Problem 8 - Prealgebra

Question

Subtract. $$-1-4$$

Step-by-Step Solution

Answer

The result of \(-1 - 4\) is \(-5\).

Step 1: Understanding the Problem

The exercise is asking us to perform a subtraction: \(-1 - 4\). We need to find the result of this operation.

Step 2: Rewriting the Expression

Subtracting a positive number is equivalent to adding a negative number. Therefore, \(-1 - 4\) can be rewritten as \(-1 + (-4)\).

Step 3: Adding Negative Numbers

To add two negative numbers, you add their absolute values and then make the result negative. Here, the absolute values are 1 and 4. So, we add them:\[1 + 4 = 5\].The sum of two negative numbers \(-1 + (-4)\) is \(-5\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers that are less than zero. They are usually represented with a minus sign (e.g., -1, -4). In mathematics, negative numbers are used to express values like debts, temperatures below freezing, or elevations below sea level. Here are some key points to remember about negative numbers:

• Negative numbers are always to the left of zero on a number line.
• The further to the left a number is, the smaller its value. For instance, -4 is less than -1.
• When you add, subtract, multiply, or divide with negative numbers, rules may change compared to positive numbers.

Understanding the nature of negative numbers is crucial. It helps us perform arithmetic operations involving them correctly.

Exploring Absolute Value

Absolute value is a mathematical concept that refers to the distance of a number from zero on the number line, regardless of direction. It is always a non-negative number. The absolute value is symbolized by two vertical bars surrounding the number, for example, \(|x|\). Let's note some important aspects of absolute value:

• The absolute value of both 4 and -4 is 4, because both are 4 units away from zero.
• For any positive number, its absolute value is the number itself: \( |4| = 4 \).
• For any negative number, its absolute value is the positive counterpart of the number: \( |-4| = 4 \).
• The absolute value of zero is 0.

Using absolute values is particularly helpful when adding or subtracting negative numbers, as they help simplify calculations by focusing solely on the magnitude.

Subtraction as Addition

In mathematics, subtraction can often be thought of as a form of addition. This means subtracting a number is the same as adding its opposite. Using this understanding, the expression \(-1 - 4\) becomes \(-1 + (-4)\), where we simply add the negative of 4, which is \(-4\). Here's why thinking about subtraction in this way can help:

• It simplifies arithmetic operations by reducing the types of operations you need to master to just addition.
• Helps in understanding how negative numbers work when combined, since it changes subtraction into adding a negative value.
• Encourages flexibility in solving problems, as it emphasizes understanding notions like opposites and transformations.

By viewing subtraction as addition, especially in dealing with negative numbers, we gain a better conceptual clarity and can solve arithmetic expressions more efficiently.