Square root of an arbitrary positive number: Determination by estimation & iteration Given an area (A) of a square, determine its width. ? Area

w0w0 w0w0 Estimate

Approximate w0w0 w0w0   ? ?  Small  neglect

Approximate w0w0 w0w0   ? ?  Small  neglect

Calculate A ≈ w w 0  + w 0  Determine , given A and w 0 w0w0 w0w0   

w0w0 w0w0    Calculate A ≈ w w 0  + w 0  A – w 0 2 = 2  w 0  Determine 

w0w0 w0w0    Calculate A – w 0 2 2w02w0  ≈ ≈

w0w0 w0w0    w 1 = w 0 +  Approximate

Iteration w 1 = w 0 +  w0w0 w0w0    Initial Estimate A – w 0 2 2w02w0  ≈ ≈

Iteration w 1 = w 0 +  w0w0 w0w0    Initial Estimate w1w1

Iteration w 1 = w 0 +  w0w0 w0w0    Initial Estimate w1w1

Iteration w 1 = w 0 +  w0w0 w0w0    Initial Estimate w1w1 1 st approximation

Iteration w 1 = w 0 +   Initial Estimate w1w1 w 2 = w 1 +    A – w 1 2 2w12w1 1 ≈1 ≈

Iteration w 1 = w 0 +   Initial Estimate w1w1 w 2 = w 1 +   w2w2

Iteration w 1 = w 0 +   Initial Estimate w1w1 w 2 = w 1 +   w2w2

Iteration w 1 = w 0 +   Initial Estimate w1w1 w 2 = w 1 +   w2w2

Numerical Values  20 = ? Estimate: 4 A – w 0 2 2w02w0  ≈ ≈ w 1 = w 0 +  Iteration = = =

Advantage Requires knowledge of fraction, decimals and math operations involving positive & negative numbers. Need not estimate “high” and “low” value. Numerical determination without the knowledge of calculus! Allows fast computation of a number!