Problem 25 - Beginning and Intermediate Algebra

Question

Add. See Examples I through 7. $$ -33+(-14) $$

Step-by-Step Solution

Answer

-47

Step 1: Understand the Problem

The problem asks us to add two negative numbers: -33 and -14. In mathematics, adding two negative numbers results in a number that is further away from zero in the negative direction.

Step 2: Identify The Operation

Both numbers are negative, and we are asked to perform an addition operation on them: \(-33 + (-14)\)

Step 3: Evaluate the Addition

When adding two negative numbers, you add their absolute values and then attach a negative sign to the result. The absolute value of -33 is 33, and the absolute value of -14 is 14. Now, calculate: \[ 33 + 14 = 47 \]

Step 4: Apply the Negative Sign

Since both numbers being added are negative, the result is also negative. Therefore, attach a negative sign to the sum obtained in Step 3: \(-47\).

Key Concepts

Absolute Value

Absolute value is a fundamental concept in mathematics that helps us understand the magnitude of a number without considering its sign. The absolute value of a number is its distance from zero on the number line, and it is always expressed as a non-negative number. For example, the absolute value of both -33 and 33 is 33.

Absolute values are crucial when dealing with negative numbers, especially in operations like addition or subtraction. To determine the absolute value of a number, simply ignore its sign. So, for any number \( a \), the absolute value \( |a| \) can be calculated as:

• If \( a \geq 0 \), then \( |a| = a \)
• If \( a < 0 \), then \( |a| = -a \)

For example, in the expression \(-33 + (-14)\), we first find the absolute values as described: \( |-33| = 33 \) and \( |-14| = 14 \). These values help us determine the sum in absolute terms, which we then adjust for the sign.

Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus sign (-). They are intriguing because their arithmetic behaves differently compared to positive numbers and zero.

When working with negative numbers, envision them on the number line to the left of zero. As you move left on this line, the numbers become smaller in value, but larger in their magnitude of negativity.

• They indicate a deficit or absence, like owing money.
• Negative numbers are crucial in contexts such as temperatures below freezing or elevations below sea level.

In mathematics, negative numbers follow specific rules, especially for operations. For instance, when we add two negative numbers like -33 and -14, we are effectively adding their distances from zero and maintaining the negative orientation.

Addition Operation

The addition operation is one of the basic arithmetic operations. When dealing with negative numbers, the process involves understanding both the signs and magnitudes of the numbers involved.

In our specific example of adding \(-33 + (-14)\), we begin by summing the absolute values of these numbers. Here’s how it works:

• Find the absolute values: \( |-33| = 33 \) and \( |-14| = 14 \).
• Add these absolute values: \( 33 + 14 = 47 \).

After calculating the sum of these absolute values, we then consider the signs. Both numbers were negative: thus, the result \( 47 \) receives a negative sign, becoming \(-47\).

This process ensures that the addition operation is consistent and logical with how negative numbers interact on the number line.