Question

What is a square of 999.9

Answer 1

The square root of 999.9 is 31.621195423323. Or, 
√999.9 = 31.621195423323

Step 1: 
 Divide the number (999.9) by 2 to get the first guess for the square root .
 First guess = 999.9/2 = 499.95.Step 2:
 Divide 999.9 by the previous result. d = 999.9/499.95 = 2.
 Average this value (d) with that of step 1: (2 + 499.95)/2 = 250.975 (new guess).
 Error = new guess - previous value = 499.95 - 250.975 = 248.975.
 248.975 > 0.001. As error > accuracy, we repeat this step again.Step 3:
 Divide 999.9 by the previous result. d = 999.9/250.975 = 3.9840621576.
 Average this value (d) with that of step 2: (3.9840621576 + 250.975)/2 = 127.4795310788 (new guess).
 Error = new guess - previous value = 250.975 - 127.4795310788 = 123.4954689212.
 123.4954689212 > 0.001. As error > accuracy, we repeat this step again.Step 4:
 Divide 999.9 by the previous result. d = 999.9/127.4795310788 = 7.8436121591.
 Average this value (d) with that of step 3: (7.8436121591 + 127.4795310788)/2 = 67.6615716189(new guess).
 Error = new guess - previous value = 127.4795310788 - 67.6615716189 = 59.8179594599.
 59.8179594599 > 0.001. As error > accuracy, we repeat this step again.Step 5:
 Divide 999.9 by the previous result. d = 999.9/67.6615716189 = 14.7779600159.
 Average this value (d) with that of step 4: (14.7779600159 + 67.6615716189)/2 = 41.2197658174(new guess).
 Error = new guess - previous value = 67.6615716189 - 41.2197658174 = 26.4418058015.
 26.4418058015 > 0.001. As error > accuracy, we repeat this step again.Step 6:
 Divide 999.9 by the previous result. d = 999.9/41.2197658174 = 24.2577797368.
 Average this value (d) with that of step 5: (24.2577797368 + 41.2197658174)/2 = 32.7387727771(new guess).
 Error = new guess - previous value = 41.2197658174 - 32.7387727771 = 8.4809930403.
 8.4809930403 > 0.001. As error > accuracy, we repeat this step again.Step 7:
 Divide 999.9 by the previous result. d = 999.9/32.7387727771 = 30.5417679156.
 Average this value (d) with that of step 6: (30.5417679156 + 32.7387727771)/2 = 31.6402703464(new guess).
 Error = new guess - previous value = 32.7387727771 - 31.6402703464 = 1.0985024307.
 1.0985024307 > 0.001. As error > accuracy, we repeat this step again.Step 8:
 Divide 999.9 by the previous result. d = 999.9/31.6402703464 = 31.6021319999.
 Average this value (d) with that of step 7: (31.6021319999 + 31.6402703464)/2 = 31.6212011732(new guess).
 Error = new guess - previous value = 31.6402703464 - 31.6212011732 = 0.0190691732.
 0.0190691732 > 0.001. As error > accuracy, we repeat this step again.Step 9:
 Divide 999.9 by the previous result. d = 999.9/31.6212011732 = 31.6211896734.
 Average this value (d) with that of step 8: (31.6211896734 + 31.6212011732)/2 = 31.6211954233(new guess).
 Error = new guess - previous value = 31.6212011732 - 31.6211954233 = 0.0000057499.
 0.0000057499 <= 0.001. As error <= accuracy, we stop the iterations and use 31.6211954233 as the square root.

So, we can say that the square root of 999.9 is 31.62119 with an error smaller than 0.001 (in fact the error is 0.0000057499). this means that the first 5 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(999.9)' is 31.62119542332326.