Problem 7 - Fundamentals of Mathematics

Question

Find the sums. $$ -2.8+(-4.6) $$

Step-by-Step Solution

Answer

The sum is \(-7.4\).

Step 1: Identify the Numbers Involved

In the given problem, the numbers involved in the sum are \(-2.8\) and \(-4.6\). Both numbers are negative.

Step 2: Addition of Negative Numbers

When adding two negative numbers, the sum is also negative. The absolute values of the numbers are summed, and the result is given a negative sign.

Step 3: Add Absolute Values

Determine the absolute values: \(|-2.8| = 2.8\) and \(|-4.6| = 4.6\). Add these absolute values: \(2.8 + 4.6 = 7.4\).

Step 4: Apply the Negative Sign

Since both original numbers were negative, the sum is negative. Therefore, add the negative sign to the result from Step 3. The sum is \(-7.4\).

Key Concepts

Understanding Absolute Value

The absolute value of a number is its distance from zero on a number line, regardless of direction. It is always a positive value or zero because it measures magnitude without considering direction. For example, the absolute value of -2.8 is written as | -2.8 | and equals 2.8. Similarly, | -4.6 | equals 4.6. Absolute values help simplify calculations involving negative numbers by allowing us to focus on the size of the numbers rather than their direction.

• Absolute value is always non-negative.
• It represents distance, not direction.
• It is useful for understanding magnitude differences.

Recognizing and using absolute values can make complex number operations, like adding negative numbers, easier to understand.

The Role of the Negative Sign

The negative sign indicates that a number is less than zero, showing its position relative to zero on a number line. Just like a positive sign, it affects how we interpret numbers, especially when performing arithmetic operations. When adding negative numbers, understanding how the negative sign alters the result is crucial:

• A negative sign before a number flips its direction from positive to negative on the number line.
• In operations like addition, it integrates into calculations by affecting the final outcome, typically resulting in a shift further from zero.
• Combining two negative numbers through addition results in a sum that is further away from zero, thus still negative.

Grasping the influence of negative signs will improve clarity, particularly when incorporating them in number operations.

Mastering the Addition of Decimals

Adding decimals involves aligning the decimal points and summing each column of numbers sequentially, similar to whole numbers. This process requires careful attention to decimal places to ensure accuracy. Let's go through the steps applied to our earlier problem of adding -2.8 and -4.6:

• First, consider the absolute values: both 2.8 and 4.6.
• Align decimal points: place 2.8 and 4.6 directly beneath each other ensuring decimal points are in a column.
• Add the numbers as if they were whole numbers - starting from right to left, carry if necessary: 2.8 + 4.6 results in 7.4.
• Add the negative sign back to the result, since both original numbers were negative: the sum is -7.4.

This method of handling decimals ensures you maintain integrity in calculations, producing precise results every time you perform decimal operations.