial Problem

1st Approximation: Calculate Square Root - Newton's Method

  • A - Initial Estimate of the square root of the radicand.

    In this example, a good initial estimate for the square root of 273 is 15

    (However, the initial estimate could be "any" number.)

  • B - Divide the radicand by the estimate in part A.

    In this example, divide 273 by 15.

    Math - Calculate Square Root without a calculator - Newton's Method - example 2 - Step 1b

  • C - 1st Approximation - refine the initial estimate by computing the Average of the estimate (from part A) and the quotient (from part B).

    In this example, find the average of 15 and 18.2.

    Math - Calculate Square Root without a calculator - Newton's Method - example 2 - Step 1c

    1st Approximation of Square Root is: Math - Calculate Square Root without a calculator - Newton's Method - example 2 - Step 1c - final



2nd Approximation: Calculate Square Root - Newton's Method

  • A - Initial Estimate for the 2nd Approximation of the square root of the radicand. Use the results of the 1st Approximation, part C, above.

    In this example, the initial estimate will be 16.6

  • B - Divide the radicand by the estimate in part A.

    In this example, divide 273 by 16.6.

    Math - Calculate Square Root without a calculator - Newton's Method - example 2 - Step 2b

  • C - 2nd Approximation - refine the initial estimate by computing the Average of the estimate (from part A) and the quotient (from part B).

    In this example, find the average of 16.6 and 16.445783.

    Math - Calculate Square Root without a calculator - Newton's Method - example 2 - Step 2c

    2nd Approximation of Square Root is: Math - Calculate Square Root without a calculator - Newton's Method - example 2 - Step 2c - final




3rd Approximation: Calculate Square Root - Newton's Method

  • A - Initial Estimate for the 3rd Approximation of the square root of the radicand. Use the results of the 2nd Approximation, part C, above.

    In this example, the initial estimate will be 16.522892

  • B - Divide the radicand by the estimate in part A.

    In this example, divide 273 by 16.522892.

    Math - Calculate Square Root without a calculator - Newton's Method - example 2 - Step 3b

  • C - 3rd Approximation - refine the initial estimate by computing the Average of the estimate (from part A) and the quotient (from part B).

    In this example, find the average of 16.522892 and 16.522531.

    Math - Calculate Square Root without a calculator - Newton's Method - example 2 - Step 3c

    3rd Approximation of Square Root is: Math - Calculate Square Root without a calculator - Newton's Method - example 2 - Step 3c - final
    (Final answer for this example)







Guess & Check Method
The Guess & Check Method used to Calculate Square Roots is exactly what the
name implies: Guess what the value of the square root might be, then multiply
your guess by itself to see if the product is equal to the original radicand
.

The three steps used to Calculate Square Roots are:


  • A - Guess - Estimate the square root of the radicand. (The initial estimate can be any number. Subsequent steps will refine the estimate.)

  • B - Multiply the estimate by itself.

  • C - Check - Compare the product obtained in step "B" (above) with the radicand.

  • D - Refine the Estimate . If the product obtained in step "B" (above) is greater than the radicand, decrease the estimate. If the product obtained in step "B" (above) is less than the radicand, increase the estimate.

  • E - Repeat steps B through D until the estimate is an accurate square root of the radicand.


Math - Calculate Square Root without a calculator - example 3 - Guess & Check Method Calculate Square Root: Math - Calculate Square Root without a calculator - example 2 - Newton-Raphson numerical process - Initial Problem

1st Approximation: Calculate Square Root - Guess & Check Method

  • A - Guess - Initial Estimate of the square root of the radicand.

    In this example, a good initial estimate for the square root of 273 is 15

    (However, the initial estimate could be "any" number.)

  • B - Multiply the estimate by itself.

    In this example, multiply 15 by 15.

    Math - Calculate Square Root without a calculator - example 3 - Guess & Check Method - Step 1b

  • C - Check - Compare the product obtained in step "B" (above) with the radicand.

    In this example, 225 is less than 273.

    Math - Calculate Square Root without a calculator - example 3 - Guess & Check Method - Step 3c

  • D - Refine the Estimate . If the product obtained in step "B" (above) is greater than the radicand, decrease the estimate. If the product obtained in step "B" (above) is less than the radicand, increase the estimate.

    In this example, since 225 is less than 273 the estimate will be increased from 15 to 16.

  • Refined estimate of Square Root is: Math - Calculate Square Root without a calculator - example 3 - Guess & Check Method - Step 3d - new estimate

2nd Approximation: Calculate Square Root - Guess & Check Method

  • A - Guess - Use the results of the 1st Approximation, part D, above..

    In this example, the estimate for the square root of 273 is raised to 16

  • B - Multiply the estimate by itself.

    In this example, multiply 16 by 16.

    Math - Calculate Square Root without a calculator - example 3 - Guess & Check Method - Step 2b

  • C - Check - Compare the product obtained in step "B" (above) with the radicand.

    In this example, 256 is still less than 273.

    Math - Calculate Square Root without a calculator - example 3 - Guess & Check Method - Step 2c

  • D - Refine the Estimate . If the product obtained in step "B" (above) is greater than the radicand, decrease the estimate. If the product obtained in step "B" (above) is less than the radicand, increase the estimate.

    In this example, since 256 is less than 273 (but closer than the original estimate) the estimate will be increased a small amount, from 16 to 16.5.

  • Refined estimate of Square Root is: Math - Calculate Square Root without a calculator - example 3 - Guess & Check Method - Step 2d - new estimate

3rd Approximation: Calculate Square Root - Guess & Check Method

  • A - Guess - Use the results of the 1st Approximation, part D, above..

    In this example, the estimate for the square root of 273 is raised to 16.5

  • B - Multiply the estimate by itself.

    In this example, multiply 16.5 by 16.5.

    Math - Calculate Square Root without a calculator - example 3 - Guess & Check Method - Step 3b

  • C - Check - Compare the product obtained in step "B" (above) with the radicand.

    In this example, 272.25 is still less than 273, but it is much closer than the other estimates.

    Math - Calculate Square Root without a calculator - example 3 - Guess & Check Method - Step 3c

  • D - Refine the Estimate . If the product obtained in step "B" (above) is greater than the radicand, decrease the estimate. If the product obtained in step "B" (above) is less than the radicand, increase the estimate.

    In this example, since 272.25 is less than 273 (but much closer than the other two estimates) the estimate should be increased slightly.

  • Refined estimate of Square Root: these steps must be
    repeated over and over again until the degree of accuracy
    required is achieved.










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