Arithmetic Magic – The Introduction

So you think you’re not good with math. Hmmm.

Well that’s something I hear all the time so you

shouldn’t feel too bad. The truth is, many, many

people, at one time or another, have felt the same way,

and have experienced something which has made them

feel bad about math and numbers in general. So the

common complaint, “I was never good at math,” or “I

hate numbers and math,” is as common as the common

cold and sometimes just as annoying. Interestingly,

what I have discovered from working with students

through the years is that this feeling of inadequacy

often stems from some negative experiences with the

subject. This might very well be the case with you.

Perhaps you had a hard time in one of your math

classes and felt very frustrated by it. Perhaps you

struggled year after year because you could’nt

understand the material being taught, or because the

pace was too fast. Perhaps you didn’t understand

because the subject matter was presented in a way that

did not get across to you. Sound familiar?

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 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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