Square Root

you've gotta guess and check. the closest squares to 85 are 81 and 100, so you know its square root is between 9 and 10, so you try 9.5 and 9.5*9.5= 90.25 try 9.3 and you get 86.49 try 9.2 and you get 84.64. since 9.2 is too low to be the actual root, and 9.3 is too high, it's obviously between those two, and since 9.3^2 is off by more than 9.2^2, you've got an answer.

The old paper and pencil way
There is a way to calculate square roots on paper and pencil using a division like technique that they used to teach in junior high math. This is a tough forum to demonstrate, but I will try to detail the technique. Section the number you want to take the square root of into pairs starting at the decimal point and going to the right and left. Your number is easy.

85 . 00 . 00 . 00 . 00

If the number was 123.4, it would look like

1 23 . 40 00

Start with the first number and find the closest integer square root. for 85 it is 9. Substract that square from the first number and write the remainder. Then bring down the next pair.

9 .
-----------------------
85 . 00 00 00 00
- 81
------
4 . 00

Take your answer so far (9), multiply it by 20 (180) and guess at how many times that goes into what is on the bottom line (400). Here's the trick: you have to put the number you determine in for the zero on the end of the 20x value. In this case, 180 goes into 400 two times, so you replace 180 with 182. Now you multiply 182 by 2 and substract from 400. Bring down the next pair while you are at it.

9 . 2
----------------------
85 . 00 00 00 00
- 81
------
4 . 00
- 3 64
-----------
36 00

Now, repeat to get as many digits as you want, multiplying by 20, replacing the digit, getting the remainder, bringing down the next pair.

9 . 2 1
----------------------
85 . 00 00 00 00
- 81
------
4 . 00
- 3 64
-----------
36 00
18 41
----------
17 59 00

ok
I learned so much through your notes. Thank you both for your help.

I'm impressed engineer - are you a surveyor?

They used to learn these sort of tricks (e.g division by mod 9 remainder check to see if you did a multiplication of large numbers correctly without a calculator).

A surveyor? Nah, I'm an ... engineer. But I love algebra and techniques for solving stuff with paper and pencil. It also helps to be able to approximate stuff in your head during meetings when we are planning out experimental approaches.