Problem 32 - Fundamentals of Mathematics

Question

Rewrite each expression in simpler form. $$ -(-4) $$

Step-by-Step Solution

Answer

The simpler form of \(-(-4)\) is 4.

Step 1: Identify the Double Negation

The given expression is \(-(-4)\). This means we need to simplify the expression by considering the effect of having a double negative sign. Remember that negating a negative number results in a positive number.

Step 2: Apply the Double Negation Rule

Applying the rule of double negation, negative of negative \(-(-4)\) results in a positive. Therefore,\(-(-4) = 4\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a key skill in mathematics that involves reducing a mathematical expression to its simplest form. This process can make calculations easier and help in solving equations more efficiently. One common scenario encountered in simplification is when there's a double negation, like in the expression \(-(-4)\). A double negation means that you have two negative signs affecting the same number. The rule here is straightforward: two negative signs cancel each other out, resulting in a positive sign. By recognizing and applying this rule, you can turn complicated-looking expressions into something much more manageable. In our example, the double negation \(-(-4)\) simplifies to simply \(4\). Understanding this concept helps in tackling more complex problems involving variables and coefficients as it builds the foundation of mathematical manipulation.

Negative Numbers

Dealing with negative numbers is an essential part of learning basic arithmetic. Negative numbers are numbers less than zero. When a negative sign is applied to a number, it essentially indicates the opposite direction on the number line from zero.
It's also important to remember the rules about combining negatives with other negative numbers. When you have two negative signs acting on the same number, as seen in our expression \(-(-4)\), these two negatives invert each other. So, mathematically speaking, negating a negative number makes it positive. For instance, when you have \(-(-4)\), these two negative signs cancel out, revealing \(4\) as the result.
Understanding how negative numbers interact is crucial not just in arithmetic, but also in algebra where you'll be handling them with variables.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. These are the operations that serve as the foundation of all mathematics, including more complex areas like algebra and calculus.
An important point to remember in arithmetic is the interplay between operations, such as how subtraction can be viewed as the addition of a negative number. For example, the expression \(3 - 4\) can also be written as \(3 + (-4)\).
Another key concept is understanding the impact of double negation, which effectively translates into the addition of a positive number. In the expression \(-(-4)\), for instance, the subtraction of a negative 4 becomes the addition of a positive 4, resulting in \(4\).
Grasping these basic operations and their nuances prepares students for solving more comprehensive problems by allowing them to rearrange and simplify expressions for easier computation.