Problem 15 - Fundamentals of Mathematics

Question

Find the sums. $$ 15+(-10) $$

Step-by-Step Solution

Answer

The sum is 5.

Step 1: Understand the Numbers

We have two numbers here: 15 and -10. The 15 is a positive number, while -10 is a negative number.

Step 2: Add Positive and Negative Numbers

When adding a positive number and a negative number, we actually perform subtraction. We subtract the smaller absolute value from the larger absolute value. So, we calculate \( 15 - 10 = 5 \).

Step 3: Determine the Sign of the Result

Since the positive number (15) has a greater absolute value than the negative number (10), the sign of the result is positive. Thus, the sum is 5.

Key Concepts

Positive and Negative Numbers

Understanding positive and negative numbers is crucial in math. These numbers help us describe values and their directions. Positive numbers are greater than zero, usually without a symbol or with a plus sign. For example, 15 is a positive number and can be also written as +15.
Negative numbers, however, are less than zero and always carry a minus sign, like -10.

• Positive numbers are located to the right of zero on the number line.
• Negative numbers are found to the left of zero.

When combining positive and negative numbers, their relationship dictates whether you will add or subtract to find a result.

Absolute Value

The absolute value of a number represents its distance from zero, without considering its sign. It is always a non-negative number. This is because distance cannot be negative.
For example, the absolute value of both 15 and -15 is 15. We write this mathematically using vertical bars, like \(|15| = 15\) and \(|-15| = 15\).

• To find the absolute value of a positive number, you just take the number itself.
• For a negative number, you drop the negative sign.

In integer addition involving a positive and a negative number, understanding absolute values helps determine which number has a greater effect, assisting us in finding the sum's sign.

Subtraction

Subtraction is the operation of finding the difference between numbers. When adding a positive number and a negative number, you essentially perform a subtraction calculation.
This process involves subtracting the smaller absolute value from the larger one to determine the size of the result.
Consider the example of adding 15 and -10.

• First, identify the absolute values: \(|15| = 15\) and \(|-10| = 10\).
• Next, subtract the smaller absolute value from the larger: \(|15 - 10| = 5\).
• Finally, since 15 has the greater absolute value, the sign of the result remains positive.

Thus, the operation \(15 + (-10)\) simplifies to \(15 - 10\), resulting in the sum of 5.