Problem 31 - Fundamentals of Mathematics

Question

For the following 6 problems, rewrite each number in simpler form. $$ -(-16) $$

Step-by-Step Solution

Answer

The simplified form is 16.

Step 1: Understand the Negative Sign

First, recognize the two negative signs before the number 16. The first negative sign directly affects the -16, making it a positive.

Step 2: Simplify Using Negative Sign Rule

Recall the rule that states that two negative signs make a positive. Thus, \[-(-16) = +(16) = 16.\]

Step 3: Finalize the Simplification

Conclude that the expression simplifies to 16, since removing the double negative simply results in a positive number.

Key Concepts

Understanding Negative Numbers

Negative numbers can sometimes be tricky, especially when you see multiple negatives together. It's helpful to think of negative signs as directions on a number line. When you have one negative, you move left; when you have two negatives together, you actually move right because two negatives make a positive.

In math, this idea is expressed through the rule: multiplying or combining two negatives results in a positive. So, if you see \(-(-16)\), you're really turning the negative direction around. This means you're effectively looking at the positive 16.

Always remember that two negatives will cancel each other out in this way when they're together.

Simplifying Expressions with Negatives

Simplifying expressions involving negative numbers follows a systematic approach. Each negative sign can be treated as a directional shift or operation.

• Identify Negatives: Start by counting the negative signs in the expression.
• Apply the Negative Rule: Use the rule that two negatives make a positive.
• Reassess the Expression: After simplification, reassess the expression for any other operations that need to be applied.

In our example, \(-(-16)\), there are two negative signs. Using the rule, it becomes a positive 16. Simplifying is like cleaning up, taking away the excess to see the core number or operation.

This rule makes dealing with negative signs easier because you can consistently apply it to unravel any combination of negatives in an expression.

Important Mathematical Rules

Many mathematical rules help define how numbers interact. Understanding them can simplify many processes. With negatives, some key rules include:

• Double Negative Rule: Two negatives combine to form a positive. This helps simplify problems where multiple negative signs appear.
• Order of Operations: Remember to follow the established order of operations (PEMDAS/BODMAS) when handling expressions to ensure accurate results. This includes tackling parentheses, exponents, and then multiplication/division followed by addition/subtraction.

Using these principles smartly can unlock efficient problem-solving techniques. In the case of negatives, always look for opportunities to apply these rules, as they can make complex expressions manageable. By aligning your steps with these rules, you can systematically and accurately simplify any expression.