My concern with the above scenario is that, uncorrected, this problem usually leads to feelings of frustration and inadequacy toward math in general. A likely—yet very undesirable—end result would be the drawing of incorrect conclusions about yourself and your abilities. Do these sound familiar?: “I’m just not good at math and never will be,” or “I’m just not smart enough to do math,” or more general and even more crippling, “I’m just not that smart.” I cannot count how many times I have heard such statements, the like of which only serve to drain the energy and stunt the educational gains of even the best of us. Unfortunately, these statements crystallize into attitudes and beliefs which become difficult to shake off. However mistaken these beliefs might be, their burdensome influence remains for a very long time— sometimes even for life. The end result—math illiteracy.

This is very unfortunate—even tragic—since the ability to work with numbers and do math are skills that have countless benefits. After all, no one should be doomed to mathematical illiteracy, especially when the alternative is so near within reach. Let’s consider the importance of basic math skills in everyday life. For one, you couldn’t count without math, pay bills and then balance a checkbook, understand basic financial matters, such as by how much money will grow at different interest rates, or even figure how much change you should get after a simple purchase transaction. What a sorry state we would all be in if we couldn’t do these simple things! Now think for a second. Suppose you could add, subtract, multiply and divide numbers so that routine calculations were no longer a bother. Numbers are now no longer your enemies but your friends. A very likely consequence would then be that the word math no longer triggers a bunch of negative impressions but rather positive ones. You now consider yourself good at a very tough subject and therefore consider yourself a “cut above the rest.” The implications of this new self-appraisal are enormous.

Furthermore, imagine having a simple method which allowed you to figure percents easily. Then you could figure discounts in a store and know exactly how much less you would pay for items that go on sale. As you will soon see, working with calculations involving percents is nothing more than understanding decimals and how to multiply numbers. Of course, rather than figure the discounts yourself you could wait until you got to the cash register to pay for the items. This would always be a possibility, but once again these methods do not only have one practical use. If they did, their value would be that much less. Percents figure in many other areas of the real world, not just in purchase transactions. In short, these newly acquired skills, whether they be in the area of multiplication and addition, or in the area of percent problems, will certainly open up all sorts of possibilities within the mathematical realm and unleash capabilities within you that you never thought you possessed. In short, this is the aim of Arithmetic Magic: to foster an appreciation of mathematics through the study of the basic arithmetical operations.

With this goal in mind, many positive things are achieved. Specifically, as you learn the techniques in this book, your new ability to work with numbers will definitely give you an advantage—an edge, sharp and cutting—to whatever you do. Aside from giving you a command over numbers, which will come in handy in daily life and serve you countless times in your days going forward, the techniques will pave the road toward new-found confidence and a boost in attitude towards your mental abilities. This—more than the computational ability—is the key payoff. Personally, it would be difficult for me to estimate the number of times I have used these techniques or the number of ways in which they serve me. In fact, I am constantly coming up with new combinations of the techniques and teaching both myself and my students new methods based on existing ones. You will find yourself doing the same as you read this book with an open and hungry mind.

For those of you who think the ability to work easily with numbers is beyond them, think again. From my experience, I am convinced that the ability to work well with numbers is not solely a byproduct of inborn talent. This is in spite of what many might think. Barring any severe mental abnormalities, an individual can improve his ability at math. Even the so-called “classified” individuals, whether they be considered “learning disabled” or afflicted with ADD (Attention Deficit Disorder)—whatever such classifications might mean—have benefitted from these techniques. Part of the inability to do math seems to be simply that, when it comes to math and working with numbers, many individuals concede defeat before even trying. Because such people “throw in the towel” before the fight is over and never attempt to do the math, they reach the mistaken conclusion that math and numbers are just not for them. An erroneous, yet very sorry, conclusion.

