Lesson 1: Introduction to Whole Numbers

New Concept

This lesson builds your foundation in algebra by exploring whole numbers. You'll learn to identify them, understand their place value, write them in words and digits, and use rounding to estimate values.

What’s next

Next, you'll work through interactive examples of naming and rounding numbers. Then, you'll solidify your skills with practice cards and challenge problems.

Property

Counting Numbers: The counting numbers start with 1 and continue. They are also called natural numbers. The notation "..." is called an ellipsis, which is another way to show "and so on," or that the pattern continues endlessly.

Whole Numbers: The whole numbers are the counting numbers and zero.

Counting numbers and whole numbers can be visualized on a number line. The point labeled 0 is called the origin. When a number is paired with a point, it is called the coordinate of the point.

Examples

• In the set {$0, \frac{3}{4}, 6, 9.5, 23, 110$}, the counting numbers are $6, 23, 110$. The whole numbers are $0, 6, 23, 110$.
• From the list {$\frac{1}{2}, 1, 4, 0, 7.8, 99$}, the counting numbers are $1, 4, 99$. The whole numbers are $0, 1, 4, 99$.
• Given the numbers {$1.5, 200, 0, 15, \frac{5}{8}$}, the counting numbers are $200, 15$. The whole numbers are $0, 200, 15$.

Explanation

Counting numbers are the numbers you use to count things, starting from 1. Whole numbers are almost the same, but they also include zero. Think of it as starting your count from nothing (zero) instead of one.

Property

Our number system is called a place value system because the value of a digit depends on its position, or place, in a number. Base-10 blocks provide a way to model place value. The blocks can be used to represent hundreds, tens, and ones. The tens rod is made up of 10 ones, and the hundreds square is made of 10 tens, or 100 ones.

Examples

• If you have 3 hundreds squares, 5 tens rods, and 2 ones blocks, the number modeled is $300 + 50 + 2 = 352$.
• A number modeled with 1 hundreds square, 0 tens rods, and 7 ones blocks represents the value $100 + 0 + 7 = 107$.
• With 4 hundreds squares, 8 tens rods, and 9 ones blocks, the total value shown is $400 + 80 + 9 = 489$.

Explanation

Imagine building numbers with blocks. A single small block is a 'one'. A rod made of 10 small blocks is a 'ten'. A flat square made of 10 rods is a 'hundred'. This shows how a digit's value grows based on its place.

Property

A place value chart is a useful way to summarize this information. The place values are separated into groups of three, called periods. The periods are ones, thousands, millions, billions, trillions, and so on. In a written number, commas separate the periods. The value of each place in the place-value chart is ten times the value of the place to the right of it.

Examples

• In the number $4,182,953$, the digit 8 is in the ten thousands place, the 9 is in the hundreds place, and the 4 is in the millions place.
• For the number $71,205,639$, the digit 7 is in the ten millions place, the 0 is in the ten thousands place, and the 3 is in the tens place.
• In $150,846,223$, the digit 1 is in the hundred millions place, the 8 is in the hundred thousands place, and the 6 is in the thousands place.

Explanation

Place value tells you the 'true' value of a digit in a number. A 5 in the tens place is worth 50, but a 5 in the millions place is worth 5,000,000. It's all about location, location, location!

Property

To write a number in words, write the number in each period followed by the name of the period without the 's' at the end. Start with the digit at the left, which has the largest place value. The commas separate the periods, so wherever there is a comma in the number, write a comma between the words. The ones period, which has the smallest place value, is not named. The word 'and' is not used when naming a whole number.

How 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.

Examples

• The number $4,281,605$ is written as four million, two hundred eighty-one thousand, six hundred five.
• The number $92,015,340$ is written as ninety-two million, fifteen thousand, three hundred forty.
• The number $7,000,500,120$ is written as seven billion, five hundred thousand, one hundred twenty.

Property

We will now reverse the process and write a number given in words as digits.

How to use place value to write a whole number:

1. Identify the words that indicate periods (e.g., million, thousand). Remember the ones period is never named.
2. Draw three blanks for each period to help visualize the places. Separate periods with commas.
3. Name the number in each period and place the digits in the correct place value position, using zeros as placeholders where necessary.

Examples

• 'Twenty-seven million, five hundred three thousand, ninety-one' is written as $27,503,091$. Notice the zeros used as placeholders.
• 'Eight billion, forty-two million, six hundred thousand, two hundred five' is written as $8,042,600,205$.
• 'Ninety-four thousand, fifty-three' is written as $94,053$.

Property

How to round a whole number to a specific place value:

1. Locate the given place value.
2. Underline the digit to the right of the given place value.
3. Determine if this digit is greater than or equal to 5.
   • Yes: add 1 to the digit in the given place value. Handle any regrouping (like 9 becoming 10).
   • No: do not change the digit in the given place value.
4. Replace all digits to the right of the given place value with zeros.

Examples

• To round $4,862$ to the nearest hundred, look at the tens digit (6). Since $6 \ge 5$, round up the hundreds digit. The result is $4,900$.
• Rounding $12,345$ to the nearest thousand means looking at the hundreds digit (3). Since $3 < 5$, the thousands digit stays the same. The result is $12,000$.
• Rounding $49,850$ to the nearest thousand involves looking at the hundreds digit (8). Since $8 \ge 5$, we add 1 to the 9, which requires regrouping. The result is $50,000$.

Explanation

Rounding simplifies a number. Look at the digit just to the right of the place you're rounding to. If that digit is 5 or more, you round up. If it's 4 or less, you keep the digit the same.