Problem 38 - Intermediate Algebra

Question

Perform the following operations with real numbers. $$ 2.73-8.14 $$

Step-by-Step Solution

Answer

The result of the operation is -5.41.

Step 1: Understand the Problem

We need to perform a subtraction operation between two real numbers: 2.73 and 8.14.

Step 2: Set Up the Equation

Write down the equation as it is presented: \( 2.73 - 8.14 \).

Step 3: Change the Order to Simplify

Reorder the subtraction by observing that subtracting a larger number from a smaller number results in a negative number: \( -(8.14 - 2.73) \).

Step 4: Perform the Subtraction

Calculate \( 8.14 - 2.73 \) by subtracting the smaller number (2.73) from the larger number (8.14).Align the decimal points and subtract:\[\begin{array}{r}8.14 \-2.73 \\hline 5.41 \\end{array}\]

Step 5: Apply Negative Sign

As our initial equation was \( 2.73 - 8.14 \), the result will be negative. Therefore, our answer from step 4 becomes \(-5.41\).

Key Concepts

Subtraction of Real Numbers

Subtraction is one of the fundamental operations in mathematics, and it can be easily performed on real numbers, which include both rational and irrational numbers. When subtracting two real numbers, the operation is quite straightforward. The general idea is to determine how much larger or smaller one number is than the other.
To subtract numbers effectively:

• Align them correctly according to their decimal points, if they have any.
• Focus on subtracting each digit from the corresponding digit in the other number.
• If the initial number is smaller than the one being subtracted, the result will be negative.

Understanding these fundamentals aids in managing various problems involving subtraction, whether you're dealing with whole numbers, fractions, or decimals.

Negative Numbers

Negative numbers are numbers less than zero. They appear frequently in mathematics, often representing values below a reference point or deficit situations. When subtracting, if the first number in the operation is smaller than the second, the result is negative.
This was evident in the operation with 2.73 and 8.14, where the smaller number was 2.73. Subtracting a larger number from a smaller one results in a negative outcome:

• Negative numbers can be thought of as owing or lacking a certain value.
• In a number line, negative numbers are positioned to the left of zero.
• They're used to represent values like debt, temperature below freezing, or below sea level measurements.

Understanding negative numbers is crucial as they frequently appear in various mathematical and real-world contexts.

Decimal Subtraction

Decimal subtraction involves numbers with fractions represented by digits following a decimal point. Performing decimal subtraction requires careful alignment of these points to ensure accurate results.
Here's how to handle decimal subtraction:

• Always align the decimal points vertically.
• If necessary, add zeroes at the end of the shorter decimal to match the length of the longer one, but ensure these adjustments don't affect the value.
• Subtract each corresponding digit, starting from the rightmost decimal place moving left, borrowing from the next if needed.

In our example with 2.73 and 8.14, aligning the decimals was critical. After computing the difference, the result was determined to be negative since the smaller number (2.73) was subtracted from the larger one (8.14). Proper handling of decimal subtraction is essential for accurate numerical computations.