Problem 38 - Fundamentals of Mathematics

Question

Find the sums in the following 27 problems. If possible, use a calculator to check each result. $$ -14+14 $$

Step-by-Step Solution

Answer

The sum of -14 and 14 is 0.

Step 1: Understanding the Problem

We need to find the sum of the numbers -14 and 14.

Step 2: Identifying Opposite Numbers

Notice that -14 and 14 are opposite numbers. Opposite numbers are numbers that have the same magnitude but different signs.

Step 3: Adding Opposite Numbers

When you add two opposite numbers, the result is always zero. So, -14 + 14 = 0.

Step 4: Verify With a Calculator

If available, use a calculator to confirm that -14 + 14 equals 0 by entering the numbers and observing the result.

Key Concepts

Opposite Numbers

In basic arithmetic, opposite numbers are pairs of values that share the same absolute value but have different signs. For example, \(-14\) and \(+14\) are opposite numbers. Despite their differences in sign, their "magnitude" or absolute value is identical. This means they are the same distance away from zero on the number line.

• Definition: Opposite numbers always pair as a negative and a positive, such as \(-x\) and \(+x\).
• Zero Result: When you add opposite numbers together, their sum will always be zero (\(-x + x = 0\)).

Recognizing opposite numbers is useful not just in theoretical mathematics but also in practical scenarios where balancing values is necessary.

Sum Calculation

Calculating sums, especially small ones, is a fundamental part of basic arithmetic. The process involves finding the total when two or more numbers are added together. For instance, when adding the opposite numbers \(-14\) and \(+14\), you can determine the sum even without doing much calculation.

• Quick Check: Use the property of opposite numbers. Since adding \(+14\) cancels out the \(-14\), directly recognize that the sum is zero.
• Manual Addition: For other numbers, list them vertically or horizontally and add each digit starting from the rightmost to the left, considering any necessary "carrying" when needed.

Developing quick recognition skills for pattern sums, like those with opposite numbers, will speed up arithmetic computations and accuracy.

Using a Calculator

While mastering basic arithmetic mentally is valuable, calculators are handy tools for confirming calculations. Calculators provide a quick way to verify results and handle complex computations that might be challenging to solve manually.

• Verification: Calculate sums manually, then use the calculator to ensure accuracy. For example, to check \(-14 + 14 \), simply input the values into the calculator to confirm the result is zero.
• Complex Problems: For sums involving larger or more complex numbers, calculators save time and reduce potential errors.

By using calculators confidently and accurately, students can double-check their work and foster confidence in their arithmetic skills.