Problem 19 - Algebra A Combined Function

Question

Add. See Examples 1 through 12,18, and 19. $$ -16+16 $$

Step-by-Step Solution

Answer

The sum of -16 and 16 is 0.

Step 1: Identify the Numbers

In the expression \(-16 + 16\)\, we have two numbers: \(-16\)\ and \(16\).\ Both numbers are integers, with \(-16\)\ being negative and \(16\)\ positive.

Step 2: Recognize Opposites

Notice that the numbers \(-16\)\ and \(16\)\ are opposites. When two numbers are opposites, their sum equals zero because they are equal in magnitude but opposite in sign.

Step 3: Calculate the Sum

To find the result of the expression, add the two numbers: \(-16 + 16 = 0\).\ Since they cancel each other out, the sum is zero.

Key Concepts

Integers

In basic arithmetic, integers are the set of whole numbers that include zero, positive numbers, and negative numbers. These are numbers without fractions or decimals. Understanding integers is foundational in mathematics, as they are used in a variety of operations:

• Positive integers (> 0) are numbers like 1, 2, 3, etc.
• Negative integers (< 0) include numbers like -1, -2, -3, etc.
• Zero (0) is an integer that is neither positive nor negative.

Integers are crucial for everyday calculations and algebraic operations. They let us express both value and direction, especially when dealing with real-world situations, like temperatures or financial transactions.

Opposites

Opposite numbers are two numbers that have the same numerical value but different signs. For example, the opposites of each other are like -5 and 5. When plotted on a number line, these numbers are equidistant from the origin (zero) but in opposite directions. The concept of opposites is important because:

• Adding opposite numbers results in zero. For instance, -16 and 16 are opposites, and their sum is zero.
• It helps in understanding symmetry in mathematics.

Recognizing opposites quickly can simplify many arithmetic operations. When you see opposite integers, you can often solve equations faster by knowing they cancel each other out.

Addition

Addition is one of the fundamental operations of arithmetic. It combines numbers into a total sum. When adding integers, remember:

• The sum of two positive numbers is positive.
• The sum of two negative numbers is negative.
• The sum of one positive and one negative number depends on their absolute values. The result takes the sign of the larger absolute value.

For opposites, such as -16 and 16, they cancel out to zero. This is because they're equal in magnitude but have different signs. Addition with integers is used in simple arithmetic and complex calculations, forming the basis for algebraic techniques.