There are several methods to calculate square roots mentally (for numbers that are not already squares). This post explains one effective method for mentally calculating the square root of 6-digit numbers to several decimal places, as in the Mental Calculation World Cup and Memoriad competitions. The method also extends to square roots of other numbers.
There is an unrelated, faster square roots method, for when you only need 3–5 digits of accuracy.
Preparation:
For this method it is helpful to know the square numbers from 31² = 961 to 99² = 9801.
Method:
For this example we will find the square root of 661062.
Step 1:
Use your knowledge of square numbers to find the first two digits of the answer.
For 661062, we see that 81² = 6561 < 6610 < 82² = 6724, so the first two digits are 81. All divisions in Step 3 and onward will use these first two digits.
Step 2:
Subtract this square (810² = 656100) from the question and divide this remainder by 20.
661062 – 656100 = 4962
4962 / 20 = 248.1
Step 3:
Divide by 81 to get the next digit of the answer, and multiply the remainder by 10.
248.1 / 81 = 3 remainder 5.1
The answer so far is now 813 and the remainder is now 5.1 × 10 = 51
Step 4:
Remove the cross products of the answer. The cross products are easiest to explain by a separate example:
Imagine that the answer so far is 987.6543
Ignore the first two digits (98) and multiply the first and last digits: 7 × 3 = 21
Then multiply the second and second-last digits: 6 × 4 = 24
Continue adding together these products until you reach the middle. If there’s an odd number of digits (e.g. like in this new example) then when you reach the middle digit, square it and halve it. In this example we have an odd number of digits in (7, 6, 5, 4, 3) so when we reach that middle number 5: 5²/2 = 12.5
In total we have 21 + 24 + 12.5 = 57.5
So, back to our original example of 661062, with answer 813…
Ignore the first two digits (81) and we only have one digit left (3) so we subtract 3²/2 = 4.5 from the remainder (51)
51 – 4.5 = 46.5
Note! Sometimes the remainder is too small, so that when the cross products are subtracted the result is negative. In this case, go back to step 3 and reduce the new digit in the answer by one. In this example we have no problem (46.5 > 0) but otherwise we would have to go back to step 3 and step 4 and do:
248.1 / 81 = 2 remainder 86.1
861 – 2²/2 = 859
Step 5:
Repeat steps 3 and 4 until you have enough accuracy:
46.5 / 81 = 0 remainder 46.5 (so new answer is 813.0…)
46.5 × 10 = 465
465 – (3 × 0) = 465
465 / 81 = 5 remainder 60 (so new answer is 813.05…)
60 × 10 = 600
600 – (3 × 5) – 0²/2 = 585
585 / 81 = 7 remainder 18 (so new answer is 813.057…)
18 × 10 = 180
180 – (3 × 7) – (0 × 5) = 159
(etc.)
Step 6:
To calculate the last digit, it is necessary also to estimate the following digit to know whether to round up or round down. In our example:
159 / 81 < 5 so we will round down and the answer (to three decimal places) is 813.057 (not 813.058)
Summary:
This algorithm seems complicated but is not too difficult to learn. For practice, browse the Memoriad or Pegasus training software listed on this website.
There’s also a similar method to mentally calculate cube roots.
If you have any questions or would like specialist coaching from myself, you’re welcome to contact me here.