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Why Study Calculus? How About Square Roots With No Calculator?

Joe Pagano

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To continue with my Why Study Calculus? series, I discuss here some interesting applications of this branch of mathematics to numbers. Numbers and the operations on them are the key to mathematics, and all higher branches are one way or another intimately linked to their inherent properties. One nice application of the Calculus to numbers is the approximation of square roots. What is more, this technique can be done without a calculator and without even a knowledge of the underlying theory.

As mentioned in one of my previous calculus articles, the two fundamental branches of this discipline are the differential and integral calculus, the former dealing with derivatives and the latter dealing with integrals. The differential branch is what gives us the ability to do such things as approximate the square roots of numbers without a calculator. What is more, this technique will allow us to approximate even cube roots and fourth roots with a high degree of accuracy.

The theory behind this method hinges upon the derivative, which in calculus, is a special kind of limit The differential, which is a quantity that approximates the derivative, particularly under certain conditions, is the mathematical tool that we employ to estimate our square root.

Specifically, the way the method works is as follows: Suppose I want to approximate the square root of 10. This is the number which when multiplied by itself will give 10 exactly. Now 10 is not a perfect square like 9. The square root of 9 is 3, since 3*3 = 9. The calculus, using the concept of the differential, tells us that the square root of 10 will be approximately equal to the square root of 9 plus 1 divided by twice the square root of 9. Since the square root of 9 is 3, what we have is that the square root of 10 is approximately 3 + 1/6 = 3.167. Take out your calculator and compute the square root of 10 (also called radical 10). You will see that radical 10 is equal to 3.162 to three decimal places. The approximation is off by less than five one-thousandths!

The method can be used for other numbers as well. Take 67. The closest perfect square to 67 is 64, the square root of which is 8. To approximate radical 67, the differential tells us to add radical 64 to the quotient of the difference of 67 and 64, which is 3, and twice radical 64. In math, radical 67 ~ radical 64 + 3/(2*radical64) which is 8 + 3/16 which is 8.1875. The true value of radical 67 is 8.185, a difference of less than three one-thousandths than that predicted by the differential.

Thus when one wonders why we study the calculus, this is one of many examples which shows the awesome power of this subject. Indeed, mathematics is full of wonders for all to enjoy. So take the plunge and start exploring. The door is open to you.

Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Under the penname, JC Page, Joe authored Arithmetic Magic, the little classic on the ABC’s of arithmetic. Joe is also author of the charming self-help ebook, Making a Good Impression Every Time: The Secret to Instant Popularity; the original collection of poetry, Poems for the Mathematically Insecure, and the short but highly effective fraction troubleshooter Fractions for the Faint of Heart. The diverse genre of his writings (novel, short story, essay, script, and poetry)—particularly in regard to its educational flavor— continues to captivate readers and to earn him recognition.

Joe propagates his teaching philosophy through his articles and books and is dedicated to helping educate children living in impoverished countries. Toward this end, he donates a portion of the proceeds from the sale of every ebook. For more information go to

See more at Help with Calculus . See more articles here (see Limits and Derivatives ).


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