Consequently, if you’ve been frustrated by math— and its fiendish counterparts, numbers—it should be refreshing to hear what I have discovered regarding this particular illness: you can do math and you can work with numbers. This is something I am sure of and, by the time you’ve read this book, something I will have proven to you. I can state this positively from experience. What you need, and what is absolutely essential, is the right approach. Time and time again, I have found that many people cannot work with numbers because they have simply convinced themselves that they are bad at math. Why? It seems that these unfortunate individuals experienced some frustration with the subject early on. Because these early failures have far outweighed any successes, these same individuals co

This is very unfortunate—even tragic—since the ability to work with numbers and do math are skills that have countless benefits. After all, no one should be doomed to mathematical illiteracy, especially when the alternative is so near within reach. Let’s consider the importance of basic math skills in everyday life. For one, you couldn’t count without math, pay bills and then balance a checkbook, understand basic financial matters, such as by how much money will grow at different interest rates, or even figure how much change you should get after a simple purchase transaction. What a sorry state we would all be in if we couldn’t do these simple things! Now think for a second. Suppose you could add, subtract, multiply and divide numbers so that routine calculations were no longer a bother. Numbers are now no longer your enemies but your friends. A very likely consequence would then be that the word math no longer triggers a bunch of negative impressions but rather positive ones. You now consider yourself good at a very tough subject and therefore consider yourself a “cut above the rest.” The implications of this new self-appraisal are enormous.

Furthermore, imagine having a simple method which allowed you to figure percents easily. Then you could figure discounts in a store and know exactly how much less you would pay for items that go on sale. As you will soon see, working with calculations involving percents is nothing more than understanding decimals and how to multiply numbers. Of course, rather than figure the discounts yourself you could wait until you got to the cash register to pay for the items. This would always be a possibility, but once again these methods do not only have one practical use. If they did, their value would be that much less. Percents figure in many other areas of the real world, not just in purchase transactions. In short, these newly acquired skills, whether they be in the area of multiplication and addition, or in the area of percent problems, will certainly open up all sorts of possibilities within the mathematical realm and unleash capabilities within you that you never thought you possessed. In short, this is the aim of Arithmetic Magic: to foster an appreciation of mathematics through the study of the basic arithmetical operations.

With this goal in mind, many positive things are achieved. Specifically, as you learn the techniques in this book, your new ability to work with numbers will definitely give you an advantage—an edge, sharp and cutting—to whatever you do. Aside from giving you a command over numbers, which will come in handy in daily life and serve you countless times in your days going forward, the techniques will pave the road toward new-found confidence and a boost in attitude towards your mental abilities. This—more than the computational ability—is the key payoff. Personally, it would be difficult for me to estimate the number of times I have used these techniques or the number of ways in which they serve me. In fact, I am constantly coming up with new combinations of the techniques and teaching both myself and my students new methods based on existing ones. You will find yourself doing the same as you read this book with an open and hungry mind.

For those of you who think the ability to work easily with numbers is beyond them, think again. From my experience, I am convinced that the ability to work well with numbers is not solely a byproduct of inborn talent. This is in spite of what many might think. Barring any severe mental abnormalities, an individual can improve his ability at math. Even the so-called “classified” individuals, whether they be considered “learning disabled” or afflicted with ADD (Attention Deficit Disorder)—whatever such classifications might mean—have benefitted from these techniques. Part of the inability to do math seems to be simply that, when it comes to math and working with numbers, many individuals concede defeat before even trying. Because such people “throw in the towel” before the fight is over and never attempt to do the math, they reach the mistaken conclusion that math and numbers are just not for them. An erroneous, yet very sorry, conclusion.

Consequently, if you’ve been frustrated by math— and its fiendish counterparts, numbers—it should be refreshing to hear what I have discovered regarding this particular illness: you can do math and you can work with numbers. This is something I am sure of and, by the time you’ve read this book, something I will have proven to you. I can state this positively from experience. What you need, and what is absolutely essential, is the right approach. Time and time again, I have found that many people cannot work with numbers because they have simply convinced themselves that they are bad at math. Why? It seems that these unfortunate individuals experienced some frustration with the subject early on. Because these early failures have far outweighed any successes, these same individuals c

Furthermore, imagine having a simple method which allowed you to figure percents easily. Then you could figure discounts in a store and know exactly how much less you would pay for items that go on sale. As you will soon see, working with calculations involving percents is nothing more than understanding decimals and how to multiply numbers. Of course, rather than figure the discounts yourself you could wait until you got to the cash register to pay for the items. This would always be a possibility, but once again these methods do not only have one practical use. If they did, their value would be that much less. Percents figure in many other areas of the real world, not just in purchase transactions. In short, these newly acquired skills, whether they be in the area of multiplication and addition, or in the area of percent problems, will certainly open up all sorts of possibilities within the mathematical realm and unleash capabilities within you that you never thought you possessed. In short, this is the aim of Arithmetic Magic: to foster an appreciation of mathematics through the study of the basic arithmetical operations.

