Question

find square root of 335 with dialong method
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Answer 1

Answer:

Step 1:

Divide the number (335) by 2 to get the first guess for the square root .

First guess = 335/2 = 167.5.

Step 2:

Divide 335 by the previous result. d = 335/167.5 = 2.

Average this value (d) with that of step 1: (2 + 167.5)/2 = 84.75 (new guess).

Error = new guess - previous value = 167.5 - 84.75 = 82.75.

82.75 > 0.001. As error > accuracy, we repeat this step again.

Step 3:

Divide 335 by the previous result. d = 335/84.75 = 3.9528023599.

Average this value (d) with that of step 2: (3.9528023599 + 84.75)/2 = 44.35140118 (new guess).

Error = new guess - previous value = 84.75 - 44.35140118 = 40.39859882.

40.39859882 > 0.001. As error > accuracy, we repeat this step again.

Step 4:

Divide 335 by the previous result. d = 335/44.35140118 = 7.5533126595.

Average this value (d) with that of step 3: (7.5533126595 + 44.35140118)/2 = 25.9523569198 (new guess).

Error = new guess - previous value = 44.35140118 - 25.9523569198 = 18.3990442602.

18.3990442602 > 0.001. As error > accuracy, we repeat this step again.

Step 5:

Divide 335 by the previous result. d = 335/25.9523569198 = 12.9082688341.

Average this value (d) with that of step 4: (12.9082688341 + 25.9523569198)/2 = 19.4303128769 (new guess).

Error = new guess - previous value = 25.9523569198 - 19.4303128769 = 6.5220440429.

6.5220440429 > 0.001. As error > accuracy, we repeat this step again.

Step 6:

Divide 335 by the previous result. d = 335/19.4303128769 = 17.2411016808.

Average this value (d) with that of step 5: (17.2411016808 + 19.4303128769)/2 = 18.3357072789 (new guess).

Error = new guess - previous value = 19.4303128769 - 18.3357072789 = 1.094605598.

1.094605598 > 0.001. As error > accuracy, we repeat this step again.

Step 7:

Divide 335 by the previous result. d = 335/18.3357072789 = 18.2703614813.

Average this value (d) with that of step 6: (18.2703614813 + 18.3357072789)/2 = 18.3030343801 (new guess).

Error = new guess - previous value = 18.3357072789 - 18.3030343801 = 0.0326728988.

0.0326728988 > 0.001. As error > accuracy, we repeat this step again.

Step 8:

Divide 335 by the previous result. d = 335/18.3030343801 = 18.3029760554.

Average this value (d) with that of step 7: (18.3029760554 + 18.3030343801)/2 = 18.3030052177 (new guess).

Error = new guess - previous value = 18.3030343801 - 18.3030052177 = 0.0000291624.

0.0000291624 <= 0.001. As error <= accuracy, we stop the iterations and use 18.3030052177 as the square root.

So, we can say that the square root of 335 is 18.303 with an error smaller than 0.001 (in fact the error is 0.0000291624). this means that the first 4 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(335)' is 18.303005217723125.