3.1 Whole Numbers and Basic Operations
Key Takeaways
- PEMDAS sets the order of operations: Parentheses, Exponents, Multiplication/Division (left to right), then Addition/Subtraction (left to right)
- Same-sign multiplication or division gives a positive result; different signs give a negative result
- To round, look at the digit immediately to the right: 5 or greater rounds up, less than 5 rounds down
- The distributive property a(b + c) = ab + ac lets you break apart multiplication for faster mental math
- Estimation by rounding gives a quick reasonableness check that catches calculator slips on the TEAS
Why Whole-Number Operations Matter on the TEAS
The ATI TEAS 7 Mathematics section contains 34 scored questions (plus unscored pretest items) answered in 57 minutes, and the largest sub-content area is Numbers and Algebra. Whole-number arithmetic is the foundation under every other topic: you cannot solve a dosage proportion or read a chart if you stumble on basic order of operations. A four-function on-screen calculator is provided, but it will faithfully compute whatever you type — so the exam really tests whether you know the correct sequence of steps, not whether you can press buttons.
Whole Numbers, Integers, and Place Value
Whole numbers are the non-negative counting numbers: 0, 1, 2, 3, 4, 5, and so on. Integers extend that set to include negatives, giving …, -3, -2, -1, 0, 1, 2, 3, ….
| Term | Definition | Example |
|---|---|---|
| Whole number | Non-negative integer | 0, 1, 42, 1000 |
| Positive integer | Whole number greater than zero | 1, 2, 3, 4 |
| Negative integer | Integer less than zero | -1, -2, -3 |
| Integer | Any positive or negative whole number, plus zero | -3, -2, -1, 0, 1, 2, 3 |
Place value tells you what each digit is worth based on its position. In 4,738 the 4 means 4,000, the 7 means 700, the 3 means 30, and the 8 means 8.
| Place | Value | Digit in 4,738 |
|---|---|---|
| Ones | 1 | 8 |
| Tens | 10 | 3 (represents 30) |
| Hundreds | 100 | 7 (represents 700) |
| Thousands | 1,000 | 4 (represents 4,000) |
Expanded form: 4,738 = 4,000 + 700 + 30 + 8. Place value is also why you line up decimal points later when adding money or lab values.
Order of Operations (PEMDAS)
When an expression mixes operations, you must evaluate it in a fixed order or you will get the wrong answer. Remember PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
| Letter | Operation | Example |
|---|---|---|
| P | Parentheses (or brackets) | (2 + 3) = 5 |
| E | Exponents (powers, roots) | 2³ = 8 |
| M/D | Multiply/Divide, left to right | 6 × 2 ÷ 3 = 4 |
| A/S | Add/Subtract, left to right | 5 + 3 - 2 = 6 |
The most common TEAS trap is the M/D and A/S tiers: multiplication and division share a rank and are done in reading order, not "all multiplication first." The same is true of addition and subtraction.
Worked Example: Evaluate 12 + 3 × 4 - 2.
- No parentheses or exponents.
- Multiply: 3 × 4 = 12, giving 12 + 12 - 2.
- Add/subtract left to right: 12 + 12 = 24, then 24 - 2 = 22. A test-taker who adds 12 + 3 first gets the wrong answer of 58.
Worked Example (parentheses + exponent): Evaluate (8 - 3)² ÷ 5 + 1.
- Parentheses: 8 - 3 = 5, giving 5² ÷ 5 + 1.
- Exponent: 5² = 25, giving 25 ÷ 5 + 1.
- Divide: 25 ÷ 5 = 5, giving 5 + 1 = 6.
Properties That Speed Up Mental Math
| Property | What it says | Example |
|---|---|---|
| Commutative | Order doesn't matter (+ and ×) | 3 + 5 = 5 + 3 |
| Associative | Grouping doesn't matter (+ and ×) | (2 + 3) + 4 = 2 + (3 + 4) |
| Distributive | Multiply across a sum | 2(3 + 4) = 2·3 + 2·4 = 14 |
| Identity | Add 0 or multiply by 1 | 5 + 0 = 5; 5 × 1 = 5 |
| Zero | Multiply by 0 | 5 × 0 = 0 |
The distributive property is the one most worth practicing because it lets you compute things like 6 × 23 as 6(20 + 3) = 120 + 18 = 138 without a calculator.
Factors and Multiples
A factor of a number divides it evenly (no remainder); a multiple is the result of multiplying that number by an integer.
- Factors of 12: 1, 2, 3, 4, 6, 12.
- Multiples of 12: 12, 24, 36, 48, ….
- A prime number has exactly two factors, 1 and itself (2, 3, 5, 7, 11, 13…).
Finding the greatest common factor (GCF) and least common multiple (LCM) is the engine behind simplifying fractions and finding common denominators in the next section.
Signed Numbers
- Addition, same signs: add and keep the sign (-4 + -3 = -7).
- Addition, different signs: subtract and take the sign of the larger absolute value (-9 + 4 = -5).
- Subtraction: add the opposite — 5 - (-3) = 5 + 3 = 8.
- Multiplication/Division: same signs → positive; different signs → negative.
| Operation | Example | Result |
|---|---|---|
| Positive × Positive | 3 × 4 | 12 |
| Negative × Negative | (-3) × (-4) | 12 |
| Positive × Negative | 3 × (-4) | -12 |
| Negative × Positive | (-3) × 4 | -12 |
Rounding and Estimation
To round: identify the target place, look at the digit just to its right, round up on 5 or more, down on less than 5, and zero out everything to the right.
Worked Example: Round 3,847 to the nearest hundred. The hundreds digit is 8; the digit to its right is 4 (< 5), so round down to 3,800.
Estimation rounds before calculating so you can sanity-check a result. To estimate 487 + 312, round to 500 + 300 = 800; the exact answer 799 is reassuringly close. On the TEAS, a fast estimate is your best defense against a mistyped calculator entry.
Evaluate: 4 + 6 ÷ 2 × 3 - 1
What is the value of (7 - 2)² - 4 × 3?
Simplify using signed-number rules: -8 - (-5).
Round 18,649 to the nearest thousand. Answer: ______
Type your answer below
Match each property of operations to its example.
Match each item on the left with the correct item on the right