This is about determining the square root of a non-perfect square. There is a method in calculus for doing this, as we know, and it is complicated. Obviously we have calculators now, so there is no need for such a method.

I thought about a different way to determine the square root of a non-perfect square (without a calculator). It is much simpler than the calculus method, which I think involves differentials. It has been some time since I took that course. I have not heard of this method I am mentioning in any math course I have taken; it is just something I thought of, although it could have been mentioned somewhere else. I tried it on the square root of 54.

So the number 54 falls between two perfect squares: 49 and 64. Where between the two numbers does it fall? It is 5 numbers after 49. There is a total of 15 numbers between 49 and 64.

The number 54 is the 5^{th}out of 15 numbers (5/15). Therefore, it is one third of the way between 49 and 64. So, according to this method, the square root of 54 should also fall one third of the way between the square roots of 49 and 64, which are 7 and 8.

The answer we get would be 7 and 1/3, or 7.33 repeating. The calculator answer is 7.348. I cannot remember what answer the calculus method produces, but it is closer than the method I mentioned. However, the simplicity of this method is to be taken into consideration. It also produces an answer that is pretty close to the calculator answer.

I tried to find a way to get the exact answer without the calculator, but I could not do it. It is also worth mentioning that this method’s accuracy increases with the increase of the number of the non-perfect square.

We can look at the square root of 78. Accoriding to this method, the answer would be 8.8235. The calculator answer is 8.8318. The difference between the two is 0.0083. This last number is less than the difference in the case of the square root of 54. The difference in that case was 0.015 (7.348-7.333).

This proves that the method’s accuracy is higher when the number of the non-perfect square is higher.

We do not need such methods anyway, since we have calculators now. But some math enthusiasts might find this interesting.