Introduction to Whole Numbers

Identify counting numbers and whole numbers

Learning algebra is similar to learning a language. You start with a basic vocabulary and then add to it as you go along. You need to practice often until the vocabulary becomes easy to you. The more you use the vocabulary, the more familiar it becomes.

Algebra uses numbers and symbols to represent words and ideas. The most basic numbers are those we use to count objects:

These are called the counting numbers (also called natural numbers). The notation “$\dots$” is called an ellipsis — a way of showing “and so on,” or that the pattern continues endlessly.

The discovery of the number zero was a big step in the history of mathematics. Including zero with the counting numbers gives a new set of numbers, the whole numbers:

Counting numbers and whole numbers can be visualized on a number line:

The numbers on the number line increase from left to right and decrease from right to left. The point labeled $0$ is called the origin. The points are equally spaced to the right of $0$ and labeled with the counting numbers. When a number is paired with a point, it is called the coordinate of the point.

Example. Which of the numbers $0, \tfrac{1}{4}, 3, 5.2, 15, 105$ are (a) counting numbers, (b) whole numbers?

(a) The counting numbers start at $1$, so $0$ is not a counting number. The counting numbers here are $3, 15,$ and $105$.

(b) The whole numbers are the counting numbers together with $0$, so the whole numbers here are $0, 3, 15,$ and $105$.

The numbers $\tfrac{1}{4}$ and $5.2$ are neither counting numbers nor whole numbers — they belong to families of numbers we will meet later.

Model whole numbers

Our number system is called a place value system, because the value of a digit depends on its position — its place — in the number. The numbers $537$ and $735$ use exactly the same digits, but they have different values because the $7$ and the $5$ sit in different places.

Money gives a familiar model of place value. Suppose a wallet contains three $100 bills, seven $10 bills, and four $1 bills. Find the total value of each kind of bill, then add:

The three digits of $374$ — the $3$, the $7$, and the $4$ — come straight from the three kinds of bills.

Base-10 blocks are another way to model place value: a single small block represents $1$, a rod of ten blocks represents $10$, and a square of one hundred blocks ($10$ rods) represents $100$. The number $138$, modeled with $1$ hundreds square, $3$ tens rods, and $8$ ones blocks, is:

Example. What number is modeled by $2$ hundreds squares, $1$ tens rod, and $5$ ones blocks?

There are $2$ hundreds, which is $200$; there is $1$ ten, which is $10$; and there are $5$ ones, which is $5$. In place value notation:

Identify the place value of a digit

A place value chart summarizes how each place in a number has a different value. The place values are separated into groups of three, called periods: ones, thousands, millions, billions, trillions, and so on. In a written number, commas separate the periods. The value of each place is ten times the value of the place to its right.

Here is the number $5{,}278{,}194$ in a place value chart:

• The digit $5$ is in the millions place. Its value is $5{,}000{,}000$.
• The digit $2$ is in the hundred thousands place. Its value is $200{,}000$.
• The digit $7$ is in the ten thousands place. Its value is $70{,}000$.
• The digit $8$ is in the thousands place. Its value is $8{,}000$.
• The digit $1$ is in the hundreds place. Its value is $100$.
• The digit $9$ is in the tens place. Its value is $90$.
• The digit $4$ is in the ones place. Its value is $4$.

Example. In the number $63{,}407{,}218$, find the place value of each of these digits:

• The $7$ is in the thousands place.
• The $0$ is in the ten thousands place.
• The $1$ is in the tens place.
• The $6$ is in the ten millions place.
• The $3$ is in the millions place.

Use place value to name whole numbers

When you write a check, you write out the number in words as well as in digits. To name a whole number in words:

1. Starting at the digit on the left, name the number in each period, followed by the period name. Do not include the period name for the ones.
2. Use commas in the number to separate the periods.

So the number $37{,}519{,}248$ is named: thirty-seven million, five hundred nineteen thousand, two hundred forty-eight. The millions period ($37$), the thousands period ($519$), and the ones period ($248$) are each named in turn, and the ones period’s name is dropped.

Notice that the word and is not used when naming a whole number.

Example. Name the number $8{,}165{,}432{,}098{,}710$ in words.

Putting it together: eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

Use place value to write whole numbers

Now reverse the process, and write a number given in words as digits:

1. Identify the words that indicate periods. (Remember, the ones period is never named.)
2. Draw three blanks to indicate the number of places needed in each period, and separate the periods by commas.
3. Name the number in each period and place the digits in the correct place value position.

Example. Write nine billion, two hundred forty-six million, seventy-three thousand, one hundred eighty-nine using digits.

The words billion, million, and thousand indicate the periods: $9$ billions, $246$ millions, $073$ thousands, $189$ ones — that is, $9{,}246{,}073{,}189$.

Notice that a zero was needed as a place-holder in the hundred thousands place (“seventy-three thousand” fills only two of the three places in the thousands period). Every period after the first must have all three places filled, using zeros as needed.

Round whole numbers

In 2013, the U.S. Census Bureau reported the population of New York State as $19{,}651{,}127$ people. Often it is enough to say the population is approximately $20$ million — close to the exact value, but not exact. The process of approximating a number is called rounding.

The number line helps you see how rounding works. Suppose we want to round $76$ to the nearest ten. Is $76$ closer to $70$ or to $80$?

$76$ is closer to $80$, so $76$ rounded to the nearest ten is $80$. Likewise $72$ is closer to $70$, so $72$ rounded to the nearest ten is $70$.

What about $75$ — exactly midway between $70$ and $80$? So that everyone rounds the same way in cases like this, mathematicians have agreed to round to the higher number: $75$ rounded to the nearest ten is $80$.

More generally, to round a number to a specific place value:

1. Locate the given place value. All digits to the left of it stay the same (unless carrying changes them — see step 3).
2. Underline the digit to the right of the given place value.
3. Is that underlined digit greater than or equal to $5$?
   • Yes — add $1$ to the digit in the given place value. (If that digit is a $9$, it becomes $0$ and the $1$ carries to the next digit left.)
   • No — leave the digit in the given place value unchanged.
4. Replace all digits to the right of the given place value with zeros.

Example. Round $843$ to the nearest ten. The tens digit is $4$; the digit to its right is $3$. Since $3 < 5$, the $4$ stays, and the $3$ becomes $0$: $843$ rounds to $840$.

Example. Round $23{,}658$ to the nearest hundred. The hundreds digit is $6$; the digit to its right is $5$. Since $5 \ge 5$, round up: the $6$ becomes $7$ and the digits to its right become zeros — $23{,}700$.

Example. Round $3{,}978$ to the nearest hundred. The hundreds digit is $9$; the digit to its right is $7$, so we round up. The $9$ becomes $0$ and the $1$ carries: $3{,}978$ rounds to $4{,}000$.

Key terms

counting numbers — the numbers $1, 2, 3, \dots$ used to count objects; also called natural numbers. whole numbers — the counting numbers together with zero: $0, 1, 2, 3, \dots$ origin — the point labeled $0$ on the number line. coordinate — the number paired with a point on the number line. place value system — our number system, in which the value of a digit depends on its position in the number. period — a group of three place values, separated by commas in a written number. rounding — the process of approximating a number to a given place value.

This section is adapted from Prealgebra 2e, Section 1.1: Introduction to Whole Numbers by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: condensed prose, redrew figures as accessible tables and inline graphics, and converted practice problems (“Try Its”) into interactive exercises with instant feedback.