Teach Your Kids Arithmetic – The Magic of “One” Numbers – Part I
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.
Teach Your Kids Arithmetic – The Magic of “One” Numbers – Part II
As a follow up to my article “The Magic of One Numbers Part I” I now continue with Part II in this fascinating series. For those who have not read the first article, please do so now so that you can better understand this one. Here I will show you a method to perform the multiplication of any two “one” numbers regardless of size. The result of such multiplication—once this method is studied and learned—can be obtained effortlessly and usually within seconds. So let’s get started.
In the first of this series, I showed how to square any number which consisted of a series of 1′s. Thus after learning this method, one could square 11 or 111. In this article, you will learn how to multiply two arbitrary “one” numbers together, such as 11 x 111. To do these multiplications, you need only learn a simple rule and the rest—well the rest—will be simply matter of fact. After thoroughly mastering these two techniques, you will be able to mesmerize people with your new-found math skill; and for those parents out there teaching these techniques to their kids, don’t be surprised if you get some phone calls from your kid’s math teachers, after your kid has demonstrated to them these powerful and novel methods.
This method is a little more involved than the squaring technique; however, with a little thought and practice, you will come to see that it really is no more difficult to master. Let us look at the example of multiplying 11 x 111. The result is 1221. The way we arrive at this result is by making some observations and then following a simple procedure. First we observe that the smaller “one” number, 11, has two 1′s. Both numbers have a total of five 1′s. The final answer will have a number of digits equal to 1 less than the total of 1′s in both numbers, or in this case 4 digits. The answer, 1221, is obtained by noticing that if we count from 1 consecutively up to the number of 1′s in the smaller “one” number and then down from that number without repeating it, we have 1 2 1, or only three digits. We need four in the answer so we insert another 2 between the 2 and 1 to get 1221. This is always the case and the number we use to “pad” the answer, so to speak, is the number which represents the number of 1′s in the smaller “one” number.
A few more examples should make this perfectly clear. Let’s look at 11 x 1,111. The total number of 1′s in both numbers is 6. So the answer will have 5 digits. Since 2 is the number of 1′s in the smaller “one” number, and if we count 1 2 1, we have only 3 digits; however, we need 2 more, so we pad the number with two more 2′s in the middle to get 1 2 2 2 1 or 12,221 as our final answer.
Take 111 x 1,111. A total of 7 1′s so our final answer will have 6 digits. Number of 1′s in the smaller number: 3. So count 1 2 3 2 1 and observe that this consumes 5 digits. We need 6 so we pad 1 more 3 in the middle to get 1 2 3 3 2 1 or 123,321. To wrap up, I’ll show one more example and then you can go off amazing your friends and family. Take 1,111 x 111,111 or one thousand one hundred eleven times one hundred eleven thousand one hundred eleven. How many total 1′s: 10. So the answer will have 9 digits. Number of 1′s in the smaller number: 4. So we count up to 4 and back from 4 to get 1 2 3 4 3 2 1 and observe that this uses 7 digits. We need 2 more so we pad with 2 more 4′s to get 1 2 3 4 4 4 3 2 1 or 123,444,321 or one hundred twenty-three million four hundred forty-four thousand three hundred twenty-one as our final answer.
What do you think now? Do you think that armed with these techniques your kids could get better math grades? I think that’s a rhetorical question. Good calculating.
Teach Your Kids Arithmetic – The Teenage Number Trick
So you think you can’t do math in your head, huh? Well, it all depends on how you do such math. After all, if given the right tools to do a job, then the job comes out right often enough. So why fret over the small stuff? Too often mathematics is made out to be some mysterious subject, only to be mastered by an esoteric group of like-minded nerds who co-habitate in some forsaken land. Well, that’s not the case and one of the things I’m fond of doing is debunking this oft-held yet misguided notion.
