Basic Math and Pre-Algebra

PART 1. The World of Numbers

Welcome to the world of numbers! Any study of mathematics begins with numbers. Your sense of how much, how many, and how big or small is critical to the work you do in math, as well as to your understanding the environment in which you function.

The world of numbers is a wide world. In this part, you’ll look at the areas of it we visit most often, but you won’t have time to explore every corner of the world. Think of this as a tour to get acquainted with numbers. You’ll learn to communicate in mathematical language and accomplish basic tasks. You’ll learn the fundamental rules and relationships of our number system.

CHAPTER 1. Our Number System

When asked to think about the word “math,” the first image most people are likely to have is one that involves numbers. This makes sense, because most of what we do in the name of math uses numbers in one way or another. Some would say that math is really about patterns, and that numbers and shapes are the vehicles, so arithmetic and geometry become two primary areas of mathematical thinking. There’s a wider world to mathematics, but you have to start somewhere, and generally you start with numbers.

In this chapter, we’ll take a look at the system of numbers we most commonly use. We’ll explore how the system works and learn to identify the value of a digit based upon its position in the number. We’ll examine how our system deals with fractions, or parts of a whole, and we’ll explain some variations in the way numbers are written, techniques to avoid long strings of zeros, and a method of writing very large and very small numbers called scientific notation.

The Counting Numbers

People have a tendency to think that our number system was always there and was always as it is now. On some level, that’s true. The desire, and need, to count things dates to early history, but how people count and what people do with numbers have changed over the years. The need to count is so fundamental that the whole system is built on the numbers people use to count. The counting numbers, also called the natural numbers, are the numbers 1, 2, 3, 4, and so on. The counting numbers are an infinite set; that is, they go on forever.

You might notice that the counting numbers don’t include 0. There’s a simple reason for that.

If you don’t have anything, you don’t need to count it. Zero isn’t a counting number, but for reasons you’ll see shortly, it’s one that is used a lot. The set of numbers 0, 1, 2, 3, 4, and so on is called the whole numbers.

DEFINITION

The counting numbers are the set of numbers {1, 2, 3, 4, ...}. They are the numbers we use to count. The counting numbers are also called the natural numbers.

The whole numbers are the set of numbers {0, 1, 2, 3, 4, ...} They are formed by adding a zero to the counting numbers.

Numbers didn’t always look like they do now. At different times in history and in different places in the world, there were different symbols used to represent numbers. If you think for a moment, you can probably identify a way of writing numbers that is different from the one you use every day. Roman numerals are an ancient system still used in some situations, often to indicate the year. The year 2013 is MMXIII, and the year 1960 is MCMLX.

Roman numerals choose a symbol for certain important numbers. I is 1, V is 5, X stands for 10, L for 50, C for 100, D for 500 and M for 1,000. Other numbers are built by combining and repeating the symbols. The 2000 in 2013 is represented by the two Ms. Add to that an X for 10 and three Is and you have 2013. Position has some meaning. VI stands for 6 but IV stands for 4. Putting the I before the V takes one away, but putting it after adds one. Roman numerals obviously did some jobs well or you wouldn’t still see them, but you can probably imagine that arithmetic could get very confusing.

MATH IN THE PAST

Ever wonder why the Romans chose those letters to stand for their numbers? They may not have started out as letters. One finger looks like an I. Hold up your hand to show five fingers and the outline of your hand makes a V. Two of those, connected at the points, look like an X and show ten.

The ancient Romans weren’t the only culture to have their own number system. There were many, with different organizing principles. The system most commonly in use today originated with Arabic mathematicians and makes use of a positional, or place value system. In many ancient systems, each symbol had a fixed meaning, a set value, and you simply combined them. In a place value system, each position represents a value and the symbol you place in that position tells how many of that value are in the number.

DEFINITION

A place value system is a number system in which the value of a symbol depends on where it is placed in a string of symbols.

The Decimal System

Our system is a positional, or place value system, based on the number 10, and so it’s called a decimal system. Because it’s based on 10, our system uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. A single digit, like 7 or 4, tells you how many ones you have.

DEFINITION

A digit is a single symbol that tells how many. It's also a word that can refer to a finger (another way to show how many).

When you start to put together digits, the rightmost digit, in the ones place, tells you how many ones you have, the next digit to the left is in the tens place and tells how many tens, and the next to the left is how many hundreds. That place is called the hundreds place. The number 738 says you have 7 hundreds, 3 tens, and 8 ones, or seven hundred thirty-eight.

The number 392,187 uses six digits, and each digit has a place value. Three is in the hundred-thousands place, 9 in the ten-thousands place, and 2 in the thousands place. The last three digits show a 1 in the hundreds place, 8 in the tens place and 7 in the ones place.

The Meaning of Digits in a Place Value System

DEFINITION

A decimal system is a place value system in which each position in which a digit can be placed is worth ten times as much as the place to its right.

Each move to the left multiplies the value of a digit by another 10. The 4 in 46 represents 4 tens or forty, the 4 in 9,423 is 4 hundreds, and the 4 in 54,631 represents 4 thousands.

If you understand the value of each place, you should be able to tell the value of any digit as well as the number as a whole.

CHECK POINT

Complete each sentence correctly.

