How to Find the Square Root of Rational Numbers
There are many problems that involve finding the square root of a number. How do you find the square root if you do not have a calculator with a square root function? There are a few simple methods you can use to quickly find the square root. These methods have been in use for thousands of years before the calculator was invented. The solutions work for rational numbers, that is numbers that are integers, or can be represented by a quotient of two integers.
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Instructions
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The square root can be found if you have access to a set of logarithmic tables. The natural log of an number x is usually denoted as ln(x). The base of the natural log is the number 2.271828... which is denoted as the letter e.
The square root of a number x is the base of natural logs, e, raised to the power of the natural log of x, divided by two. This can be expressed as:
square root of x = e^(ln(x) / 2).
This looks complicated, but logarithmic tables typically have the values for ln(x) and e^x for a large range of numbers. So you just need to look up the value of ln(x) in the table, then divide the value you found in the table by 2. Lets call the value ln(x)/2 = a. Next look up the value of e^a. This is your answer.
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The square root of a number can also be found by successive approximations. If you use a simple formula, you can improve your results quickly.
If you need to find x, which is the square root of y, then choose a value of x that is approximately the square root of y. To begin with, you might choose a value that is based on a known root close to the value of y. The approximation formula we are using is x' = (x + y/x) / 2, where y is the number we need to find the root of, and x is the number we are guessing the root to be.
Compute x' = (x + y/x) / 2. Use this value of x' to compute y and see how close the solution is. If not sufficiently close, then use this value of x' in the same formula to compute a new x' and try again. For instance, for the square root of 8, guess a solution of x = 2.6:
x' = (x + y/x) / 2 = (2.6 + 8/2.6) / 2 = 2.8384615
So x' = 2.8384615 becomes our first approximation of the square root of 8. Because x' is not very close to our first guess of x = 2.6, we need to repeat the process in our formula to get a result closer to the real square root of 8.
Now compute x' = 2.8384615 and y/x' = 8 / 2.8384615 = 2.81842818.
Compare the values of x' and y/x'. You can see that they are close, but accurate to only one decimal place. Try another iteration, using the values just computed in the equation x' = (x + y/x')/2.
x' = (2.8384615 + 2.81842818) / 2 = 2.8284448
The actual answer is 2.8284271... so already we are accurate to 4 decimal places. The process can be continued for more accuracy if needed. The values of x' and y/x' will eventually converge, that is, eventually x' will equal y/x'. At that point, there can be no increase in accuracy.
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The number we are seeking the root of (the radicand), can be a perfect square. Numbers that are perfect squares have integer roots. For instance, 9 is a perfect square with a root of 3.
If the radicand ends in a 2, 3, 7, or 8, then we know that the number is not a perfect square. All numbers that are not perfect squares have irrational roots. That is, the root of these numbers will be a decimal that can only be approximated. Pi is famously an irrational number, as is the square root of 2.
You can guess the number to begin the approximation of a root, if you know the perfect square that is smaller than the radicand (y in the formula). The closer the initial guess, the less times you need to compute the approximation formula.
For instance, say we need to find the square root of 21. We know that 16 is a perfect square with a root of 4. We also know the the next perfect square is 25, with a root of 5. Because 16 is less than 21, use the root of 16, x = 4, to begin the computation.
x' = ( x + y/x) / 2
x' = ( 4 + 21/4 ) / 2 = 4.625
x' = (4.625 + 21/4.625) / 2 = 4.58277
x' = (4.58277 + 21/4.58277) / 2 = 4.582575, which is the answer to 6 decimal places.
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Tips & Warnings
The methods described find the principal square root, that is the positive square root, of a rational number greater than zero.
An integer is a number that does not have a decimal, such as 2 or 2315.
A quotient is a fraction such as 4/3 or 576/1024.
References
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