Problem 37 - Elementary Algebra

Question

Find the sums. \((-6)+(-20)\)

Step-by-Step Solution

Answer

Answer: The sum of -6 and -20 is -26.

Step 1: Identifying the numbers

We have two negative numbers: \((-6)\) and \((-20)\).

Step 2: Adding the numbers

Since both numbers are negative, we add their absolute values and keep the negative sign. So, \((-6) + (-20) = -(6+20)\).

Step 3: Calculating the sum

Now, we add the absolute values of the numbers (6 and 20) to get the final result. \(-(6+20) = -(26)\). So, the sum of \((-6)\) and \((-20)\) is \(-26\).

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. You can think of them like debts or temperatures below freezing. They're written with a minus sign, for example,

• -1
• -6
• -20

These numbers exist on the left side of zero on the number line. For instance, |-6| and |-20| are examples of negative numbers we encountered. When dealing with negative numbers, it’s important to remember:

• They decrease in value as they get further from zero. This means \(-20\) is smaller than \(-6\) even though \(|-20|\) is a larger non-negative number.
• Adding two negative numbers will result in a more negative number. Consider the temperature: \(-6^ ext{°C}\) added to \(-20^ ext{°C}\) makes it even colder, reaching \(-26^ ext{°C}\).

Understanding negative numbers is essential for managing everyday situations involving loss or decrease.

Absolute Value

The absolute value of a number is its distance from zero on a number line, regardless of direction. It’s noted using vertical bars like this: \(|-6|\).
Here are some important points about absolute value:

• The absolute value of any number is always non-negative.
• The absolute value of -6 is 6 because it is 6 units away from zero.
• The absolute value of -20 is 20 since it stands 20 units from zero.

When adding negative numbers like \(-6\) and \(-20\), we look at their absolute values (6 and 20). Adding these values gives us 26, and then we return to the negative sign to show direction. Hence, \((-6) + (-20) = -(6 + 20)\), which equals \(-26\).
In summary, absolute value helps us focus on the size of numbers when combining them, especially when negative signs come into play.

Arithmetic Operations

Arithmetic operations are essential building blocks in mathematics. They include addition, subtraction, multiplication, and division. When working with integers, understanding how these operations function helps avoid errors. Here's a closer look:

• Addition: When adding integers, it’s important to consider their signs. Two negative integers added together increase their ‘negativeness’ as shown in \((-6) + (-20)\). Adding means combining their absolute values, leading to \(-26\).
• Subtraction: This is often seen as adding the opposite. Subtracting a larger number from a smaller one results in a negative number, which flips depending on the integers' absolute values.
• Multiplication: Two negatives multiplied together result in a positive. This is contrary to addition, where the result stays negative.
• Division: Similar to multiplication, dividing two negatives results in a positive. But division by zero is undefined.

Understanding these operations, especially addition, is crucial when performing calculations with negative and positive numbers. Knowing the outcomes and patterns leads to more accurate and faster problem-solving in daily life.