Whole Numbers and Operations

Let’s be honest: when someone says “math,” most people don’t think of counting apples or splitting a pizza. They think of complicated symbols, confusing formulas, and that sinking feeling from a test they’d rather forget. But here’s a secret that might surprise you: every single piece of mathematics—from simple arithmetic to rocket science—is built on the same foundation you’ve been using since you were a kid.

Whole numbers and basic operations are that foundation. You already use them every day without even thinking about it. When you check the time, count your change, or figure out how many slices of pizza each person gets, you’re doing exactly the math we’re going to explore in this chapter. We’re just going to give you some vocabulary, fill in any gaps, and make sure you feel rock-solid confident about these fundamentals.

Think of this chapter as tuning up a car before a road trip. You might already know how to drive, but making sure everything is running smoothly will make the journey ahead much easier.

Core Concepts

What Are Whole Numbers?

Whole numbers are exactly what they sound like—numbers that are complete, with no fractions or decimals. They start at zero and go on forever:

$$0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, \ldots$$

These are the counting numbers plus zero. They’re called “whole” because they represent complete units—whole apples, whole dollars, whole people (you can’t have 2.7 friends, after all).

Place Value: The Secret to Reading Big Numbers

Here’s something powerful: we can write any number, no matter how enormous, using just ten symbols (0 through 9). The trick is place value—the position of each digit tells us its value.

Consider the number 5,847. Reading from right to left:

• The 7 is in the ones place: $7 \times 1 = 7$
• The 4 is in the tens place: $4 \times 10 = 40$
• The 8 is in the hundreds place: $8 \times 100 = 800$
• The 5 is in the thousands place: $5 \times 1000 = 5000$

Add them up: $5000 + 800 + 40 + 7 = 5847$

Each position is worth ten times more than the position to its right. This pattern continues forever: ten-thousands, hundred-thousands, millions, billions, and beyond.

Position	Value	Example in 3,456,789
Ones	1	9
Tens	10	80
Hundreds	100	700
Thousands	1,000	6,000
Ten-thousands	10,000	50,000
Hundred-thousands	100,000	400,000
Millions	1,000,000	3,000,000

The Four Operations

You’ve been doing these since elementary school, but let’s make sure we’re speaking the same language:

Addition (+) combines quantities. When you add $3 + 5$, you’re asking “what’s the total?”

Subtraction (−) finds the difference. When you calculate $8 - 3$, you’re asking “how much is left?” or “how much more?”

Multiplication (×) is repeated addition. $4 \times 3$ means “add 4 three times” or “add 3 four times”—either way, you get 12.

Division (÷) splits into equal groups. $12 \div 4$ asks “if I split 12 into 4 equal groups, how many in each group?”

Order of Operations: PEMDAS

Here’s where things get interesting. What’s the answer to this?

$$3 + 4 \times 2$$

Is it 14 (adding first, then multiplying)? Or 11 (multiplying first, then adding)?

The answer is 11, and here’s why: mathematicians agreed long ago on a specific order for doing operations, so everyone gets the same answer. This order is often remembered as PEMDAS:

1. Parentheses (do what’s inside first)
2. Exponents (powers, like $2^3$)
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

A popular way to remember this: “Please Excuse My Dear Aunt Sally.”

Important note: Multiplication and division have the same priority—you do them left to right, not multiplication before division. Same goes for addition and subtraction.

Notation and Terminology

Let’s build your math vocabulary:

Term	Meaning	Example
Addend	A number being added	In $3 + 5 = 8$, both 3 and 5 are addends
Sum	The result of addition	In $3 + 5 = 8$, the sum is 8
Minuend	The number being subtracted from	In $9 - 4 = 5$, the minuend is 9
Subtrahend	The number being subtracted	In $9 - 4 = 5$, the subtrahend is 4
Difference	The result of subtraction	In $9 - 4 = 5$, the difference is 5
Factor	A number being multiplied	In $6 \times 7 = 42$, both 6 and 7 are factors
Product	The result of multiplication	In $6 \times 7 = 42$, the product is 42
Dividend	The number being divided	In $20 \div 4 = 5$, the dividend is 20
Divisor	The number you divide by	In $20 \div 4 = 5$, the divisor is 4
Quotient	The result of division	In $20 \div 4 = 5$, the quotient is 5

You don’t need to memorize all these right away—they’ll become natural as you practice. But knowing them helps you understand instructions and communicate about math clearly.

