Problem 13 - Prealgebra

Question

Draw a number line from 10 to 10 and use it to add the following numbers. $$-4+(-5)$$

Step-by-Step Solution

Answer

The result of adding -4 and -5 is -9.

Step 1: Prepare the Number Line

First, we need to draw a number line that covers the numbers from -10 to 10 at least, since we are dealing with negatives. Draw a horizontal line and mark equally spaced points on it, labeling each point with numbers from -10 to 10.

Step 2: Locate the Starting Point

Locate -4 on the number line. This is your starting point for the addition.

Step 3: Determine the Direction and Movement

Since we are adding a negative number (-5), we will move 5 steps to the left from -4. Moving left indicates subtraction or adding a negative value.

Step 4: Count the Steps

From the point at -4, count 5 steps to the left: - From -4 to -5 (1 step), - From -5 to -6 (2 steps), - From -6 to -7 (3 steps), - From -7 to -8 (4 steps), - From -8 to -9 (5 steps).

Step 5: Identify the Final Position

After counting 5 steps to the left, you land on -9 on the number line. This is the endpoint of the addition.

Key Concepts

Negative Numbers

Negative numbers are essential when dealing with values less than zero. Think of them as numbers that move you to the left on the number line. They are often used to represent losses, debts, or decreases.
For example, in temperature, below zero degrees Celsius, the numbers are negative. Instead of getting warmer, it becomes colder. Negative numbers have a negative sign (-) in front of them, indicating their direction on the number line.
When adding negative numbers, you're often moving further into negatives. It's like taking steps backwards. In the problem \(-4 + (-5)\), both numbers are negative, so you will move left on the number line for both of them.

Integer Operations

Understanding integer operations is vital for performing calculations involving whole numbers. Integers include both positive numbers, negative numbers, and zero. They allow us to perform basic arithmetic smoothly across both directions of the number line.

• Positive integers are greater than zero and increase in value the more you have.
• Negative integers are less than zero and decrease in value the more you have.
• Zero is unique as it represents a neutral point; it's neither positive nor negative.

When adding integers with different signs, the operation depends on whether you're adding a positive to a negative or vice versa. The rule for adding two negative integers, as in our exercise, means you add their absolute values and keep the result negative.
In this case, \(-4\) added to \(-5\) is just like adding \(4 + 5 = 9\), but since both numbers are negative, the result is \(-9\).

Basic Arithmetic

Basic arithmetic refers to the fundamental operations used in mathematics, which include addition, subtraction, multiplication, and division. These operations form the foundation of more complex mathematical concepts and problem-solving techniques.

• **Addition** combines numbers together to find a total.
• **Subtraction** removes one number from another to find the difference between them.
• **Multiplication** is a shortcut for repeated addition.
• **Division** splits a number into specified equal parts.

In our specific problem involving \(-4 + (-5)\), we use addition to combine negative values. The operation performed is straightforward once you visualize it on a number line, highlighting how moving in the negative direction impacts the outcome.
Although the technique seems simple, mastery of number line addition is a crucial stepping stone in learning how to proficiently handle integers and advance further into mathematical concepts.