Question

Find the sum. $$ -3+(-6)+(-2) $$

Short answer

-11

Step-by-step solution

Understand the concept of adding negative numbers

When adding negative numbers, imagine owing money. If a person has -3 dollars, that means they owe someone 3 dollars. So, when they owe an additional 6 dollars (-6) and 2 more dollars (-2), the total amount they owe increases.

Add each number

To find out the total amount owed, add the three amounts together. The sum of -3, -6, and -2 equals -3 - 6 - 2.

Calculate the sum

Performing this operation gives a sum of -11. This means that in total, they owe 11 dollars.

Key concepts

Sum of Integers

The process of adding numbers, or integers, is a fundamental skill in mathematics. An integer can be positive, negative, or zero. When we sum integers, we are combining them to find their total amount. Trying to visualize integers as items or money can help you understand the process. For example, if you have 3 apples and you add another 2 apples, you will have a total of 5 apples. Similarly, if you owe someone \(3 and you borrow \)2 more, you now owe \(5 in total.

Let's improve our understanding by applying this concept specifically to negative numbers. If you start with -3 dollars (owing \)3) and then borrow \(6 and \)2 more, the amounts you owe are getting stacked together, resulting in a larger debt. Mathematically, this is represented as \( -3 + (-6) + (-2) \), which combines all the amounts owed. By sequentially adding each negative number, we proceed towards computing the total sum, which in this case is \( -11 \). This result communicates that the collective debt, or the sum of the integers involved, is $11.

Negative Number Arithmetic

When dealing with negative number arithmetic, it's important to remember that adding a negative number is the same as subtracting its positive counterpart. For example, \( 5 + (-3) \) is the same as \( 5 - 3 \), and both result in 2. This is because the negative sign indicates a direction opposite to positive numbers on the number line. So when you're moving to the left on the number line (adding negative numbers), it's equivalent to subtracting from your original position.

Understanding Negative Sums

If you are adding several negative numbers, like in our exercise \( -3 + (-6) + (-2) \), think of it as increasing the distance in the negative direction. Each number you add pushes you further left on the number line. Therefore, when these individual movements are combined, they result in a more significant single movement in the negative direction, yielding a larger negative sum. It's the accumulation of debt, not the creation of wealth. By adding negative numbers, we are essentially compounding the amounts that are owed or decreasing.

Basic Algebra

Basic algebra involves understanding and working with variables and constants to solve equations. However, before diving into complex equations, it's crucial to master the basic operations with numbers, especially when these operations involve negative numbers. If you can navigate the rules of arithmetic with negative numbers, you'll find it easier to manage them within algebraic expressions later on.

Building Blocks

Think of negative numbers as the building blocks that will help you understand expressions and equations in algebra. Consider a simple algebraic expression such as \( x - y \). If both \( x \) and \( y \) were negative, understanding how to add or subtract negative numbers would directly apply here. For instance, if \( x = -3 \) and \( y = -6 \) and you were asked to calculate \( x - y \), this would become \( -3 - (-6) \), which simplifies to \( -3 + 6 \) according to the rules of negative number arithmetic, resulting in a positive 3.

Understanding these foundational concepts is crucial for tackling algebraic problems. Furthermore, these principles help in solving more complex equations involving negative numbers, preparing you for a deeper exploration of algebra.