Problem 25 - Fundamentals of Mathematics

Question

Find the sums in the following 27 problems. If possible, use a calculator to check each result. $$ (-3)+(-12) $$

Step-by-Step Solution

Answer

The sum of \(-3 + (-12)\) is \(-15\).

Step 1: Identify the Values

We have two values to add: - The first value is \(-3\).- The second value is \(-12\).

Step 2: Understand the Operation

The operation we are performing is addition, and both numbers are negative. In this scenario, we need to add their absolute values and then give the result a negative sign.

Step 3: Add the Absolute Values

Calculate the sum of the absolute values: \(3 + 12 = 15\).

Step 4: Apply the Negative Sign

Since both original numbers were negative, the result is also negative: \(-15\).

Step 5: Verify with a Calculator

Use a calculator to check:Enter \(-3\) + \(-12\) = \(-15\).The calculated result matches our solution.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are represented with a minus sign in front of them, like \( -3 \) or \( -12 \). In everyday life, they can denote losses or debt, like owing money. When adding negative numbers, imagine moving left on a number line. Each step to the left takes you further into negative territory. If you keep adding negative numbers, you continue moving left, deepening the negative value.

Exploring Absolute Values

The absolute value of a number is its distance from zero on the number line, regardless of direction. It’s always a non-negative value. For instance, both \( |3| \) and \( |-3| \) have an absolute value of 3. Absolute values are crucial in addition problems involving negatives. Here, you ignore the negative sign while summing up the values, then apply the negative sign back to the sum if both numbers were negative. This ensures the correct magnitude while preserving the sign of the original numbers.

Efficient Calculator Usage

A calculator is a handy tool for verifying sums, particularly when dealing with complex integers or when you're unsure of manual calculations. It's a good practice to double-check your work. To use your calculator for adding negative numbers, enter each number with its sign. For example, to verify \( -3 + (-12) = -15 \), input exactly as it appears: \-3\ + \-12\. Ensure you include negative signs for each number. It's essential to enter the sequence correctly to receive an accurate output.

Finding the Sum of Integers

Adding integers involves determining their net value, which could be positive, negative, or zero. When both numbers are negative, as in \( -3 \) and \( -12 \), consider their absolute values. Calculate the sum of the absolute values \( 3 + 12 = 15 \), then apply a negative sign to the result, because both integers are negative. This method is reliable: add magnitudes first when negative, then correct the sign of the result. This ensures your final answer reflects the true sum of the integers involved.