Problem 24 - Prealgebra

Question

Combine the following by using the rule for addition of positive and negative numbers. $$-5+(-4)$$

Step-by-Step Solution

Answer

-5 + (-4) = -9.

Step 1: Understand the Signs

The first step is to identify the signs of the numbers in the expression \(-5 + (-4)\). Here, \(-5\) is a negative number and \(-4\) is also a negative number. When dealing with addition and subtraction, understanding the signs helps us determine how to combine these numbers.

Step 2: Addition of Negative Numbers

When adding two negative numbers together, the sum is also negative. This is because adding a negative value increases the negative magnitude. Therefore, combining two negative numbers involves adding their absolute values (ignoring the negative sign) and then attaching a negative sign to the result.

Step 3: Calculate the Result

Take the absolute values of \(-5\) and \(-4\), which are \(5\) and \(4\), respectively. Add these values: \(5 + 4 = 9\). Since both original numbers were negative, attach a negative sign to the sum: \(-9\).

Key Concepts

Negative Numbers

Negative numbers are a critical part of mathematics. They help us represent values less than zero, which is essential for various real-life situations like temperature below zero or owing money. Negative numbers have a "-" sign before them to differentiate from positive numbers. For instance, in the expression \(-5 + (-4)\) both numbers, \(-5\) and \(-4\), are negative.
Negative numbers are just as important and useful as positive numbers. They allow us to fully describe values and changes in contexts that go below zero.

Absolute Value

The absolute value of a number measures its distance from zero on the number line, ignoring its sign. This concept helps when adding numbers, especially negative ones. For example, the absolute value of both \(-5\) and \(-4\) is positive because it counts steps away from zero without considering direction.

• For \(-5\), the absolute value is 5.
• For \(-4\), the absolute value is 4.

When we need to add negative numbers, we find their absolute values first, add these, and then place the negative sign back, because the directions we moved from zero were both negative.

Addition Rules

When you're adding numbers, knowing whether they are positive or negative is key. Here's why:

• Adding positive numbers gives you a more positive result.
• Adding a positive number and a negative number means you find the difference between them and keep the sign of the larger absolute value.
• Adding two negative numbers involves adding their absolute values and keeping the negative sign.

These rules help provide a straightforward way to handle expressions like \(-5 + (-4)\), where adding two negative numbers requires combining their absolute values and then attaching the negative sign to the final sum, resulting in \(-9\). This creates an easy checklist to follow whenever you're working with additions involving different integer signs.