Problem 26 - Fundamentals of Mathematics

Question

Find the sums in the following 27 problems. If possible, use a calculator to check each result. $$ (-6)+(-20) $$

Step-by-Step Solution

Answer

-26

Step 1: Understand the Problem

We need to find the sum of two negative integers: -6 and -20. When we add two negative numbers, the result is a negative number with an absolute value equal to the sum of their absolute values.

Step 2: Add the Absolute Values

First, take the absolute values of the two numbers. The absolute value of -6 is 6, and the absolute value of -20 is 20. Add these absolute values together: 6 + 20 = 26.

Step 3: Determine the Sign of the Sum

Since both numbers we are adding are negative, their sum will also be negative. Therefore, the sum is -26.

Key Concepts

Absolute Value

In mathematics, the absolute value of a number refers to its distance from zero on a number line, regardless of its direction. It is always a non-negative number. For any given number \( x \), its absolute value is denoted by \( |x| \). Consider the number -6 as an example. The absolute value of -6 is simply 6, because it is 6 units away from 0 on the number line.

• Absolute values are useful when dealing with real numbers.
• They provide clarity in operations like addition and subtraction, especially with negative numbers.

In our example problem, finding the absolute values of -6 and -20, then summing them, shows us the total magnitude without concern for the direction (or sign). It's a great way to simplify calculations before considering the sign of the final result.

Sum of Integers

The sum of integers involves adding together two or more numbers. When we talk about integers, we're referring to whole numbers that can be positive, negative, or zero. Calculating the sum is one of the fundamental operations in mathematics and can be applied to solve a multitude of problems.
When adding two negative integers, as in the exercise with \((-6) + (-20)\), there are a few important things to note:

• Both numbers have negative signs.
• Their sum will also be negative.
• The magnitude of the sum is determined by the absolute values of the integers.

Calculations follow a straightforward path: convert the negative integers to their absolute values, sum these values, and then reapply a negative sign to reflect the original numbers' nature.

Negative Integers

Negative integers are numbers less than zero, represented with a minus sign (\(-\)). They are an essential part of the integer family, which also includes zero and positive integers. Negative integers are useful in various real-world contexts, representing values below a defined baseline, such as temperatures below freezing or debts in accounting.

• When adding negative integers, the result is more negative if no positive numbers counterbalance.
• The sum's magnitude increases as you add more negative integers.
• Understanding the nature of negative integers helps to predict the result quickly—if all numbers are negative, expect a negative sum.

In our example, adding -6 and -20 made use of these properties. By identifying both numbers as negative, we anticipated a negative sum. The operation confirmed this with an expected result: -26. It's like combining debts; the total amount of debt increases.