Square Root

I learned how to get the square root of a number without a calculator in grade school. I have never really tried to figure out the logic of how the method works. All I know is that if I follow the steps to the letter, I manage to get the square root of a number as if I had picked up a calculator and pushed that special button.

1. First group the numbers to be calculated in pairs from right to left


2. Think of a number (A) whose square is nearest the first group of digits on the left. Write that number on top of the square root sign above the leftmost set of digits


3. Calculate the square of A. Place the answer right below the leftmost set of digits just as you would when doing long division


4. Subtract the result of Step 3 from the first group of digits


5. Multiply 20 by the number/s on top


6. Bring down the next pair of numbers (as you would when doing long division)


7. Think of a number (B) which when added to step 5 then multiplied by the same number (B) would give a product closest to the resulting number in Step 6. Write that number on top of the square root sign to the right of any number already there.


8. Repeat steps 2 - 7 until all sets of digits have gone through the calculations

Let us try this with an easy number --> What is the square root of 484?

Grouping the number 484 by 2's results in 4-84. We all know that the sqare root of 4 is equal to 2. So A=2 and that goes on TOP of the square root sign.

Then, as in division, we put 4 under 4, and get its difference which is zero.

Step 5. We multiply 20 by the number on top of the square root sign. 20 x 2 = 40.

Step 6. We bring the 84

Step 7. B = 2 such that 42 x 2 = 84 which is great because that means we're done since 84 - 84 = 0.

In conclusion, the square root of 484 is 22. That was simple enough (I hope).

Here is a more complicated number. What is the square root of 106,276?

Grouping the number 106,276 by 2's results in 10-62-76. The closest square root to 10 is 3 (since 3 x 3 = 9). So A=3 and that goes on TOP of the square root sign.

Then, as in division, we put 9 under 10, and get its difference. 10-9 = 1. We're in step 5 now. Multiply 20 by the number on top of the square root sign. 20 x 3 = 60.

Step 6. We bring the next set of numbers down to get 162.

Step 7. B = 2 such that 62 x 2 = 124 which is closest to 162 without going over. (163 x 3 = 489, way over 162!). So now there is a 3-2 on top of the square root sign, and a difference of 38 when 124 is subtracted from 162.

Step 8. We repeat the steps and calculate 20 x 32 = 640. We bring down 76. This time, B = 6 such that 646 x 6 = 3876.

Fortunately the example is a perfect square. And the square root of 106,276 is 326.

That concludes our lesson for the day.

(No, I'm not quitting my day job just yet. Soon though, I'll think of an alternate career ... probably not as a Math teacher, though.)