Problem 14 - Algebra and Trigonometry

Question

\(13-20=\) Evaluate the expression. $$ \left(\begin{array}{l}{8} \\ {3}\end{array}\right) $$

Step-by-Step Solution

Answer

The result of the subtraction is \(-7\).

Step 1: Identify the Problem

The equation presents a simple subtraction problem: \(13 - 20\). Since 20 is larger than 13, the result will be a negative number.

Step 2: Subtraction Process

To solve \(13 - 20\), we find the difference between the two numbers. Considering the absolute values, we subtract the smaller number from the larger: \(20 - 13 = 7\).

Step 3: Apply Negative Sign

Since the subtrahend (20) is greater than the minuend (13), the result of \(20 - 13 = 7\) must be negative, leading to the final solution: \(-7\).

Key Concepts

Understanding Subtraction

Subtraction is one of the four basic arithmetic operations. It is the process of taking away one number from another. By understanding subtraction, you can solve a wide variety of mathematical problems. Here’s how the subtraction of two numbers works:

• **Minuend**: The number from which another number (the subtrahend) is to be subtracted. In our example, this is 13.
• **Subtrahend**: The number that is to be subtracted. In our example, this is 20.
• **Difference**: The result of subtracting one number from another. This is what we are trying to find through subtraction.

When you perform subtraction and the subtrahend is larger than the minuend, the result is a negative number. We first subtract the smaller number from the larger and then apply a negative sign to the result.
This ensures you get the correct difference. So, to solve \(13 - 20\), as explained in the problem, we calculate \(20 - 13\) which gives 7, then apply a negative sign: \(-7\). It’s essential to understand the role these components play in subtraction to quickly identify and solve mathematical problems, especially those involving negative results.

Negative Numbers Explained

Negative numbers are numbers that are less than zero. They are represented with a "-" sign in front. Negative numbers can be strange at first since we're used to dealing with positive numbers in most daily situations.
Here are some key points about negative numbers:

• **Representation**: A negative number is written by placing a minus sign before a positive number, like \(-7\).
• **Comparison**: Any negative number is always less than a positive number and also less than zero.
• **Arithmetic**: When you subtract a larger number from a smaller one, the result is negative, which is what happened in our problem of \(13 - 20 = -7\).

Understanding negative numbers is crucial for solving problems involving subtraction where the subtrahend is larger than the minuend. As we see in our solution, when the value we are taking away is greater, the overall result dips below zero, indicating a negative number.

Grasping Absolute Value

The absolute value of a number is the distance of the number from zero on the number line, without considering the direction. For example, both 7 and -7 have an absolute value of 7. In mathematical terms, the absolute value of \(-a\), denoted as \(|a|\), is simply \(a\).
Here are the facts about absolute value:

• **Non-negative**: Absolute value is always non-negative because it measures magnitude without direction.
• **Notation**: Represented by vertical bars, for example, \(|-7| = 7\).
• **Usage**: In our subtraction problem, the absolute values help determine the magnitude of the difference without worrying about whether either number is negative.

When solving problems such as \(13 - 20\), the absolute values help you first determine the number magnitude (i.e., \(20\) minus \(13\) equals \(7\)), and then you apply the negative sign because the original operation results in a drop below zero. This concept facilitates understanding and managing operations with negative numbers effectively.