Number theory is a branch of mathematics which treats of numbers and the various properties that come out of working with these most interesting mathematical entities. All mathematics has its roots in numbers, for without a start in the domain of these “creatures,” mathematics would never build and ramify into its many distinct branches. Indeed just the knowledge and in-depth understanding of those curious numbers we call primes form the linchpin of internet security and moreover internet commerce. Yes, that’s right. Without a knowledge of prime numbers (the primes, by the way, are numbers which are only divisible by the number 1 and the number itself; thus 3 and 5 are primes because they can only be divided evenly by 1 and themselves), secure internet transactions would not be possible.
Thus understanding numbers and how to work with them via arithmetical operations form the foundations for all of mathematics. In this article, I want to give you a tool that you can pass to your children. This tool will give them the ability to multiply what I call any two “teenage numbers.” Teenage numbers are simply the numbers from 13-19—teenagers. (The method also works if we include 11 and 12, but for these poor souls the name of the method doesn’t apply as they are “pre-teens.” The method I present here is actually a default case of the 2 by 2 Cross Multiplication Technique which I teach in my Wiz Kid series. For more information on this go to my website and contact me directly.)
At any rate, the way to obtain the product from multiplying any two “teenage numbers” is as follows, in which we use 13 x 15 as the model:
Add the two digits in the ones columns of the numbers. Thus 3 + 5 = 8. Add this to 10 and add a 0 to the result. Thus 10 + 8 = 18, and 18 with a 0 added to the end of it is 180. Multiply the digits in the ones column from the two numbers and add the result to the previous step. You now have your answer. Thus 3 x 5 = 15 and 180 + 15 = 195. That’s it. With a little practice you can beat the calculator every time. Guaranteed.
Let’s try one more and you be the judge. Take 14 x 18. Now 4 + 8 = 12; 12 + 10 = 22 and 22 with a 0 at the end is 220; 8 x 4 = 32 and 220 + 32 = 252. Presto! Imagine how your children’s teachers will react when your eight and nine year olds are doing this in math class. I think the method speaks for itself. Till next time, happy multiplying.
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.
Teach Your Kids Arithmetic – The Magic of “One” Numbers – Part II
As a follow up to my article “The Magic of One Numbers Part I” I now continue with Part II in this fascinating series. For those who have not read the first article, please do so now so that you can better understand this one. Here I will show you a method to perform the multiplication of any two “one” numbers regardless of size. The result of such multiplication—once this method is studied and learned—can be obtained effortlessly and usually within seconds. So let’s get started.
In the first of this series, I showed how to square any number which consisted of a series of 1′s. Thus after learning this method, one could square 11 or 111. In this article, you will learn how to multiply two arbitrary “one” numbers together, such as 11 x 111. To do these multiplications, you need only learn a simple rule and the rest—well the rest—will be simply matter of fact. After thoroughly mastering these two techniques, you will be able to mesmerize people with your new-found math skill; and for those parents out there teaching these techniques to their kids, don’t be surprised if you get some phone calls from your kid’s math teachers, after your kid has demonstrated to them these powerful and novel methods.
This method is a little more involved than the squaring technique; however, with a little thought and practice, you will come to see that it really is no more difficult to master. Let us look at the example of multiplying 11 x 111. The result is 1221. The way we arrive at this result is by making some observations and then following a simple procedure. First we observe that the smaller “one” number, 11, has two 1′s. Both numbers have a total of five 1′s. The final answer will have a number of digits equal to 1 less than the total of 1′s in both numbers, or in this case 4 digits. The answer, 1221, is obtained by noticing that if we count from 1 consecutively up to the number of 1′s in the smaller “one” number and then down from that number without repeating it, we have 1 2 1, or only three digits. We need four in the answer so we insert another 2 between the 2 and 1 to get 1221. This is always the case and the number we use to “pad” the answer, so to speak, is the number which represents the number of 1′s in the smaller “one” number.
