Question

346 square root using diagonal method ​

Answer 1

Answer: 18.601075237738

Step-by-step explanation:

Step 1:

Divide the number (346) by 2 to get the first guess for the square root .

First guess = 346/2 = 173.

Step 2:

Divide 346 by the previous result. d = 346/173 = 2.

Average this value (d) with that of step 1: (2 + 173)/2 = 87.5 (new guess).

Error = new guess - previous value = 173 - 87.5 = 85.5.

85.5 > 0.001. As error > accuracy, we repeat this step again.

Step 3:

Divide 346 by the previous result. d = 346/87.5 = 3.9542857143.

Average this value (d) with that of step 2: (3.9542857143 + 87.5)/2 = 45.7271428572 (new guess).

Error = new guess - previous value = 87.5 - 45.7271428572 = 41.7728571428.

41.7728571428 > 0.001. As error > accuracy, we repeat this step again.

Step 4:

Divide 346 by the previous result. d = 346/45.7271428572 = 7.5666218876.

Average this value (d) with that of step 3: (7.5666218876 + 45.7271428572)/2 = 26.6468823724 (new guess).

Error = new guess - previous value = 45.7271428572 - 26.6468823724 = 19.0802604848.

19.0802604848 > 0.001. As error > accuracy, we repeat this step again.

Step 5:

Divide 346 by the previous result. d = 346/26.6468823724 = 12.9846334428.

Average this value (d) with that of step 4: (12.9846334428 + 26.6468823724)/2 = 19.8157579076 (new guess).

Error = new guess - previous value = 26.6468823724 - 19.8157579076 = 6.8311244648.

6.8311244648 > 0.001. As error > accuracy, we repeat this step again.

Step 6:

Divide 346 by the previous result. d = 346/19.8157579076 = 17.4608511879.

Average this value (d) with that of step 5: (17.4608511879 + 19.8157579076)/2 = 18.6383045478 (new guess).

Error = new guess - previous value = 19.8157579076 - 18.6383045478 = 1.1774533598.

1.1774533598 > 0.001. As error > accuracy, we repeat this step again.

Step 7:

Divide 346 by the previous result. d = 346/18.6383045478 = 18.5639202918.

Average this value (d) with that of step 6: (18.5639202918 + 18.6383045478)/2 = 18.6011124198

Step 8:

Divide 346 by the previous result. d = 346/18.6011124198 = 18.6010380558.

Average this value (d) with that of step 7: (18.6010380558 + 18.6011124198)/2 = 18.6010752378 (new guess).

Error = new guess - previous value = 18.6011124198 - 18.6010752378 = 0.000037182.

0.000037182 <= 0.001. As error <= accuracy, we stop the iterations and use 18.6010752378 as the square root.

So, we can say that the square root of 346 is 18.601 with an error smaller than 0.001 (in fact the error is 0.000037182). this means that the first 4 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(346)' is 18.601075237738275.