SQUARE ROOTING FOR SPECIAL PEOPLE!!!

How to calculate square roots by hand;

Can you imagine being stuck out at sea needing to find the square root of 527? Well, your dream has come true. Read on!

√527.00

1. You’re going to consider this number as pairs of numbers starting from the left of the decimal point. So in this example we will consider 5 first, then 27, then two 0s, etc.

2. Starting with 5 find the largest number that can be squared and less than 5 (in this case 2 because 2 squared = 4).

2

√527.00

4

127

3. Treat this like a division problem so our solution will be growing at the top where the 2 is. 2 squared is 4 so you put the 4 under the 5. You then subtract the 4 from the 5, and you get 1. Drop down the next two digits (27) so the next number we will consider is 127.

4. You now multiply our solution so far by 2, (in this case, you will get 4 because 2 x 2 = 4). We want to now think about what the largest number is that we can put in the blank below, (where the n is) so that 4n x n < 127. In this case, let’s try 3 first; 43 x 3 = 129, which is a little too big. So, we’ll try 2; 42 x 2 = 84. Two seems to be the largest value for n to work.

2

√527.00

4

127 4n

5. So now we add the digit 2 to our solution so it becomes 22 (below). Subtract the quotient (42x2=84)from 127 to get 43. Now, pull down two more numbers (the two zeros after the decimal point).

22

√527.00

4

127 4n

84

4300

6. Double 22, our current solution, to get 44. What value for p below will produce 44p x p < 4300? Let’s try p = 9. 449 x 9 = 4041.

22

√527.00

4

127 4n

84

4300 44p

4041

7. Put the 9 next to our solution to get 229. Double it (229 x 2 = 458), and bring down two more zeros. [We know the answer will be 22.9___ but for right now we will ignore the decimal point.]

229

√527.00

4

127 4n

84

4300 44p

4041

25900 458q

8. What value for q will make 458q x q < 25900? The answer is 5 (4585 x 5 = 22925). Is this starting to sound familiar? Add the 5 to our growing solution and subtract 22925 from 25900. Let’s do one more.

2295

√527.00

4

127 4n

84

4300 44p

4041

25900 458q

22925

2975

9. Double our current solution (which = 4590). Bring down 2 more zeros. What value for r (below) will make 4590r x r < 297500? The answer is 6 (45906 x 6 = 275436). Add a 6 to our solution, and a decimal point between the 22 and the 9. Thus, 22.956 is our solution! When you calculate the square root of 527 with a calculator, you get 22.95648057. So our estimate is quite good! You can iterate this process over and over to get a more precise answer.

22.956

√527.00

4

127 4n

84

4300 44p

4041

25900 458q

22925

297500 4590r