Problem 33 - Fundamentals of Mathematics

Question

For the following 6 problems, rewrite each number in simpler form. $$ -[-(-20)] $$

Step-by-Step Solution

Answer

The simplified form of -[-(-20)] is -20.

Step 1: Understanding the problem

We need to simplify the expression \[-[-(-20)].\] This expression involves nested negative signs and requires careful attention to the order of operations.

Step 2: Solve inner expression

The inner expression is \[-20.\] This is simply the number -20. This represents a negative value.

Step 3: Apply first outer '-' sign

Now we take the negative of the value obtained in the previous step, \[-(-20).\] Applying the '-' sign to -20 results in +20. So, this part simplifies to 20.

Step 4: Apply second outer '-' sign

Finally, we have \[-(20).\] This is simply -20 by negating the positive 20 we obtained previously. Hence, the original expression simplifies to -20.

Key Concepts

Order of Operations

When faced with a mathematical expression involving several operations, it is crucial to follow the correct order to arrive at the right answer. This is commonly known as the Order of Operations. You might recall this as PEMDAS or BIDMAS, which stands for Parentheses/Brackets, Exponents/Indices, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
This order ensures that everyone interprets and solves expressions the same way, leading to consistent results. In our original exercise, the expression involves nested negative signs, similar to how parentheses are usually handled. To simplify, we work from the inside out:

• First, resolve the innermost operation, which in our case, involves dealing with the inner negative sign.
• Then, proceed outward, applying each subsequent operation one at a time.

This approach ensures clarity in expressions, especially when multiple operations are involved, and helps us avoid misunderstanding complex expressions.

Negative Numbers

Negative numbers can sometimes confuse students, particularly when they are nested or occur multiple times within an expression. Simply put, a negative number is any number less than zero, often represented with a '-' sign. Understanding how to deal with negative numbers is essential in algebra and many areas of mathematics.
Here's what to remember:

• A negative times a negative is a positive. For example, \(-(-20) = 20\).
• However, a negative times a positive remains negative, so applying another negative to 20 gives us back -20 as in our exercise.

When simplifying nested negatives, work from the innermost part outward, methodically converting and applying negative signs as needed.

Mathematical Notation

Mathematical notation is a language all on its own. It uses symbols to represent numbers, operations, and other mathematical concepts. These symbols convey a lot of information quickly and concisely, provided you understand their meanings and how to interpret them.
For example, consider our exercise with the expression \(-[-(-20)]\):

• The negative sign \(-\) indicates a reversal of the sign; it can turn a positive into a negative or a negative into a positive, depending on how many times it occurs.
• Brackets or parentheses, as seen here, denote which operations should be completed first, similar to prioritizing tasks in a list.

Understanding and using these notations correctly helps in expressing complex ideas clearly and avoiding errors in calculations. Mastery of mathematical notation is foundational for success in higher levels of math and science.