√30.25 solve with division methods
Answers
We must set an error for the final result. Say, smaller than 0.01. In other words we will try to find the square root value with at least 1 correct decimal places.
Step 1:
Divide the number (30.25) by 2 to get the first guess for the square root .
First guess = 30.25/2 = 15.125.
Step 2:
Divide 30.25 by the previous result. d = 30.25/15.125 = 2.
Average this value (d) with that of step 1: (2 + 15.125)/2 = 8.5625 (new guess).
Error = new guess - previous value = 15.125 - 8.5625 = 6.5625.
6.5625 > 0.01. As error > accuracy, we repeat this step again.
Step 3:
Divide 30.25 by the previous result. d = 30.25/8.5625 = 3.5328467153.
Average this value (d) with that of step 2: (3.5328467153 + 8.5625)/2 = 6.0476733577 (new guess).
Error = new guess - previous value = 8.5625 - 6.0476733577 = 2.5148266423.
2.5148266423 > 0.01. As error > accuracy, we repeat this step again.
Step 4:
Divide 30.25 by the previous result. d = 30.25/6.0476733577 = 5.0019235846.
Average this value (d) with that of step 3: (5.0019235846 + 6.0476733577)/2 = 5.5247984711 (new guess).
Error = new guess - previous value = 6.0476733577 - 5.5247984711 = 0.5228748866.
0.5228748866 > 0.01. As error > accuracy, we repeat this step again.
Step 5:
Divide 30.25 by the previous result. d = 30.25/5.5247984711 = 5.4753128387.
Average this value (d) with that of step 4: (5.4753128387 + 5.5247984711)/2 = 5.5000556549 (new guess).
Error = new guess - previous value = 5.5247984711 - 5.5000556549 = 0.0247428162.
0.0247428162 > 0.01. As error > accuracy, we repeat this step again.
Step 6:
Divide 30.25 by the previous result. d = 30.25/5.5000556549 = 5.4999443457.
Average this value (d) with that of step 5: (5.4999443457 + 5.5000556549)/2 = 5.5000000003 (new guess).
Error = new guess - previous value = 5.5000556549 - 5.5000000003 = 0.0000556546.
0.0000556546 <= 0.01. As error <= accuracy, we stop the iterations and use 5.5000000003 as the square root.
So, we can say that the square root of 30.25 is 5.5 with an error smaller than 0.01 (in fact the error is 0.0000556546). this means that the first 4 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(30.25)' is 5.5.