How do you find the square root of 0.14?

Exactly is impossible as it is an irrational number.

However, I can find an approximation by:

• Using my calculator

Depending upon which calculator I either press:

• [√][0][.][1][4][=]; or
• [0][.][1][4][√]

• Using my slide rule

Line the cursor up on the D (or C) scale on 14 (=0.14 x 100) and read the [approximation of the] square root of 14 off the A (or B) scale and divide by 10 to get the square root of 0.14

• Using logs

Using logs to any base, find the logarithm of 0.14 (either using tables, slide rule or calculator), divide it by 2 and then find the anti-logarithm of that (again using tables, slide rule or calculator)

• By a kind of "long" division

1. Write the number in pairs of digits from the right hand end starting with the units; write the decimal point and write the decimal digits after the decimal point in pairs (adding an extra zero to make the last pair a pair)
2. write extra pairs of zeros after the last pair after the decimal point to the required degree of accuracy;
3. Find the largest square that is not greater than the first pair of digits;
4. write the square root of this above the pair and write the square below the pair and subtract (just like normal long division);
5. Bring down the next pair of digits;
6. Double the result so far (ignoring the decimal point), multiply it by 10 and write it as the divisor next to the result of step 4;
7. Replace the last zero digit with a digit so that when the divisor is multiplied by this digit, it is the largest number not greater than the result of step 5;
8. write this under the result of step 5 and subtract;
9. repeat from step 5 until all the digits have been used up and the result of the subtraction is zero (and there are only pairs of zeros after the decimal point in the original number remaining), or the required degree of accuracy has been found.

Doing the "long" division for 0.14, calculating to 5 dp to get result rounded to 4 dp:

...............0 . . 3 . 7 . 4 . 1 . 6

.............---------------------

.............| 0 . 14 00 00 00 00

.............. 0 ............................. largest square ≤ 0 is 02 = 0; put 0 in result (over 0)

.............. - ............................. write 0 underneath and subtract;

......... 3 | 0 . 14 ...................... bring down next pair of digits; double 0 and x10 to get 0

..................... 9 ...................... 3 x 3 = 9, 4 x 4 = 16, so make units 3, put in result (over 14)

................... --- ..................... write 9 underneath and subtract;

....... 67 | ..... .5 00 ................. bring down next pair of digits; double 3 and x 10 to get 60

......................4 69 ................. 67 x 7 = 469; making the units 7, put in result (over 00);

..................... ----- ................. write 469 underneath and subtract;

..... 744 | ......... 31 00 ............ bring down next pair of digits; double 37 and x 10 to get 740

........................ 29 76 ............ 744 x 4 = 2976; making the units 4, put in result (over 00);

........................ ------ ............ write 2976 underneath and subtract;

... 7481 | ........... 1 24 00 ....... bring down next pair of digits; double 374 and x10 to get 7480

............................. 74 81 ....... 7481 x 7481 = 7481; making the units 1, put in result (over 00)

............................. ------ ....... write 7481 underneath and subtract;

74826 | ............... 49 19 00 .. bring down next pair of digits; double 3741 and x10 to get 74820

............................. 44 89 56 .. 74826 x 6 = 448956; making the units 6, put in result (over 00)

............................. --------- .. write 448956 underneath and subtract;

............................... 4 29 44 .. stop as to get ans to 4 dp work to 5 dp and round using 5th dp.

The final subtraction is not really necessary as only the 5th decimal place digit is required to round to 4 dp

To 4 dp, √0.14 ≈ 0.3742