Problem 90 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ 3-14 $$

Step-by-Step Solution

Answer

The result of \(3 - 14\) is \(-11\).

Step 1: Understand the Operation

The problem provided is a simple subtraction operation: \(3 - 14\). This means you are subtracting 14 from 3.

Step 2: Perform the Subtraction

Since we cannot subtract a larger number (14) from a smaller number (3) without resulting in a negative number, find the result by calculating: \(3 - 14 = -11\).

Step 3: Result Interpretation

The result of the subtraction is \(-11\), meaning the original three units are short by eleven units to reach 14.

Key Concepts

Subtraction

Subtraction is one of the four basic arithmetic operations. It involves taking away a certain number from another number. In the exercise given, you see the subtraction of 14 from 3, or in other terms, how much more 14 is than 3.
To perform subtraction, start with the minuend (the number you subtract from) and subtract the subtrahend (the number you are taking away). The result is called the difference. Here, 3 is the minuend, and 14 is the subtrahend.

• Structure of Subtraction: Minuend - Subtrahend = Difference
• Example: In the equation 3 - 14, 3 is diminished by 14, resulting in a negative difference.

Understanding subtraction is crucial as it forms the basis for algebraic manipulation and deepens comprehension of number systems in mathematics.

Negative Numbers

Negative numbers are numbers less than zero. When subtracting a larger number from a smaller one, as in the case of 3 - 14, you end up with a negative number. Negative numbers lie to the left of zero on the number line, indicating values less than zero. When interpreting these values:

• A negative number indicates a deficiency, distance below a boundary, or reverse expectation. In practical terms, if you have only 3 units but need 14, you are missing 11 units, hence the result -11.
• Negative numbers are used in various fields such as economics, where deficits occur, or temperatures below freezing point.

Handling negative numbers can initially seem tricky, but it becomes intuitive with practice, especially in arithmetic operations like subtraction.

Integer Arithmetic

Integer arithmetic refers to mathematical operations involving integers, which include positive numbers, negative numbers, and zero. This concept is foundational in mathematics, ensuring students understand how to handle numbers across the spectrum from negative through positive.
In the realm of integer arithmetic, the rules slightly adjust, notably with operations like subtraction:

• Subtraction Rule: When subtracting, if the second number is larger, the result will be negative. For example, 3 - 14 results in -11.
• Addition of Negatives: Adding a negative is akin to subtracting a positive, thus 3 + (-14) delivers the same answer.

By comprehending integer arithmetic, students develop an analytical mindset crucial for higher mathematics fields such as algebra, calculus, and beyond. Thorough familiarity with these rules aids in solving complex problems seamlessly.