Question

find tha square root of 1.89 by
long division method
​

Answer 1

Answer:

Step 1:

Divide the number (1.89) by 2 to get the first guess for the square root .

First guess = 1.89/2 = 0.945.

Step 2:

Divide 1.89 by the previous result. d = 1.89/0.945 = 2.

Average this value (d) with that of step 1: (2 + 0.945)/2 = 1.4725 (new guess).

Error = new guess - previous value = 0.945 - 1.4725 = 0.5275.

0.5275 > 0.001. As error > accuracy, we repeat this step again.

Step 3:

Divide 1.89 by the previous result. d = 1.89/1.4725 = 1.2835314092.

Average this value (d) with that of step 2: (1.2835314092 + 1.4725)/2 = 1.3780157046 (new guess).

Error = new guess - previous value = 1.4725 - 1.3780157046 = 0.0944842954.

0.0944842954 > 0.001. As error > accuracy, we repeat this step again.

Step 4:

Divide 1.89 by the previous result. d = 1.89/1.3780157046 = 1.3715373444.

Average this value (d) with that of step 3: (1.3715373444 + 1.3780157046)/2 = 1.3747765245 (new guess).

Error = new guess - previous value = 1.3780157046 - 1.3747765245 = 0.0032391801.

0.0032391801 > 0.001. As error > accuracy, we repeat this step again.

Step 5:

Divide 1.89 by the previous result. d = 1.89/1.3747765245 = 1.3747688925.

Average this value (d) with that of step 4: (1.3747688925 + 1.3747765245)/2 = 1.3747727085 (new guess).

Error = new guess - previous value = 1.3747765245 - 1.3747727085 = 0.000003816.

0.000003816 <= 0.001. As error <= accuracy, we stop the iterations and use 1.3747727085 as the square root.

So, we can say that the square root of 1.89 is 1.37477 with an error smaller than 0.001 (in fact the error is 0.000003816). this means that the first 5 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(1.89)' is 1.374772708486752.