1.1 Introduction to Whole Numbers

Understanding Whole Numbers

Whole numbers are the set of numbers starting at zero and going up forever: 0, 1, 2, 3, 4, and so on. They're the foundation for almost everything else in math, from basic arithmetic to algebra. This section covers how whole numbers are structured, how to represent them in different ways, and how to work with them.

Counting vs. Whole Numbers

Counting numbers (also called natural numbers) start at 1 and go up indefinitely: 1, 2, 3, 4, ... These are the numbers you'd naturally use to count objects.

Whole numbers include all counting numbers plus zero: 0, 1, 2, 3, 4, ... The only difference is that zero is included. Zero represents the absence of quantity or a starting point, like the beginning of a number line.

Visual Representations of Whole Numbers

There are a few common ways to picture whole numbers:

• Base-ten blocks use physical shapes to represent values: single unit cubes (ones), rods of 10 units (tens), flats of 100 units (hundreds), and large cubes of 1,000 units (thousands). So the number 243 would be 2 flats, 4 rods, and 3 unit cubes.
• Dot arrays show numbers as groups of dots arranged in organized rows and columns. For example, 12 can be shown as 3 rows of 4 dots.
• Number lines show whole numbers as evenly spaced points on a horizontal line, increasing from left to right. Numbers farther to the right are always greater.

Place Value in Multi-Digit Numbers

Our number system is built on place value, which means a digit's value depends on its position within the number. Each position is worth 10 times more than the position to its right.

Reading from right to left, the positions are: ones, tens, hundreds, thousands, ten-thousands, and so on.

To find the value a digit contributes, multiply the digit by its place value. For example, in the number 5,678:

$5{,}678 = 5 \times 1{,}000 + 6 \times 100 + 7 \times 10 + 8 \times 1$

The 5 isn't just "five." It represents 5,000 because it sits in the thousands place.

Forms of Whole Number Notation

You can write the same number in three different forms:

• Standard form uses digits with commas separating every three digits from right to left: 1,234,567
• Expanded form breaks the number into the sum of each digit's value: $1{,}234{,}567 = 1{,}000{,}000 + 200{,}000 + 30{,}000 + 4{,}000 + 500 + 60 + 7$
• Word form writes the number out in words: one million, two hundred thirty-four thousand, five hundred sixty-seven

Being able to convert between these three forms shows that you understand the structure of the number, not just how to read it.

Rounding for Estimation

Rounding simplifies a number so it's easier to work with, especially for estimating answers or checking whether your calculations are reasonable.

To round a number to a specific place value:

1. Find the digit in the place you're rounding to.

2. Look at the digit directly to its right.

   • If that digit is 5 or greater, round up (increase your target digit by 1).
   • If that digit is less than 5, round down (keep your target digit the same).

3. Replace all digits to the right of your target place with zeros.

Example: Round 5,678 to the nearest thousand.

• The digit in the thousands place is 5.
• The digit to its right is 6, which is 5 or greater, so you round up.
• Result: 6,000

Comparing and Ordering Whole Numbers

To compare two whole numbers, start at the leftmost digit and work your way right. The first place where the digits differ tells you which number is larger.

For example, comparing 4,372 and 4,519: both start with 4 in the thousands place, but in the hundreds place, 3 < 5, so 4,372 < 4,519.

Three symbols express these relationships:

• Less than (<): $a < b$ means $a$ is smaller than $b$
• Greater than (>): $a > b$ means $a$ is larger than $b$
• Equal to (=): $a = b$ means both values are the same

To order a set of numbers, compare them in pairs using this method, then arrange them from least to greatest (or greatest to least).

Basic Operations with Whole Numbers

The four basic operations each do something different:

• Addition combines two or more numbers to find their sum (e.g., $12 + 8 = 20$).
• Subtraction finds the difference between two numbers (e.g., $20 - 8 = 12$).
• Multiplication is repeated addition of the same number (e.g., $4 \times 3$ means $4 + 4 + 4 = 12$).
• Division splits a number into equal groups (e.g., $12 \div 3 = 4$ means 12 split into 3 equal groups of 4).

Properties of Whole Numbers

• Even numbers are divisible by 2 with no remainder (0, 2, 4, 6, 8, ...). They always end in 0, 2, 4, 6, or 8.
• Odd numbers are not evenly divisible by 2 (1, 3, 5, 7, 9, ...). They always end in 1, 3, 5, 7, or 9.

A quick way to tell: just look at the last digit of any whole number. That single digit determines whether the entire number is even or odd.