Problem 30 - Fundamentals of Mathematics

Question

For the following 6 problems, rewrite each number in simpler form. $$ -(-2) $$

Step-by-Step Solution

Answer

The simplified form of \(-(-2)\) is 2.

Step 1: Understand the Double Negative

When you see a double negative in a mathematical expression, such as \(-(-2)\), it means that you need to negate the number twice. Each negation changes the sign of the number.

Step 2: Apply the First Negation

The expression \(-2\) indicates that the 2 is negative. This means we are one step away from the positive equivalent.

Step 3: Apply the Second Negation

Applying the negation twice, the negative of a negative number becomes positive. Thus, \(-(-2)\) simplifies to \(2\).

Key Concepts

Simplifying Expressions

When we talk about simplifying expressions, we are referring to the process of reducing a mathematical expression to its simplest form. This involves performing operations like addition, subtraction, multiplication, or division that make the expression easier to understand and work with.

In the case of the expression \[ -(-2) \], simplifying involves identifying and removing unnecessary operations. Double negatives, such as two negative signs as found here, can complicate the expression. By recognizing their effect, you can rewrite the expression in a simpler, more concise form.

Here are some steps you typically follow when simplifying expressions:

• Look for operations, like double negatives, that can be simplified.
• Apply rules of arithmetic to consolidate or eliminate_terms.
• Represent the result in its simplest numerical or algebraic form.

In our example, \[-(-2)\] simplifies to \[2\], which is a clearer and more straightforward representation of the value.

Negation

Negation in mathematics is essentially the process of changing the sign of a number. In other words, negating a number converts a positive number to a negative, and a negative number to a positive.

In the expression \[-(-2)\], understanding negation is crucial. The first negation changes positive \[2\] to negative \[-2\]. The second negation reverses this effect, turning the expression back to positive again. This is why we end up with simply \[2\].

It can be helpful to consider basic rules of negation:

• The negation of a positive number is negative.
• The negation of a negative number is positive.
• Negating a number twice restores its original value.

These rules allow us to simplify expressions efficiently by understanding how sign changes affect numbers.

Mathematical Sign Change

A sign change involves switching a number's positive or negative sign. This occurs in various scenarios, such as when we perform negation. In equations or expressions, sign changes have significant roles in determining the final result.

In the case of \[-(-2)\], the double negative leads to two sign changes:

• The first minus sign transforms \[2\] into \[-2\].
• The second minus sign switches \[-2\] back to \[2\].

Each sign change plays a crucial role in interpreting the outcome of the expression.

Understanding the effect of each sign flip helps to clarify why the expression simplifies to a positive \[2\]. In essence, every negative sign encountered in such scenarios flips the current sign of the number.