---
title: "Integer Arithmetic in Elementary Algebra"
description: "Integer arithmetic is adding, subtracting, multiplying, and dividing positive and negative whole numbers in Elementary Algebra, using sign rules and zero correctly."
canonical: "https://fiveable.me/elementary-algebra/key-terms/integer-arithmetic"
type: "key-term"
subject: "Elementary Algebra"
unit: "Unit 1"
---

# Integer Arithmetic in Elementary Algebra

## Definition

Integer arithmetic is the set of rules for adding, subtracting, multiplying, and dividing integers, which are positive whole numbers, negative whole numbers, and zero. In Elementary Algebra, you use these sign rules to get correct answers fast.

## What It Is

Integer arithmetic is the set of rules you use when you work with positive integers, negative integers, and zero in Elementary Algebra. It covers addition, subtraction, multiplication, and division, and the main job is keeping track of signs so your answer matches the number pattern.

For addition and subtraction, sign matters more than the numbers themselves at first. If two integers have the same sign, you add their absolute values and keep that sign. If they have different signs, you subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value. That is why 8 + (-3) becomes 5, while -8 + 3 becomes -5.

Subtraction can feel trickier because it changes into adding the opposite. So 6 - 4 is easy, but 6 - (-4) means 6 + 4. A lot of mistakes happen when you treat the second minus sign like a negative number instead of a subtraction symbol. A quick habit that helps is rewriting subtraction as “add the additive inverse” before you calculate.

Multiplication and division follow a different sign pattern. Numbers with the same sign give a positive result, and numbers with different signs give a negative result. So (-3)(-5) = 15, but 12 ÷ (-3) = -4. The size of the answer comes from the arithmetic facts, while the sign comes from the rule.

Division also has one extra restriction: you cannot divide by zero. In Elementary Algebra, that matters because zero is a number you can add, subtract, and multiply with, but not use as a divisor. If a problem gives you an expression like 18 ÷ 6, integer arithmetic gives an integer answer, but not every division problem does. Sometimes the quotient is not an integer, and that is still outside the basic integer-arithmetic rules you are practicing here.

A compact example shows the whole process: -7 + 12 - 5 = 5 - 5 = 0. First compare absolute values, then follow the sign rule, then keep simplifying step by step. That kind of careful sign work shows up everywhere in algebra, especially when expressions have multiple negative signs or when you are combining like terms.

## Why It Matters

Integer arithmetic is the number skill behind a lot of first-year algebra work. If you can handle signs cleanly, you can simplify expressions, solve equations, and check whether an answer makes sense without getting stuck on the arithmetic itself.

It also connects directly to number line thinking. When you move right, you add positive amounts. When you move left, you add negative amounts or subtract positive amounts. That same idea shows up in temperature changes, bank account changes, elevation above and below sea level, and any problem where a quantity moves above or below zero.

In Elementary Algebra, integer arithmetic is often the first place where students see why symbols matter. A minus sign can mean subtraction, or it can mean a negative number, and those are not the same thing. Being able to tell the difference keeps you from losing points on otherwise simple problems.

This skill also supports later topics like solving linear equations, working with variables, and factoring. If you can compute with integers confidently, you have a much easier time focusing on the algebra instead of getting distracted by sign errors.

## Connections

### [Positive Integers](/elementary-algebra/key-terms/positive-integers)

Positive integers are the counting numbers you already know, like 1, 2, 3, and so on. In integer arithmetic, they act as the reference point for sign rules, because adding or multiplying by a positive number keeps the direction of the value the same. They are the numbers you compare negative integers against when you decide which absolute value is larger.

### Negative Integers

Negative integers are the whole numbers less than zero, like -1, -2, and -3. They are the reason sign rules matter in algebra, because subtraction, multiplication, and division can all produce negative results. A lot of integer arithmetic comes down to deciding whether the final answer should move left on the number line or stay below zero.

### Absolute Value

Absolute value is the distance a number is from zero, without thinking about sign. In integer arithmetic, you often use absolute value to find the size of an answer before choosing its sign. That is especially useful for addition and subtraction, where you compare magnitudes first and then apply the correct sign.

### [Additive Inverse](/elementary-algebra/key-terms/additive-inverse)

The additive inverse of a number is the value that adds to it and gives zero. For example, the additive inverse of 7 is -7. This idea makes subtraction easier to rewrite, because a problem like 9 - 4 can be thought of as 9 + (-4). It is one of the cleanest ways to avoid sign confusion.

## On the AP Exam

A problem set usually asks you to simplify integer expressions, compute with positive and negative numbers, or choose the correct sign for an answer. You might see a multi-step expression where one mistake with a negative sign throws off the whole result, so the real task is showing careful sign control, not just fast calculation.

You may also be asked to interpret a real-world context, like temperature change or account balance, and translate it into integer arithmetic. In those questions, the point is to decide whether the situation means add, subtract, multiply, or divide, then apply the correct sign rule. If your class uses quick quizzes or exit tickets, these often test whether you can tell the difference between subtraction and adding a negative.

## Key Takeaways

- Integer arithmetic means working with positive integers, negative integers, and zero using addition, subtraction, multiplication, and division.
- For addition and subtraction, compare absolute values and then choose the correct sign, rather than guessing from the operation alone.
- For multiplication and division, numbers with the same sign give a positive answer, and numbers with different signs give a negative answer.
- Subtraction can be rewritten as adding the additive inverse, which makes many sign problems easier to manage.
- Zero can be added, subtracted, and multiplied with, but it cannot be used as a divisor.

## FAQs

### What is integer arithmetic in Elementary Algebra?

Integer arithmetic is the set of rules for adding, subtracting, multiplying, and dividing integers in Elementary Algebra. You use sign rules to keep track of whether the answer is positive, negative, or zero. It is the basic number skill behind a lot of algebraic simplification.

### How do you add and subtract integers?

When integers have the same sign, add their absolute values and keep that sign. When they have different signs, subtract the smaller absolute value from the larger one and keep the sign of the number with the larger absolute value. For subtraction, it often helps to rewrite the problem as adding the opposite.

### Why do two negative numbers make a positive in multiplication?

The sign rules for multiplication and division follow a pattern: same signs give positive, different signs give negative. So a negative times a negative is positive, and a negative divided by a negative is also positive. This is a rule you apply directly in algebra, even if the reason feels abstract at first.

### What is a common mistake with integer arithmetic?

One common mistake is mixing up subtraction with a negative number. For example, 6 - (-4) is not 2, it is 10 because subtracting a negative is the same as adding a positive. Another frequent error is forgetting that the sign rule for multiplication and division is different from the rule for addition and subtraction.

## Related Study Guides

- [1.3 Add and Subtract Integers](/elementary-algebra/unit-1/3-add-subtract-integers/study-guide/Fjqcg9R1kqamM5f2)

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