Examples

Key Properties and Rules

These properties aren’t just abstract rules—they’re tools that make calculations easier and help you check your work.

Commutative Property

For addition: $a + b = b + a$

For multiplication: $a \times b = b \times a$

You can add or multiply in any order. This is why $3 + 7$ equals $7 + 3$, and $4 \times 5$ equals $5 \times 4$.

Does NOT work for subtraction or division: $5 - 3 \neq 3 - 5$

Associative Property

For addition: $(a + b) + c = a + (b + c)$

For multiplication: $(a \times b) \times c = a \times (b \times c)$

When adding or multiplying three or more numbers, you can group them however you want. This lets you choose the easiest grouping.

Example: $25 \times 4 \times 7 = (25 \times 4) \times 7 = 100 \times 7 = 700$

Distributive Property

$$a \times (b + c) = a \times b + a \times c$$

Multiplication distributes over addition (and subtraction). This is the property behind the mental math trick in Example 4.

Example: $6 \times 102 = 6 \times (100 + 2) = 600 + 12 = 612$

Identity Properties

Additive identity: $a + 0 = a$ (adding zero changes nothing)

Multiplicative identity: $a \times 1 = a$ (multiplying by one changes nothing)

Zero Property of Multiplication

$$a \times 0 = 0$$

Any number multiplied by zero equals zero.

Rounding and Estimation

Rounding simplifies numbers by replacing them with nearby “round” numbers. This is incredibly useful for:

• Quick mental calculations
• Checking if your answer makes sense
• Real-world situations where exact answers aren’t needed

To round a number:

1. Find the place value you’re rounding to
2. Look at the digit to its right
3. If it’s 5 or more, round up; if it’s less than 5, round down

Example: Round 7,849 to the nearest hundred.

• The hundreds digit is 8
• The digit to its right is 4
• Since 4 < 5, we round down: 7,800

Example: Round 7,849 to the nearest thousand.

• The thousands digit is 7
• The digit to its right is 8
• Since 8 ≥ 5, we round up: 8,000

Real-World Applications

Managing Money

You have $50. You want to buy a book for $12, a notebook for $7, and pens for $4. How much will you have left?

$$50 - (12 + 7 + 4) = 50 - 23 = 27$$

You’ll have $27 left.

Planning Time

Your flight boards at 2:15 PM. Security takes about 45 minutes, parking takes 20 minutes, and you want 30 minutes of buffer time. What time should you leave home?

Total time needed: $45 + 20 + 30 = 95$ minutes (or 1 hour 35 minutes)

Count back from 2:15 PM: You should leave by 12:40 PM.

Splitting Costs

Three roommates split a $2,847 monthly rent. How much does each person pay?

$$2847 \div 3 = 949$$

Each roommate pays $949.

Estimation at the Store

Your cart has items priced at $3.49, $7.99, $12.75, $4.29, and $8.50. You have $40. Is that enough?

Round and estimate: $3 + 8 + 13 + 4 + 9 = 37$

Yes, $40 should be enough (actual total is $37.02).

Sports Statistics

A basketball player scored 23, 18, 31, 15, and 28 points in five games. What’s the average?

$$\frac{23 + 18 + 31 + 15 + 28}{5} = \frac{115}{5} = 23$$

Average: 23 points per game.

Self-Test Problems

Test yourself with these problems. Try to solve each one before checking the answer.

Summary

You’ve just reviewed the fundamental building blocks of all mathematics:

• Whole numbers are complete numbers (0, 1, 2, 3, …) with no fractions or decimals

• Place value is the position system that lets us write any number using just ten digits (0-9)

• The four operations are addition, subtraction, multiplication, and division—each with its own vocabulary (sum, difference, product, quotient)

• Order of operations (PEMDAS) ensures everyone calculates the same way: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right)

• Key properties make math easier:

  • Commutative: order doesn’t matter for + and ×
  • Associative: grouping doesn’t matter for + and ×
  • Distributive: $a(b + c) = ab + ac$
  • Identity: adding 0 or multiplying by 1 changes nothing

• Rounding and estimation help you check answers and make quick calculations

• These skills appear everywhere: money, time, cooking, sports, shopping—you’re already using them daily

The concepts in this chapter might seem basic, but they’re the foundation everything else builds upon. When you’re comfortable here, you’re ready for anything that comes next. And remember: if you ever feel stuck in a later chapter, coming back to these fundamentals is always a smart move.

You’ve got this. Let’s keep building.