Arithmetic Explained: From Memorization to Understanding

If you've forgotten how to multiply or do long division, let alone work with percentages, this book is for you. Arithmetic Explained begins at the beginning, with numbers and place notation, then explains arithmetic, step by step, from addition through subtraction to multiplication and division before finally ending with percents.

Arithmetic Explained is for adults who have forgotten, or never really learned, how to do arithmetic; the only mathematics the vast majority of us actually need. By the end of the book, not only will you remember what you likely learned as a child, but you'll understand the why behind the steps, something children are seldom taught.

Each chapter details the necessary steps, warns of pitfalls and offers plenty of examples.

1. Numbers - The book begins at the beginning asking "what is a number?" Answering that question isn't as straightforward as it might appear at first. Our struggle with the idea of number will lead us to a deeper appreciation of the power and utility of our current number system.
2. Addition - Addition is the easiest of the big four operations, so we begin with it. The chapter introduces addition as grouping together and as motion along a number line.
3. Subtraction - The opposite of addition, of grouping together, is taking away, which we call "subtraction." This chapter builds on addition and teaches us the meaning and mechanics of subtraction.
4. Mixed-Sign Numbers - Some arithmetic books shy away from the concept of negative numbers. We won't. The number line shows us how to deal with negative numbers.
5. Multiplication - This chapter introduces multiplication as repeated addition, then builds the (often dreaded) multiplication table. Once we have the multiplication table, we tear it apart to see that a handful of simple rules reduces the table to a manageable size. With the multiplication table out of the way, we dive into the multiplication algorithm, the recipe we use to multiply two numbers.
6. Division - If multiplication is repeated addition, then division is repeated subtraction. Division is the most complex of the big four operations, but by now, we're up to the challenge. As with multiplication, I'll go beyond the how and explain the why as well. Knowing why makes it easier to see and remember the how.
7. More Division - Division needs another chapter. This one is it.
8. Fractions - Fractions are mysterious beasts, masters of disguise. In this chapter, we explore fractions and learn how to work with them in all their guises.
9. Powers - In arithmetic, multiplying a number by itself repeatedly is known as raising the number to a power.
10. Decimals - Decimal numbers are numbers that combine a whole number with a fraction but use place notation to write both parts. This chapter introduces decimal numbers, along with what they mean when written and where to find them on the number line.
11. Percents - Fractions typically live between 0 and 1 on the number line. Fractions can be written as decimal numbers, but it's inconvenient to work with numbers like 0.17, so people since ancient times have written such numbers as fractions with a denominator of 100, i.e., by the hundred, so they can talk about 17 percent instead of 0.17. Don't worry if none of that makes sense. Making sense of it is precisely what this chapter sets out to do.
12. Epilogue, Resources, And Cheat Sheet