Add and Subtract Integers
How to Master Integer Operations with This Calculator
You have the tool right above you, but understanding why the numbers behave the way they do is the key to mastering algebra. This calculator doesn’t just give you the answer; it shows you the step-by-step logic needed to solve equations involving positive and negative numbers.
Whether you are checking homework, balancing a ledger, or just trying to visualize how “minus a negative” works, this guide covers the essential rules of integer arithmetic.
How to Use This Tool
- Enter your equation: You can type simple integers (e.g.,
-5) or complex expressions with parentheses (e.g.,-10 + (3 - -4)). - Hit Calculate: The tool will break down the signs, simplify parentheses first (following the order of operations), and present the final result.
What Exactly is an Integer?
Before we add or subtract, we need to define our terms. In mathematics, an integer is a whole number that can be positive, negative, or zero. Integers do not include fractions or decimals.
- Positive Integers: Numbers greater than zero ($1, 2, 3, \dots$).
- Negative Integers: Numbers less than zero ($-1, -2, -3, \dots$).
- Zero: The neutral center point.
Mathematicians use the symbol Z to represent the set of all integers. Think of them like steps on a staircase: you can go up (positive), go down (negative), or stand still (zero), but you cannot stand on “step 2.5”.
The Rules of Addition
Adding integers is intuitive when the signs are the same, but it often trips students up when the signs are mixed.
1. Adding Numbers with the Same Sign
If the signs are the same, the numbers work together.
- Rule: Add the absolute values (ignore the signs for a moment) and keep the sign.
- Example (Positive): 5 + 3 = 8
- Example (Negative): -5 + (-3) = -8
- Think of it logically: If you borrow $5 (negative) and then borrow another 3 (negative), your total debt grows to $8.
2. Adding Numbers with Different Signs
When signs clash (one positive, one negative), they fight against each other.
- Rule: Subtract the smaller absolute value from the larger one. The answer keeps the sign of the “heavier” number (the one with the larger absolute value).
- Example A: -10 + 4 = -6
- Here, 10 is bigger than 4. Since the 10 is negative, the negative side wins.
- Example B: 10 + (-4) = 6
- Here, the positive 10 is stronger than the negative 4. The result is positive.
The Rules of Subtraction
Subtraction is the trickiest part of integer math. The secret? Don’t subtract.
In algebra, subtraction is actually just adding the opposite. This is often taught using the “Keep-Change-Change” (KCC) method.
How “Keep-Change-Change” Works
To solve A – B:
- Keep the first number exactly as it is.
- Change the subtraction sign (–) to an addition sign (+).
- Change the sign of the second number (positive becomes negative, or negative becomes positive).
Case 1: Subtracting a Positive
- Problem: 5 – 8
- Rewrite it: 5 + (-8)
- Solve: The signs are different. The difference between 8 and 5 is 3. Since 8 is the “heavier” number and it is negative, the answer is -3.
Case 2: Subtracting a Negative (The Double Negative)
This is the most common error point. Why does minus-minus equal plus?
- Problem: 10 – (-5)
- Rewrite it: 10 + (+5)
- Solve: 15
- The Analogy: Think of a negative sign as “dirt.” If you have a pile of dirt (a negative value) and you take away (subtract) that dirt, you are making the hole smaller effectively adding ground back. Or, think of it as finance: Subtracting a debt is the same as gaining money.
The Number Line
If you get stuck, draw a horizontal line with 0 in the middle.
- Positive numbers live to the right.
- Negative numbers live to the left.
Operations as Movement:
- + (positive): Move to the Right.
- – positive): Move to the Left.
- + (negative): Move to the Left (same as subtracting).
- – (negative): Move to the Right (same as adding).
Examples of Integer Math
Why do we need negative numbers? They model real-world scenarios where “less than nothing” exists.
| Scenario | Positive Integer (+) | Negative Integer (-) | Example Calculation |
| Finance | Income / Deposit | Debt / Withdrawal | You have $50, you spend $70. 50 – 70 = -20 (Overdraft) |
| Temperature | Above Zero | Below Zero | It’s -5⁰C and drops 10⁰. -5 – 10 = -15⁰C |
| Elevation | Above Sea Level | Below Sea Level | A diver is at -20ft and rises 5ft. -20 + 5 = -15ft |
| Sports (Golf) | Over Par (Bad) | Under Par (Good) | Score is -2, then bogies (+1). -2 + 1 = -1 |
FAQs
Q.1 What is the “Absolute Value”?
A: Absolute value is the distance a number is from zero, regardless of direction. It is always positive. For example, the absolute value of -5 is 5, written as |-5| = 5. This concept is crucial when determining which sign “wins” during addition.
Q2. Can I multiply and divide integers here?
A: This specific calculator is optimized for Addition and Subtraction. However, the rules for multiplication are simpler:
- Same signs = Positive Result (+ times + or – times -).
- Different signs = Negative Result (+ times -).
Q3. Why does the calculator use parentheses?
A: In math syntax, we generally don’t like two operator signs touching. Instead of writing 5 + – 3, we write$5 + (-3) to make it clear that the negative sign belongs to the 3. Our calculator handles both formats, but using parentheses helps prevent mistakes.
Sources: CalculatorSoup, SnapXam, Symbolab, Calculator.io, Omni Calculator, MathFraction, AllMath, Siyavula.