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Chapter 5: Problem 138

Explain how to add integers.

Short Answer

Expert verified
When adding integers, if they are both positive, just add them. If one is positive and one is negative, subtract the smaller one from the larger one and carry the sign of the larger one. If both are negative, add them and carry the negative sign.

Step by step solution

01

Understand positive integers addition

Adding two positive integers is straightforward. The numbers are simply added together. For example, the equation \(3 + 2\) equals \(5\).
02

Understand negative integers addition

When you have to add a negative number to a positive number, consider the negative number as a debt. For example, \(5 + (-3)\), means you have 5 and you owe 3, so you have \(5 - 3 = 2\) left.
03

Understanding the addition of two negative integers

If you are adding two negative integers together, consider both as debts. For example, \(-3 + (-2)\) would be equivalent to owing 3 and then owing 2 more; the total debt will be \( -3 - 2 = -5 \) .

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Positive Integers
Positive integers are numbers greater than zero. They are represented on the right side of the number line, starting from 1 and extending to infinity. Positive integers are the numbers we frequently use in everyday situations like counting objects or measuring quantities. When you add two positive integers, it's like combining two groups of objects together.
For instance:
  • Adding 3 apples to 2 apples gives you 5 apples.
  • Using the number line, if you start at 0 and move 3 steps to the right, then 2 more steps to the right, you'll land on 5.
This is a straightforward process because you are simply increasing the total amount.
Negative Integers
Negative integers are numbers less than zero. They are located on the left side of the number line and are used to represent values like temperature below zero or financial debt. Negative integers have a minus sign, such as -1, -2, or -50.
Adding negative integers can be understood with respect to the position on the number line:
  • If you add two negative numbers, you're essentially moving further away from zero on the negative side.
  • Think of adding -3 and -2 as taking 3 steps to the left of zero, then 2 more steps to the left, ending at -5.
This helps visualize the result of combining negative amounts.
Integer Operations
Integer operations consist of addition, subtraction, multiplication, and division involving whole numbers, which include both positive and negative integers as well as zero. When dealing with integer addition, several rules apply that make calculations easier.
For addition:
  • Adding two positive integers yields a larger positive integer: e.g., 4 + 6 = 10.
  • Adding two negative integers results in a more significant negative integer: e.g., -3 + -7 = -10.
  • When you add a positive integer to a negative integer, you reduce the value of the positive number: e.g., 5 + (-2) = 3.
  • Adding zero to any integer leaves the integer unchanged: e.g., 7 + 0 = 7.
These operations are fundamental for solving more complex math problems.
Debt Analogy
The debt analogy offers a practical way to understand adding integers, particularly when involving negative numbers. Imagine positive integers as money you have and negative integers as money you owe.
Consider these examples to clarify:
  • In the case of 5 + (-3), you can think of having $5 but owing $3, leaving you with $2.
  • When adding two negative integers like -4 + (-6), it is like owing $4 and then owing another $6, making your total debt $10.
  • If you add two positive integers such as 7 + 8, it's like having $7 and then having an additional $8, resulting in $15 overall.
This analogy simplifies the concept by associating numbers with real-world financial transactions, making it easier for learners to grasp.

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