A few more examples should make this perfectly clear. Let’s look at 11 x 1,111. The total number of 1′s in both numbers is 6. So the answer will have 5 digits. Since 2 is the number of 1′s in the smaller “one” number, and if we count 1 2 1, we have only 3 digits; however, we need 2 more, so we pad the number with two more 2′s in the middle to get 1 2 2 2 1 or 12,221 as our final answer.
Take 111 x 1,111. A total of 7 1′s so our final answer will have 6 digits. Number of 1′s in the smaller number: 3. So count 1 2 3 2 1 and observe that this consumes 5 digits. We need 6 so we pad 1 more 3 in the middle to get 1 2 3 3 2 1 or 123,321. To wrap up, I’ll show one more example and then you can go off amazing your friends and family. Take 1,111 x 111,111 or one thousand one hundred eleven times one hundred eleven thousand one hundred eleven. How many total 1′s: 10. So the answer will have 9 digits. Number of 1′s in the smaller number: 4. So we count up to 4 and back from 4 to get 1 2 3 4 3 2 1 and observe that this uses 7 digits. We need 2 more so we pad with 2 more 4′s to get 1 2 3 4 4 4 3 2 1 or 123,444,321 or one hundred twenty-three million four hundred forty-four thousand three hundred twenty-one as our final answer.
What do you think now? Do you think that armed with these techniques your kids could get better math grades? I think that’s a rhetorical question. Good calculating.
Teach Your Kids Arithmetic – The Teenage Number Trick
So you think you can’t do math in your head, huh? Well, it all depends on how you do such math. After all, if given the right tools to do a job, then the job comes out right often enough. So why fret over the small stuff? Too often mathematics is made out to be some mysterious subject, only to be mastered by an esoteric group of like-minded nerds who co-habitate in some forsaken land. Well, that’s not the case and one of the things I’m fond of doing is debunking this oft-held yet misguided notion.
Number theory is a branch of mathematics which treats of numbers and the various properties that come out of working with these most interesting mathematical entities. All mathematics has its roots in numbers, for without a start in the domain of these “creatures,” mathematics would never build and ramify into its many distinct branches. Indeed just the knowledge and in-depth understanding of those curious numbers we call primes form the linchpin of internet security and moreover internet commerce. Yes, that’s right. Without a knowledge of prime numbers (the primes, by the way, are numbers which are only divisible by the number 1 and the number itself; thus 3 and 5 are primes because they can only be divided evenly by 1 and themselves), secure internet transactions would not be possible.
Thus understanding numbers and how to work with them via arithmetical operations form the foundations for all of mathematics. In this article, I want to give you a tool that you can pass to your children. This tool will give them the ability to multiply what I call any two “teenage numbers.” Teenage numbers are simply the numbers from 13-19—teenagers. (The method also works if we include 11 and 12, but for these poor souls the name of the method doesn’t apply as they are “pre-teens.” The method I present here is actually a default case of the 2 by 2 Cross Multiplication Technique which I teach in my Wiz Kid series. For more information on this go to my website and contact me directly.)
At any rate, the way to obtain the product from multiplying any two “teenage numbers” is as follows, in which we use 13 x 15 as the model:
Add the two digits in the ones columns of the numbers. Thus 3 + 5 = 8. Add this to 10 and add a 0 to the result. Thus 10 + 8 = 18, and 18 with a 0 added to the end of it is 180. Multiply the digits in the ones column from the two numbers and add the result to the previous step. You now have your answer. Thus 3 x 5 = 15 and 180 + 15 = 195. That’s it. With a little practice you can beat the calculator every time. Guaranteed.
Let’s try one more and you be the judge. Take 14 x 18. Now 4 + 8 = 12; 12 + 10 = 22 and 22 with a 0 at the end is 220; 8 x 4 = 32 and 220 + 32 = 252. Presto! Imagine how your children’s teachers will react when your eight and nine year olds are doing this in math class. I think the method speaks for itself. Till next time, happy multiplying.