Problem 13 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each sum without the use of a number line. $$-30+(-30)$$

Step-by-Step Solution

Answer

-60

Step 1: Understanding what adding two negative numbers means

Adding two negative numbers together is equivalent to further going down the negative number scale. Negative numbers get 'larger' (or 'more negative') as we move left on the number line (though we are not using a number line for this problem, it's good to visualize). So, adding -30 and -30 means getting more negative by 30 twice.

Step 2: Performing the addition

To add -30 and -30, simply add the magnitudes together (that's 30 + 30 = 60), and then apply the negative sign to the result, because we are getting 'more negative' by both amounts. So, the sum is -60.

Key Concepts

Integer Addition

Adding integers, particularly negative ones, might seem tricky at first. But with practice, it becomes second nature. When two negative numbers are added, their influence intensifies the negativity. Imagine moving further left on a number line if you were using one - adding two negative numbers means going further into the negatives.

For example, in our problem with \(-30 + (-30)\), consider the following:

• The negative sign indicates a direction opposite to positive, moving left on the imaginary number line when visualized.
• Adding two negative integers means increasing the 'negative distance' without reversing the direction.

Thus, the sum \(-30 + (-30)\) results in \(-60\), doubling the negativity effectively.

Arithmetic Operations

Arithmetic operations with negative numbers follow simple rules but require careful attention. Each operation has unique features to remember, especially when negatives are involved.

When dealing with addition:

• Identify if you are dealing with positive or negative integers.
• Adding two positives gives a positive, while two negatives give a negative.
• If the signs differ, subtract the smaller magnitude from the larger, and keep the sign of the larger magnitude.

In our example of \(-30 + (-30)\), both numbers are negative. The operation becomes straightforward: sum the absolute values \(30 + 30 = 60\), then apply the negative, resulting in \(-60\).

Understanding these operations simplifies calculations involving any integers, improving math fluency.

Mathematical Reasoning

Mathematical reasoning helps develop a logical approach to solving problems like integer addition. It involves analyzing the situation and making sound conclusions based on concepts you've learned.

For example, when faced with \(-30 + (-30)\):

• Recognize that adding two negatives will yield a more significant negative value.
• Consider the principle that the number further from zero in the negative direction has a larger absolute value.
• Calculate step-by-step by first working out the absolute values and then applying the negative sign.

This reasoned approach ensures accuracy and builds confidence in handling various mathematical problems. It's about seeing beyond the calculations to understand their underlying logic and implications.