Problem 22 - Elementary Algebra

Question

How should the real numbers be read ? (Write in words.) $$ -(-4) $$

Step-by-Step Solution

Answer

Question: Simplify the expression "-(-4)" using the double-negative rule. Answer: 4

Step 1: Understanding double negatives

In mathematics, a double negative means multiplying two negative numbers. When multiplying two negative numbers, the result is a positive number. This is because a negative number is the opposite of a positive number, and when two opposites are multiplied, the result is positive.

Step 2: Applying the double-negative rule to the expression

We have the expression "-(-4)". The expression inside the parenthesis is negative, and it's enclosed by another negative sign. To simplify this expression, we will apply the double-negative rule we discussed in step 1. Multiplying a negative number by another negative number results in a positive number. Therefore, we can rewrite the expression as: $$ -(-4) = (+4) $$ The result is a positive number. So, when reading the real numbers in the given expression, you can say, "Negative of negative four equals positive four."

Key Concepts

Double Negatives

Dealing with double negatives in mathematics is quite straightforward once you get the hang of it. When you see two negative signs in succession, think about it like an English sentence: "the opposite of the opposite." Just as in English where a double negative can imply a positive meaning, in math, "negative of a negative" turns the expression into a positive number.

• For example, in our expression \(-(-4)\), the first negative sign flips the -4 to its opposite, which is 4.
• Thus, \(-(-4)\) becomes \(+4\). The two negatives cancel each other out, resulting in a positive.

So, understanding double negatives is simply about recognizing the transformation from a set of negatives to a positive through multiplication. It's a handy rule that works every time!

Negative Numbers

Negative numbers are fundamental in mathematics and serve as a means of describing values less than zero. They often represent deficits or losses in real-world contexts.

• An easy way to think about negative numbers is to imagine a thermometer or a number line. If zero is your starting point, negative numbers are below or to the left of zero, representing colder temperatures or diminished values.
• In expressions like \(-4\), you are simply indicating a value 4 less than zero. It's like owing someone 4 dollars.

When dealing with negative numbers in calculations, remember that multiplying or dividing by a negative flips the sign of the result.
This understanding is crucial in working with more complex expressions, ensuring you apply the right logic to get the accurate outcome.

Reading Mathematics

Reading mathematics can feel like deciphering a new language, but it doesn't have to be daunting. To become comfortable, treat the symbols and numbers as if they were words.
This approach allows for better comprehension and execution of mathematical processes.

• When you see an expression like \(-(-4)\), read it in plain language. It could be stated as "the negative of negative four" or "the opposite of negative four."
• Breaking it down verbally can help you visualize the transformation happening within the expression.

Approaching math with the mindset of reading and interpreting a language can build confidence and clarity. You'll start seeing patterns and recognizing what operations or transformations are necessary, making mathematics a more approachable subject.