With this goal in mind, many positive things are achieved. Specifically, as you learn the techniques in this book, your new ability to work with numbers will definitely give you an advantage—an edge, sharp and cutting—to whatever you do. Aside from giving you a command over numbers, which will come in handy in daily life and serve you countless times in your days going forward, the techniques will pave the road toward new-found confidence and a boost in attitude towards your mental abilities. This—more than the computational ability—is the key payoff. Personally, it would be difficult for me to estimate the number of times I have used these techniques or the number of ways in which they serve me. In fact, I am constantly coming up with new combinations of the techniques and teaching both myself and my students new methods based on existing ones. You will find yourself doing the same as you read this book with an open and hungry mind.

For those of you who think the ability to work easily with numbers is beyond them, think again. From my experience, I am convinced that the ability to work well with numbers is not solely a byproduct of inborn talent. This is in spite of what many might think. Barring any severe mental abnormalities, an individual can improve his ability at math. Even the so-called “classified” individuals, whether they be considered “learning disabled” or afflicted with ADD (Attention Deficit Disorder)—whatever such classifications might mean—have benefitted from these techniques. Part of the inability to do math seems to be simply that, when it comes to math and working with numbers, many individuals concede defeat before even trying. Because such people “throw in the towel” before the fight is over and never attempt to do the math, they reach the mistaken conclusion that math and numbers are just not for them. An erroneous, yet very sorry, conclusion.

Consequently, if you’ve been frustrated by math— and its fiendish counterparts, numbers—it should be refreshing to hear what I have discovered regarding this particular illness: you can do math and you can work with numbers. This is something I am sure of and, by the time you’ve read this book, something I will have proven to you. I can state this positively from experience. What you need, and what is absolutely essential, is the right approach. Time and time again, I have found that many people cannot work with numbers because they have simply convinced themselves that they are bad at math. Why? It seems that these unfortunate individuals experienced some frustration with the subject early on. Because these early failures have far outweighed any successes, these same individuals c

With this goal in mind, many positive things are achieved. Specifically, as you learn the techniques in this book, your new ability to work with numbers will definitely give you an advantage—an edge, sharp and cutting—to whatever you do. Aside from giving you a command over numbers, which will come in handy in daily life and serve you countless times in your days going forward, the techniques will pave the road toward new-found confidence and a boost in attitude towards your mental abilities. This—more than the computational ability—is the key payoff. Personally, it would be difficult for me to estimate the number of times I have used these techniques or the number of ways in which they serve me. In fact, I am constantly coming up with new combinations of the techniques and teaching both myself and my students new methods based on existing ones. You will find yourself doing the same as you read this book with an open and hungry mind.

For those of you who think the ability to work easily with numbers is beyond them, think again. From my experience, I am convinced that the ability to work well with numbers is not solely a byproduct of inborn talent. This is in spite of what many might think. Barring any severe mental abnormalities, an individual can improve his ability at math. Even the so-called “classified” individuals, whether they be considered “learning disabled” or afflicted with ADD (Attention Deficit Disorder)—whatever such classifications might mean—have benefitted from these techniques. Part of the inability to do math seems to be simply that, when it comes to math and working with numbers, many individuals concede defeat before even trying. Because such people “throw in the towel” before the fight is over and never attempt to do the math, they reach the mistaken conclusion that math and numbers are just not for them. An erroneous, yet very sorry, conclusion.

