Problem 42 - Prealgebra

Question

Add the following numbers left to right. $$-27+(-56)+(-89)$$

Step-by-Step Solution

Answer

The sum of the numbers is -172.

Step 1: Add the first two numbers

Let's start by adding the first two numbers together: \(-27 + (-56)\). Adding two negative numbers means we add their absolute values and keep the negative sign. So, \(27 + 56 = 83\). Thus, \(-27 + (-56) = -83\).

Step 2: Add the result to the last number

Now, take the result from the first step and add it to the last number in the sequence: \(-83 + (-89)\). Again, add the absolute values: \(83 + 89 = 172\). So, \(-83 + (-89) = -172\).

Key Concepts

Understanding Negative Numbers

Negative numbers are essential to grasp for mastering integer arithmetic. They represent values less than zero. In the context of a number line, these numbers are found to the left of zero. Negative numbers give us a way to discuss deficiencies or reductions. For instance, if you owe $5, your financial position is -5 dollars.

• A number is negative if it carries a minus sign (-) in front of it.
• The farther a negative number is from zero, the smaller its value in real terms, but it appears larger when considering size.

Recognizing negative numbers is crucial in both real-world contexts, such as temperatures or financial debts, and in mathematical operations.

Decoding Absolute Value

Understanding the absolute value is crucial when dealing with both positive and negative numbers. The absolute value of a number is its distance from zero on the number line, regardless of its direction. It tells us how much "quantity" a number has, stripping away any negative sign it may carry.

• The absolute value of a number is always a non-negative number.
• For example, the absolute value of -10 is 10, denoted as \(|-10| = 10\).

In practical terms, absolute value is helpful as it allows us to measure "magnitude" without concern for direction, which is pivotal when performing operations that depend on size rather than direction.

The Process of Adding Negative Numbers

Adding negative numbers can initially seem tricky, but it's ultimately about combining values while considering their direction. When you add negative numbers:

• You find the sum of their absolute values.
• The result retains the negative sign since you're effectively combining deficits.

Let's say you add two negative numbers like -3 and -7:- First, find the absolute values: 3 and 7.- Add these values: 3 + 7 = 10.- Since both numbers were negative, the result is -10.We apply the same logic to the provided problem: add the absolute values of each number and keep the negative sign, as seen in calculating \(-27 + (-56) = -83\) and continuing with \(-83 + (-89) = -172\). Understanding this process helps in overcoming challenges associated with negative figures in larger mathematical operations.