Problem 23 - Elementary Algebra

Question

How should the real numbers be read ? (Write in words.) $$ -(-1) $$

Step-by-Step Solution

Answer

Answer: The expression -(-1) can be read as "the negative of negative one" or "the opposite of negative one." The simplified form of the expression is 1.

Step 1: Understanding the expression

In the given expression, -(-1), we have a double negation. The negative sign inside the brackets is applied to -1, and the negative sign outside the brackets is applied to the whole expression.

Step 2: Reading the expression in words

The expression $$-(-1)$$ can be read as "the negative of negative one" or "the opposite of negative one." In either case, the meaning is that we are negating the negative value, which results in a positive value.

Step 3: Simplifying the expression

Since a double negative results in a positive value, the simplified expression is $$1$$.

Key Concepts

Understanding Double Negation

Double negation in mathematics might seem tricky at first, but it's actually quite simple. When you have two negative signs applied to a number or variable, they cancel each other out. This is commonly encountered in expressions like \(-(-1)\).

In this expression, the first negative sign turns a positive into a negative, while the second negative changes a negative back into a positive.
Think of it as the mathematical equivalent of reversing direction twice on a road: you end up going the original way you started.

Key details to remember:

• Two negatives make a positive.
• Applicable to real numbers and variables.
• Think of it as "undoing" a negation.

Approaching double negation with these points in mind can help demystify complex expressions and provide clarity in solving problems.

Simplifying Expressions

Expression simplification is a crucial skill in algebra. It involves transforming an expression into a simpler or more concise form. In the case of expressions like \(-(-1)\), simplifying is about recognizing patterns and applying rules.

The expression begins with a double negation, which we previously noted cancels itself out, leaving us with a positive result: 1.
Steps to simplify any expression include:

• Identify and resolve any double negations.
• Perform arithmetic operations as necessary.
• Combine like terms, if applicable.
• Check and verify the simplified expression is equivalent to the original.

Facilitating these steps helps make complex equations manageable, ensuring accuracy and enhancing problem-solving skills.

Working with Negative Numbers

Negative numbers are integral to mathematics. They extend the number line below zero and are essential in various contexts, from accounting to physics.

When dealing with negative numbers, it’s important to understand the effect of negative signs. A single negative sign changes a positive number to its opposite, a negative one. For example, the number \(-1\) is simply negative one.
Double negatives, as mentioned before, will revert the number to its positive form.

Important facts about negative numbers:

• Represent a loss or absence of quantity.
• Used in temperature scales, elevation measurements, etc.
• Follow specific arithmetic rules (e.g., multiplying two negatives results in a positive).

Handling negative numbers confidently enables you to solve a wide range of mathematical problems effectively.