All the Math You'll Ever Need. Steve Slavin
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months to see tangible results. You won't even have to wait weeks. In just a few days your friends and colleagues will notice your new mathematical muscles. So don't delay another minute. Turn to Chapter 2 and just watch those brain cells start to grow.
Acknowledgments
From Steve: Many thanks are due, so I'd like to name names. My longtime editor at Wiley, Judith McCarthy, made hundreds of suggestions to improve and update the book. Authors often hate to change even a word, but Judith's editing has made this second edition a much smoother read. Claire McKean did a thorough copyedit, catching dozens of errors that made it through the first edition, and Benjamin Hamilton supervised the production of the book from copyediting through page proofs.
I owe a large debt of gratitude to my family, especially to my nephews, Jonah and Eric Zimiles. Jonah provided me with a blow-by-blow critique of the strengths and weaknesses of my previous book, Economics: A Self-Teaching Guide (Wiley, 1988), on which I was able to build while writing this book. And Eric, after having read that book, recognized its format lent itself best to my writing style and encouraged me to write another book. Eric's daughters, Eleni, 11, Justine, 7, and Sophie, 5, have contributed to the new edition by helping me with my math whenever I happened to get stuck.
My father, Jack, a retired math teacher, provided inspiration of another kind. As the oldest living academic perfectionist, he upholds such unattainable standards that one cannot help but feel tolerance for one's own shortcomings and those of just about everyone else. And finally, I wish to thank my sister, Leontine Temsky, for her rationality and common sense in the most uncommon and irrational of times.
From Carolyn: Every opportunity to work with the incredible folks at Wiley has been a pleasure. This project was no exception. My thanks to go Christine O'Connor, Tom Dinse, and Riley Harding for providing the vision for the project, giving me the freedom to make it my own, and guiding me through every step. I'm grateful also to copyeditor Julie Kerr, whose keen eye and infinite patience make the final product so much better.
1 Getting Started
Far too many Americans are mathematically illiterate. Although many of these people are college graduates, they have trouble doing simple arithmetic. One cannot help but wonder how so many people managed to get so far without having mastered basic arithmetic. Math phobia seems to have become fashionable. People who would never think it amusing to claim not to be able to read or write chuckle as they announce, “I can't do math.”
We all have to deal with numbers sometime—in banking, on taxes, in choosing a mortgage. Like it or not, numbers are an important part of our lives, and the importance of numerical literacy is increasing in finance, economics, science, government, and more. It is time math stopped intimidating us.
What we'll be doing in this book is going back to basics. We'll focus on the multiplication table. You'll need to memorize it. If you need an even more basic text, you can refer to one such as Quick Arithmetic: A Self-Teaching Guide, 3rd edition by Robert A. Carmen and Marilyn J. Carmen (Wiley, 2001).
In All the Math You'll Ever Need, the use of complex formulas is generally avoided. Although such formulas have an honored place in mathematics, they rarely need to be memorized. The ones that are used frequently work their way into memory. The others can be looked up when they're needed.
Finally, the use of technical terms is minimized whenever possible. Having the vocabulary to describe mathematical ideas and operations accurately is important to learning but you don't need a lot of fancy language for that. There are no quadratic formulas, logarithmic tables, integrals, or derivatives, and there are only a handful of very simple graphs.
This book was designed to be explored without ever using a calculator or computer. Don't get nervous. You will not be asked to throw away your calculator. Just put it in a safe place for now, to be taken out and used only on proper occasions. A calculator is most effectively used for three tasks: (1) to do calculations that need to be done rapidly, (2) to do repetitive calculations, and (3) to do sophisticated calculations that would take a great deal of time to do without a calculator. Calculators and computers are fast and as accurate as their users allow them to be. Typos are a thing, even on calculators. You need to first know what you want to ask the calculator to do, and then have enough math knowledge to decide if the answer it gives you makes sense.
The trick is to use our calculators for these specific tasks and not for arithmetic functions that we can do in our heads. So put away your calculator and start using your innate mathematical ability.
2 Essential Arithmetic
Every number system (and, yes, there are or have been others) is made up of a set of symbols that we call numbers and one or more operations you can perform with them. Those operations make up what we call arithmetic. The basic operation in our number system is addition, the act of putting together. The other operations—multiplication, subtraction, division—are related to, or built from, addition.
1 ADDITION
Addition is, at its heart, about counting. If you have 6 pair of shoes and you buy 3 new pairs, counting will tell you that you now have 9 pairs. You added 6 + 3 and got an answer of 9 by counting. After a while you don't have to count every time, because you get to know that 6 + 3 = 9.
You store a lot of addition facts like that in your memory, but there's a limit to how much memorization can help. You probably know that 4 + 8 = 12, but you're unlikely to memorize the answer to 5,387 + 9,748. Adding larger numbers requires a little more information about our number system.
Place Value
Our number system is a place value system, meaning that the value of a numeral depends on the place it sits in. In the number 444 each 4 has a different meaning. The 4 on the right is in the ones place so it represents 4 ones or simply 4. The 4 on the left is in the hundreds place and represents 4 hundreds or 400. The middle 4 is in the tens place so it represents 4 tens or 40. The number 444 is a shorthand for 400 + 40 + 4.
That expanded form, 400 + 40 + 4, helps to explain how we add large numbers. We add the ones to the ones, the tens to the tens, the hundreds to the hundreds and on up in the place value system. If you need to add 444 + 312, think:
Add the 4 ones and the 2 ones to get 6 ones, the 4 tens with 1 ten to get 5 tens and the 4 hundreds with 3 hundreds to get 7 hundreds. Now that would look like this:
You're probably thinking that you could just write the numbers underneath one another in standard form and add down the columns, and you'd be absolutely correct.
The reason to think about it in expanded form, at least for a few minutes, comes up when you have to add something like 756 + 968. The basic rule is the same.