Consequently, if you’ve been frustrated by math— and its fiendish counterparts, numbers—it should be refreshing to hear what I have discovered regarding this particular illness: you can do math and you can work with numbers. This is something I am sure of and, by the time you’ve read this book, something I will have proven to you. I can state this positively from experience. What you need, and what is absolutely essential, is the right approach. Time and time again, I have found that many people cannot work with numbers because they have simply convinced themselves that they are bad at math. Why? It seems that these unfortunate individuals experienced some frustration with the subject early on. Because these early failures have far outweighed any successes, these same individuals c

Consequently, if you’ve been frustrated by math— and its fiendish counterparts, numbers—it should be refreshing to hear what I have discovered regarding this particular illness: you can do math and you can work with numbers. This is something I am sure of and, by the time you’ve read this book, something I will have proven to you. I can state this positively from experience. What you need, and what is absolutely essential, is the right approach. Time and time again, I have found that many people cannot work with numbers because they have simply convinced themselves that they are bad at math. Why? It seems that these unfortunate individuals experienced some frustration with the subject early on. Because these early failures have far outweighed any successes, these same individuals come under the mistaken notion that they cannot do math. This situation now sets in motion a negative cycle. These early failures—or “lack of successes”—snowball to the point of no return and form a downward spiral. Once this happens, the individual is then certain that math is not a fun subject and one to be avoided at all costs. After all, you can’t do something if you believe you can’t.

Admittedly, mathematics is not an easy subject. Math has its roots in numbers and although the more abstract branches of this subject might find themselves far removed from numbers, the foundation in numbers is still there. You cannot escape them. There is even a whole branch of mathematics devoted to the study of numbers and number properties called Number Theory. It would seem then that the royal road to understanding math lies in understanding numbers and that the two were inseparable. I say royal because even such famous mathematicians as Pythagoras—you know the one with that famous theorem in geometry— have written that numbers and counting formed the foundation of all mathematics. Moreover, since math—with its many branches and applications— serves as one of the principle tools to explaining the world around us, and since these things we call numbers find themselves intricately woven into all of mathematics, a good understanding of them can certainly help pave the way to a better understanding of mathematics in general. Even doing calculus requires that you understand the basic arithmetic properties of numbers. As though this were not all, one very real and definite benefit in learning and studying mathematics comes from the educational gain made in rational and logical thinking. Now that’s a nice payoff!

What I have laid out here in four simple chapters is the basis for understanding numbers. This is done through a study of the four basic arithmetic operations: addition, subtraction, multiplication, and division. As you will soon see, each of these operations finds itself related by special properties to the other. This relationship allows us to simplify even further the study of these basic operations. I do use some basic math terminology but I deliberately avoid getting weighted down in special names. Far more important than memorizing any terminology is understanding and learning the techniques and knowing how and when to utilize the methods.

As someone who struggled with math—who understands the frustrations and feelings of inadequacy that this subject can rain upon you—I put this book together with the confident hope that it will help you gain a whole new perspective on math and numbers. This new attitude might foster a desire for further study, and this would ultimately lead to a higher rate of math proficiency. I am delighted to tell you that in overcoming my frustrations toward math and numbers, I have benefited enormously from gains made in selfconfidence towards this subject and many others. I now approach any subject matter or challenge— whether personal or academic—with a feeling of “I can do it.” If after reading this, your attitude towards math and numbers has changed to a more positive and less fearful one, then my goal will have been accomplished.

One final note that should set you in a very positive frame of mind before reading is the following: the arithmetic techniques presented here are not only empowering but also mind stimulating. They are confidence boosting. Now I am quite aware that in the high technology world of today, calculators are readily available and found everywhere; even cellular phones have these built-in devices ready to serve you. But the truth of the matter is that even though technology is everywhere, the use of such technology is sometimes limited. The point of this discussion is that there is something special about knowing how to do something, and this is particularly true when you can do something that others think is hard. Such as mathematics. The other point is that knowledge in and of itself has a very rewarding aspect to it, one that often cannot be thoroughly evaluated or appraised. So you are not learning these techniques solely to do away with your calculator, as this device will always have its uses. You are learning them so that you come to understand that what you thought was undo-able is actually do-able and what you thought might be impossible is actually possible. This fact is very empowering indeed and can serve as the very bridge to higher achievement.

So let’s get going and begin our journey into Arithmetic Magic. May you never be the same again.

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