Problem 32 - Algebra for College Students

Question

Perform the following operations with real numbers. $$-16.3+19.6$$

Step-by-Step Solution

Answer

The result of \(-16.3 + 19.6\) is \(+3.3\).

Step 1: Identify the Numbers

The problem is asking us to perform addition with two real numbers: \(-16.3\) and \(+19.6\). One of these numbers is negative, and the other one is positive.

Step 2: Apply Addition with Different Signs

When adding a negative number and a positive number, we find the difference between their absolute values, which is similar to subtraction, and then assign the sign of the number with the larger absolute value. Here, the absolute values are \(16.3\) and \(19.6\).

Step 3: Calculate the Difference

Subtract the smaller absolute value from the larger absolute value: \[ 19.6 - 16.3 = 3.3 \]

Step 4: Determine the Sign of the Result

Since \(+19.6\) has a larger absolute value than \(-16.3\), the final result takes the sign of \(+19.6\), which is positive.

Key Concepts

Addition of Real Numbers

When we talk about the addition of real numbers, we consider all possible numbers on the number line. This includes both positive and negative numbers, fractions, and whole numbers. Adding real numbers can have different rules depending on the signs.

• If both numbers are positive, you simply add their values normally and the result is positive.
• If both numbers are negative, add their absolute values and the result is negative.
• If one number is negative and the other is positive, subtract the smaller absolute value from the larger one. The result will have the sign of the number with the larger absolute value.

It's important to understand these rules, as they are fundamental to arithmetic operations with real numbers.

Absolute Value

Absolute value is a concept that signifies the distance of a number from zero on the number line, without considering the direction. It is always a non-negative number. For example, the absolute value of both \( -5 \) and \( 5 \) is \( 5 \).To denote absolute value, we use vertical bars around the number, like this: \( |x| \). Some key properties of absolute value include:

• For any positive number \( a \, \ |a| = a \).
• For any negative number \( a \, \ |a| = -a \).
• The absolute value of zero is zero, \( |0| = 0 \).

Absolute value helps us determine how to add numbers with different signs by allowing us to focus only on their magnitude.

Negative Numbers

Negative numbers are those less than zero and they lie to the left of zero on the number line. They are usually represented with a minus sign, e.g., \( -3, -10.5 \), etc.When working with negative numbers, keep these principles in mind:

• Adding two negative numbers will result in a negative number. For example, \( -3 + -5 = -8 \).
• Subtracting a negative number is equivalent to adding its positive counterpart. For example, \( 5 - (-3) = 5 + 3 = 8 \).
• Multiplying or dividing two negative numbers results in a positive number, while multiplying or dividing a positive number by a negative number results in a negative number.

Understanding negative numbers is critical in performing arithmetic operations, especially when combined with positive numbers.

Positive Numbers

Positive numbers are greater than zero and they extend to the right of zero on the number line. They are generally represented without any sign. For instance, 4, 12.7, and so on.Working with positive numbers is quite straightforward:

• Addition: When you add any positive numbers together, the result is always positive. For example, \( 3 + 5 = 8 \).
• Subtraction: When you subtract a smaller positive number from a larger one, the result remains positive. For example, \( 10 - 3 = 7 \).
• Multiplication and Division: Multiplying or dividing positive numbers results in a positive number.

Positive numbers are the foundation of arithmetic operations and ensure that calculations can be managed easily without having to worry about direction on the number line.