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Chapter 10: Problem 13

Add. Do not use the number line except as a check. $$ 17+(-17) $$

Short Answer

Expert verified
The sum of 17 and -17 is 0.

Step by step solution

01

- Identify the Numbers

Recognize the two numbers involved in the addition problem: 17 and -17.
02

- Understand the Concept of Opposites

Recall that adding a number to its opposite (or additive inverse) results in zero. In this case, 17 and -17 are opposites.
03

- Perform the Addition

Compute the sum of 17 and -17. Since they are opposites, the sum is zero: \[ 17 + (-17) = 0 \]
04

- Check Using the Number Line (Optional)

As a check, place 17 on the number line, and then move 17 units to the left (adding -17). You will end up at zero, confirming the sum.

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Key Concepts

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

Opposites in Addition
In math, every number has an opposite, also known as its additive inverse. Opposites are numbers that, when added together, result in zero. For example, the opposite of 17 is -17. When you add 17 and -17 together, the sum is zero. This is because they cancel each other out:
  • Positive and negative numbers are opposites.
  • The sum of a number and its opposite is always zero.
Understanding opposites in addition is crucial when working with integers because it helps simplify many problems. You see this concept clearly in the equation \(17 + (-17) = 0\). The positive 17 and the negative 17 are opposites, and their sum equals zero.
Additive Inverse
The additive inverse of a number is the number you add to it to get zero. For any number \(a\), the additive inverse is \(-a\).
To understand this better, consider these points:
  • The additive inverse of 5 is -5.
  • The additive inverse of -8 is 8.
Here's why this works: Adding a number and its additive inverse can be thought of as moving in opposite directions on the number line. If you start at zero and move to 17, then move back to -17, you'll be back at zero. So, \(17 + (-17) = 0\) is a perfect example of how the concept of additive inverse works in addition problems.
Number Line
Using a number line can make understanding addition and subtraction of integers much easier.
Here's how you can use a number line to check the addition of integers:
  • Start at 0 on the number line.
  • Move to the right if you're adding a positive number.
  • Move to the left if you're adding a negative number.
For the problem \(17 + (-17)\), you would start at 0, move 17 units to the right to reach 17, then move 17 units back to the left because you are adding -17.
You end up back at zero, which confirms that the sum is \(0\). This visual tool is especially helpful for checking your work and ensuring you understand the addition of integers.

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