Where Did Negative Numbers Come From?
Have you ever checked the weather and seen a temperature like −5°F? That little minus sign tells you it's below zero. But for thousands of years, people didn't believe numbers below zero could even exist! The story of integers (whole numbers that can be positive, negative, or zero) is a long journey through history.
Ancient Greek mathematicians actually rejected the idea of negative numbers. They thought math was about measuring real things — and how could you have less than nothing? It took centuries before people realized negative numbers were useful for tracking debts, temperatures, and much more.
So here's the big question this lesson answers: How do we add, subtract, multiply, and divide integers correctly every single time? On the SHSAT, you'll see integer problems everywhere. Let's build a rock-solid foundation.
Core Principles of Integer Operations
Before we jump into calculations, let's nail down the key ideas. An integer is any whole number — positive, negative, or zero. Examples include −7, 0, 3, and 42. Fractions like ½ and decimals like 2.5 are not integers.
Absolute Value
Sign Rules for Multiplication & Division
Additive Inverse
Subtraction = Adding the Opposite
Order of Operations Still Applies
The Number Line — Your Best Friend
The number line is the most powerful visual tool for understanding integers. It places zero in the center, with positive numbers stretching to the right and negative numbers stretching to the left. Let's look at how addition and subtraction work on it.
Here's the big idea: adding a positive number moves you right, and adding a negative number moves you left. Since subtraction is the same as adding the opposite, subtracting a positive moves you left, and subtracting a negative moves you right. If you can picture the number line in your head, you can check your answers quickly on the SHSAT.
The Rules — Written as Formulas
Let's organize the rules for all four operations into clear formulas. These are the rules you should memorize for the test.
Addition Rules
Subtraction Rule
Multiplication & Division Sign Rules
Sign Rules — A Complete Visual Guide
The sign rules can feel confusing at first, but they follow a clean pattern. The diagram below shows every combination of signs for all four operations. Study it like a cheat sheet!
| Operation | Example | Step-by-Step | Result |
|---|---|---|---|
| Addition (same signs) | (−3) + (−8) | Add: 3 + 8 = 11; both negative → −11 | −11 |
| Addition (diff. signs) | 7 + (−10) | Subtract: 10 − 7 = 3; 10 is bigger and negative → −3 | −3 |
| Subtraction | 4 − (−6) | Rewrite: 4 + 6 = 10 | 10 |
| Multiplication | (−5) × (−7) | Multiply: 5 × 7 = 35; same signs → positive | 35 |
| Division | (−24) ÷ 6 | Divide: 24 ÷ 6 = 4; different signs → negative | −4 |
Worked Example — Multi-Step Integer Problem
On the SHSAT, you'll often see problems that combine multiple integer operations. Let's work through one step by step.
Common Mistakes and How to Avoid Them
Integer operations are straightforward once you know the rules, but there are traps that catch students on the SHSAT every year. Let's look at the most common mistakes and how to fix them.
| Common Mistake | What Students Do Wrong | How to Fix It |
|---|---|---|
| Subtracting a negative | Write 5 − (−3) = 2, forgetting to flip the sign. | Rewrite as 5 + 3 = 8. Subtracting a negative always becomes addition. |
| Wrong sign on the answer | Write (−4) + 9 = −5 instead of +5. | The larger absolute value is 9 (positive). The answer must be positive: +5. |
| Multiplying before parentheses | In 2 × (3 + 5), they do 2 × 3 first to get 6 + 5 = 11. | Parentheses first! 3 + 5 = 8, then 2 × 8 = 16. |
| Confusing −x² with (−x)² | Think −3² = 9, but it actually equals −9. | −3² means −(3²) = −9. Only (−3)² = 9 because the parentheses include the negative. |
| Not going left to right | In 12 ÷ 3 × 2, they do 3 × 2 first to get 12 ÷ 6 = 2. | Multiplication and division are equal priority — go left to right. 12 ÷ 3 = 4, then 4 × 2 = 8. |
Connecting Integers to Bigger Ideas
Integer operations are the building blocks for nearly everything else in math. Once you master them, you're ready to tackle more advanced topics. Let's see how integers connect to what's coming next.
| Concept Now (Integers) | Where It Leads |
|---|---|
| Adding and subtracting integers | Solving one-step and two-step equations (e.g., x + 5 = −3) |
| Multiplying and dividing integers | Working with algebraic expressions (e.g., −2(x − 4)) |
| Order of operations with integers | Evaluating complex expressions and functions |
| Number line understanding | Coordinate plane graphing (x-axis and y-axis use integers) |
| Sign rules for multiplication | Understanding slopes (positive vs. negative) in linear equations |
On the SHSAT specifically, integer operations show up in arithmetic problems, algebra problems, and even geometry problems (when coordinates or measurements are negative). The sign rules you learn here will be used over and over. Think of integers as the foundation of every math topic on the test.
Practice Problems
Time to test yourself! These five problems go from easy to challenging. Try each one before reading the answer. Write out your work on scrap paper, just like you would on test day.
Integer Operations — Key Concepts Review
Integers are whole numbers that can be positive, negative, or zero. For addition, same signs mean you add the absolute values and keep the sign; different signs mean you subtract the absolute values and take the sign of the larger one. Subtraction is simply rewritten as adding the opposite: a − b = a + (−b). For multiplication and division, same signs give a positive result and different signs give a negative result.
Always follow PEMDAS (order of operations) when a problem has more than one step. Watch out for common traps like subtracting a negative (which becomes addition) and the difference between −x² and (−x)². Use the number line to visualize problems and double-check your signs. These skills are the foundation for algebra, coordinate geometry, and nearly every other SHSAT math topic.