Problem 63 - Prealgebra

Question

Simplify each of the following. $$-(-2)$$

Step-by-Step Solution

Answer

The expression simplifies to 2.

Step 1: Understanding the Negative Sign

The expression given is $$-(-2).$$ Here, the negative sign outside the parentheses applies to the expression inside, which is $$(-2).$$ This means we are looking at the opposite (or negative) of $$-2.$$

Step 2: Applying Double Negative Rule

In mathematics, negating a negative number results in a positive number. Thus, $$-(-2)$$ becomes $$+2.$$ This is because two negatives cancel out to make a positive.

Key Concepts

Double Negative Rule

The concept of the double negative rule might seem tricky at first, but it's actually quite simple when you break it down. In the world of mathematics, when you apply a negative sign to an already negative number, you are essentially canceling out the negatives, resulting in a positive number. This is because two negatives, when multiplied or applied in sequence, create a positive.

To illustrate this with our example, the expression \(-(-2)\) indicates that we need to find the opposite of negative 2. Here, the outer negative sign turns the direction around once again, making our final outcome positive. So, changing \(-(-2)\) to a positive 2 is exactly the action of this rule.

Think of it this way: if someone says "not unhappy," it means "happy." Similarly, negating a negative number makes it positive.

Simplifying Expressions

Simplifying expressions is a very important skill in mathematics. It involves reducing expressions to their simplest form. Such expressions could include operations with variables, numbers, or a combination of both.

Consider the simple expression \(-(-2)\). Initially, it might appear complex, featuring not just numbers but operations with them. However, by applying known rules, such as the double negative rule, we can reduce it to its simplest form of \(2\).

Simplifying involves steps such as:

• Recognizing and applying arithmetic rules like the double negative rule.
• Performing any indicated operations like multiplication or addition where applicable.
• Concluding with the most reduced form of the expression, without unnecessary symbols.

Mastering such techniques allows for easier manipulation and understanding of more complex problems.

Basic Arithmetic Operations

Basic arithmetic operations form the foundation of all mathematical computations. These include addition, subtraction, multiplication, and division. Each operation involves specific rules and properties. In the expression \(-(-2)\), we primarily focus on subtraction and its relationship to the double negative rule.

Subtraction in this context can be viewed as the addition of a negative number's opposite — or making it positive. Hence, \(-(-2)\) is akin to finding the opposite of \(-2\), which we know results in \(2\) through the application of arithmetic rules.

A few key points on basic arithmetic:

• Subtraction of a negative number equates to additional positives.
• Consistency with the rules, like negatives canceling out, keeps computation correct.
• Understanding these fundamental operations simplifies learning more advanced math.

By grasping these basics, students build a solid foundation for tackling more challenging mathematical concepts.