Problem 44 - Fundamentals of Mathematics

Question

Determine each value. $$ -(-|90|) $$

Step-by-Step Solution

Answer

The value is 90.

Step 1: Simplify the Absolute Value

The absolute value of any number is the number without regard to its sign. Thus, calculate the absolute value of 90 by removing any negative sign, if any, resulting in 90.\[ |90| = 90 \]

Step 2: Apply the Negative Sign

Next, apply the negative sign outside the absolute value. This involves changing the positive value to its negative form.\[ -(90) = -90 \]

Step 3: Apply the Outer Negative Sign

Finally, apply the outer negative sign to the result from Step 2. This converts the negative value back to positive.\[ -(-90) = 90 \]

Key Concepts

Negative Numbers

Negative numbers are numbers with a value less than zero. They are represented by a minus (-) sign placed in front of them. Here's how to understand and work with negative numbers:

• Identity: The operation of negative numbers is to decrease or go opposite to positive numbers.
• Arithmetic: When adding a negative number, you essentially subtract its absolute value, and when subtracting a negative, you add its absolute value.

It is crucial to remember that the negative of a negative number results in a positive. For instance, \(-(-x) = x\). Working with negative numbers is simple if you keep track of their signs and know how they interact with positive numbers.

Order of Operations

The order of operations is essential when solving mathematical expressions. It tells you which calculations to perform first to arrive at the correct result. Think of it as the grammar rules of math:

• PEMDAS/BODMAS: This acronym helps remember the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• Importance: Properly following these rules ensures that everyone solves expressions the same way, avoiding confusion and errors.

In the given exercise, the absolute value and negatives must be resolved following PEMDAS. First, evaluate the absolute value, then handle the negative signs in their order.

Mathematical Notation

Mathematical notation is a standardized set of symbols used to express numbers, functions, and operations clearly. Understanding these symbols is like learning a language that enables us to convey and solve mathematical problems effectively:

• Absolute Value: The symbol \(|x|\) represents the distance of a number from zero on the number line, irrespective of direction, meaning it is always positive or zero.
• Negative Sign: The negative sign \(-\) denotes numbers less than zero but also is used in operations to denote subtraction or the opposite of a number or a value.
• Nested Operations: Parentheses in expressions indicate which operations to perform first.

In the problem provided, understanding and applying mathematical notation correctly transformed a negative nested inside an absolute value into a positive number by recognizing the rules that govern these symbols.