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 Question from mariana, a student: I have read various articles on how to find the square root of irrational numbers and every article out there seems to be very confusing. i read you answer to LUKOW about irrational numbers and i am still quite confused. Say i want to find the square root of 326. i know that it is between 18 and 19 because 18 is the square root of 324 and 19 is the square root of 361 im just very confused about the rest of the process. Please help! ( if possible i would appreciate two examples. thanks)

Hi Mariana,

I can't find the answer to LUKOW that you mentioned so I am not sure what you find confusing in this answer. If you send the web address I will try to help.

The specific question you asked is to find the square root of 326. We know that 326 is not a perfect square so it's square root is irrational. The proof of this fact is similar to the proof that the square root of 6 and the square root of 3 are irrational. You can find a decimal approximation to the square root of 326 and you have a good start. Since $18^2 = 324$ and $19^2 = 361$ you know that the square root of 326 is between 18 and 19. Let's try 18.5. $18.5^2 = 342.25$ which is larger than 325 so the square root of 326 is between 18 and 18.5.

What about 18.25? $18.25^2 = 333.0625$ which is again larger than 326 so now we know that the square root of 326 is between 18 and 18.25. Again, find the middle of this interval, 18.125 to see if the square root of 326 is between 18 and 18.125 or between 18.125 and 18.25.

You can continue this process as many times as you like, each time finding an interval, half the length of the previous interval, with rational endpoints and which contains the square root of 326.

There are other procedures for approximating the square root of a number that are more efficient than the method I suggested above, more efficient in that they take fewer steps to obtain the same level of approximation.

I hope this helps,
Penny

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