Do you know that numbers are the key to life? Indeed numbers form the foundations of mathematics from number theory all the way up to partial differential equations. Without these curious creatures, we could not calculate, estimate, or compute (as in computer); nor could we transact any of the business that occurs daily throughout the known world. Yet most people find these most interesting entities just a bothersome part of life. Why should this be so? Well maybe a person’s ambivalence towards numbers derives from childhood frustrations experienced when struggling to learn the basics of arithmetic and the rote calculations associated with this discipline.
Lacking an appreciation for numbers can be a sad state of affairs for any person since such a condition often leads to mathematical illiteracy. As parents we should try to instill a love and curiosity for numbers into the minds of our children so that they never arrive at this blighted condition. One way we can do this is by teaching our children early on the amazing properties of numbers and some of the magical calculations that can be performed with them. Indeed the Teenage Number Trick, the Nifty Five Square Technique (see my other articles on these topics), and the techniques laid out in this article should suffice to arouse and stimulate curiosity even in the most torpid learners. Thus here we go on to introduce a calculation that permits one to square any “one” number with blinding speed. In a future article we will learn how to multiply any two arbitrary “one” numbers with no difficulty at all.
But in order to begin this short educational journey, we need to understand what we are talking about in the first place. What is a “one” number. Take a stab at it before reading further. Okay, now the definition. A “one” number is simply a number whose digits consist entirely of 1's. Thus 1 11 111 1111 11111 are all “one” numbers. Because 1 is such a special number—in fact one of the most important constants in mathematics, much like water is one of the most important compounds in chemistry—its presence as the sole digit in forming numbers—the “one” numbers—makes squaring these numbers such an easy task. Watch what I mean.
To multiply the number 11 by itself, something called “squaring 11" or taking the square of 11, we obtain 121. Stepping up to 111, squaring we have 12321. If we square 1111, we get 1234321. I think by now the pattern has become clear to you and we can thus form a rule for squaring any “one” number: write down the digits consecutively from 1 to the number represented by the number of 1's forming the “one” number and then count backwards from this number to 1. So in 1111, we have four 1's. To square this number write the numbers from 1 to 4 consecutively, and then back from 4 to 1 so as not to repeat the 4 twice: 1 2 3 4 3 2 1 to get 1234321 or one million two hundred and thirty-four thousand three hundred and twenty-one. That’s all there is to it. Want a bigger example. Try 11,111 x 11, 111. How many 1's? Five. So we get 123454321 or one hundred twenty-three million four hundred fifty-four thousand three hundred twenty-one.
If you show your kids how to do these calculations so that they can multiply and square numbers which produce results in the hundreds of millions, do you think they might have a new outlook on mathematics and their ability to master this subject? I think you’ll agree that the answer to this question is self-evident. Stay tuned for Part II which is even more amazing than this one.
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 http://www.mathbyjoe.com