Problem 19 - Beginning and Intermediate Algebra

Question

Add. See Examples I through 7. $$ -16+16 $$

Step-by-Step Solution

Answer

The sum is 0.

Step 1: Identify the Numbers

Look at the given exercise and identify the numbers involved, which are \(-16\) and \(16\).

Step 2: Look for Inverse Pairs

Notice that \(-16\) and \(16\) are inverse pairs, meaning they are equal in magnitude but opposite in sign.

Step 3: Add the Numbers

Add the numbers together by calculating \(-16 + 16 = 0\). The sum of a number and its inverse is always zero.

Key Concepts

Inverse Pairs

Inverse pairs are an interesting concept in mathematics. They consist of two numbers that have the same absolute value but opposite signs. You could think of them as being mirror images of each other on the number line. For example,

• -16 and 16
• -5 and 5
• -8 and 8

These pairs are special because when you add them together, they cancel each other out and make zero. This is because the negative effectively "undoes" the positive. This is a very useful property in many areas of mathematics, especially when solving equations or simplifying expressions. It's like having a balance scale where each number offsets the other to make it level at zero.

Sum of Integers

The sum of integers can come in various forms, but the basic idea is to add up all the numbers. When dealing with positive and negative integers, it helps to first identify whether some of the numbers can cancel each other out due to inverse pairs,

• like
  • -16 and 16
  • making zero

This simplifies the calculation without the need for lengthy addition. When summing up a list of integers, always look out for such pairs—they make the task much easier. Remember to combine all like signs separately, adding the positive numbers and the negative numbers first before combining the results. This method efficiently reduces the complexity of adding several integers.

Zero Property of Addition

The zero property of addition is a fundamental rule in arithmetic. Simply put, adding zero to any number doesn’t change the number. This makes zero the "identity" element for addition. The concept extends to the fact that the sum of two inverse pairs is zero. For instance, with

• -16 and 16
• the sum is zero

This rule is applicable in many mathematical scenarios, serving as a handy shortcut for simplifying calculations. This property helps in understanding that any number added to its inverse results in zero, aiding both in mental math and larger algebraic calculations. By using this concept, you can simplify many mathematical operations, making problem-solving faster and more intuitive.