1. In the number 3,492, the 9 is worth _______________.

2. In the number 45,923,881, the 5 is worth _______________.

3. In the number 842,691, the 6 is worth _______________.

4. In the number 7,835,142, the 3 is worth _______________.

5. In the number 7,835,142, the 7 is worth _______________.

When you read a number aloud, including an indication of place values helps to make sense of the number. Just reading the string of digits “three, eight, two, nine, four” tells you what the number looks like, but “thirty-eight thousand, two hundred ninety-four” gives you a better sense of what it’s worth.

In ordinary language, the ones place doesn’t say its name. If you see 7, you just say “seven,” not “seven ones.” The tens place has the most idiosyncratic system. If you see 10, you say “ten,” but 11 is not “ten one.” It’s “eleven” and 12 is “twelve,” but after that, you add “teen” to the ones digit. Sort of. You don’t have “threeteen,” but rather “thirteen.” You do have “fourteen” but then “fifteen.” The next few, “sixteen,” “seventeen,” “eighteen,” and “nineteen” are predictable.

When the tens digit changes to a 2, you say “twenty” and 3 tens are “thirty,” followed by “forty,” “fifty,” “sixty,” “seventy,” “eighty,” and “ninety.” Each group of tens has its own family name, but from twenty on, you’re consistent about just tacking on the ones. So 83 is “eighty-three” and 47 is “forty-seven.” And the hundreds? They just say their names.

Larger numbers are divided into groups of three digits, called periods. A period is a group of three digits in a large number. The ones, tens and hundreds form the ones period. The next three digits are the thousands period, then the millions, the billions, trillions, and on and on.

WORLDLY WISDOM

In the United States, you separate periods with commas. In other countries, like Italy, they're separated by periods, and in others, like Australia, by spaces.

You read each group of three digits as if it were a number on its own and then add the period name. The number 425 is “four hundred twenty-five,” so if you had 425,000, you’d say “four hundred twenty-five thousand.” The number 425,000,000 is “four hundred twenty-five million,” and 425,425,425 is “four hundred twenty-five million, four hundred twenty-five thousand, four hundred twenty-five.”

CHECK POINT

6. Write the number 79,038 in words.

7. Write the number 84,153,402 in words.

8. Write “eight hundred thirty-two thousand, six hundred nine” in numerals.

9. Write “fourteen thousand, two hundred ninety-one” in numerals.

10. Write “twenty-nine million, five hundred three thousand, seven hundred eighty-two” in numerals.

Powers of Ten

Each place in a decimal system is ten times the size of its neighbor to the right and a tenth the size of its neighbor to the left. As you move through a number, there are a whole lot of tens being used. You can write out the names of the places in words: the hundredths place or the ten-thousands place. You can write their names using a 1 and zeros: the 100 place or the 10,000 place. The first method tells you what the number’s name sounds like, and the other helps you have a sense of what the number will look like.

You can keep moving into larger and larger numbers, and the naming system keeps going with the same basic pattern. The problem is that those numbers, written in standard notation, take up lots of space and frankly, don’t always communicate well. In standard notation, one hundred trillion is 100,000,000,000,000. Written that way, most of us just see lots of zeros, and it’s hard to register how many and what they mean.

There’s a shortcut for writing the names of the places called powers often. All of the places in our decimal system represent a value that’s written with a 1 and some zeros. The number of zeros depends on the place. The ones place is just 1—no zero. The tens place is 10, a 1 and one zero. The hundreds place is 100, a 1 and two zeros. The thousands place has a value of 1,000 or a 1 and three zeros, and on it goes.

To write powers of ten in a more convenient form, you use exponents. These are small numbers that are written to the upper right of another number, called the base, and tell how many of that number to multiply together.

If you want to show 3 x 3, you can write 3². In this case, 3 is the base number and 2 is the exponent. This notation tells you to use two 3s and multiply them. We’ll look at exponents again in a later chapter, but for now we’re going to take advantage of an interesting result of working with tens.

DEFINITION

The expression power of ten refers to a number formed by multiplying a number of 10s. The first power of ten is 10. The second power of ten is 10 x 10 or 100, and the third power of 10 is 10 x 10 x 10 or 1,000.

An exponent is a small number written to the upper right of another number, called the base. The exponent tells how many of that number should be multiplied together. You can write the third power of 10 (10 is 10 x 10 x 10) as 10³. In this case, 10 is the base number and 3 is the exponent.

When you multiply tens together, you just increase the number of zeros. 10 x 10 = 100, 100 x 10 = 1000. Each time you multiply by another ten, you add another zero. Look at a place value, count the number of zeros in the name, and put that exponent on a 10, and you have the power-of-ten form of that place value.

Powers of 10

Using this system, a million, which you write as 1,000,000 in standard notation, has 6 zeros after the 1, so it would be 10⁶. One hundred trillion is 100,000,000,000,000 or a 1 followed by 14 zeros. You can write one hundred trillion as 10¹⁴, which is a lot shorter.

CHECK POINT

11. Write 10,000 as a power of ten.

12. Write 100,000,000,000 as a power of ten.

13. Write 10⁷ in standard notation.

14. Write 10¹² in standard notation.

15. Write 10⁵ in standard notation.