Problem 9 - Fundamentals of Mathematics

Question

Perform the indicated subtractions. $$ 1-(-12) $$

Step-by-Step Solution

Answer

The result is 13.

Step 1: Understand the Expression

The expression given is: \(1 - (-12)\). This is a subtraction problem where you are subtracting a negative number.

Step 2: Convert Subtraction of Negative to Addition

Remember that subtracting a negative is the same as adding its positive. Therefore, \(1 - (-12)\) becomes \(1 + 12\).

Step 3: Perform the Addition

Now simply add the numbers: \(1 + 12 = 13\).

Key Concepts

Negative Numbers

In mathematics, negative numbers are numbers less than zero. They are used to denote values below a certain reference point, often represented with a minus sign in front, like \(-1\), \(-12\), and so forth. Negative numbers are particularly useful when dealing with losses, depths below sea level, or temperatures below freezing. Even everyday situations like owing money in a bank account can involve negative numbers. They act as opposites to positive numbers and follow specific rules when it comes to arithmetic operations.

• Signs: A negative number has a "-" sign in front of it.
• Comparison: Negative numbers are always less than zero; for instance, \(-3\) is less than \(-2\).
• Usage: Often used in contexts where something is being taken away or is below a particular level.

Understanding how to handle negative numbers is essential in mathematics, making them a critical concept in many areas of study.

Addition

When you perform addition, you are essentially finding the total, or the sum, of two or more numbers. It's one of the basic operations in arithmetic and is usually the second step in problems involving subtraction of negative numbers. Let's explore a few fundamental aspects of addition:

• Commutative Property: This property tells us that numbers can be added in any order. For example, \(a + b = b + a\).
• Associative Property: When adding more than two numbers, you can group them differently without changing the outcome, e.g., \((a + b) + c = a + (b + c)\).
• Identity Property: Adding zero to any number doesn't change the number, meaning \(a + 0 = a\).

Breaking down subtraction of negative numbers into addition can sometimes simplify the calculation process. Recall that \(1 - (-12)\) turns into the addition problem \(1 + 12\). This transformation helps to understand and solve such problems.

Mathematical Operations

In mathematics, the four essential operations are addition, subtraction, multiplication, and division. These basic mathematical operations are foundational for solving various types of problems. Each has its own rules and properties that make problem-solving straightforward and effective.

• Addition: Combining numbers to get a sum.
• Subtraction: Finding the difference between numbers.
• Multiplication: Repeated addition of the same number; for example, \(4 \times 3 = 4 + 4 + 4\).
• Division: Splitting a number into equal parts or groups; e.g., \(12 ÷ 3 = 4\).

Understanding these operations, especially how they interact with negative numbers and each other, assists in performing calculations accurately. In the exercise, subtraction was handled by converting it into an addition problem, showing the interplay between different mathematical operations. This knowledge is invaluable, making math problems much easier to tackle!