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

2. w0w0 w0w0 Estimate

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

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

5. Calculate A ≈ w 0 2 + w 0  + w 0  Determine , given A and w 0 w0w0 w0w0   

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

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

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

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

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

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

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

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

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

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

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

17. Numerical Values  20 = ? Estimate: 4 A – w 0 2 2w02w0  ≈ ≈ w 1 = w 0 +  Iteration 0 4.000000 1 4.500000 = 4.000000 + 0.500000 2 4.472222 = 4.500000 + -0.027778 3 4.472136 = 4.472222 + -0.000086

18. 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!