Problem 68 - Prealgebra and Introductory Algebra

Question

Simplify. $$-(29)$$

Step-by-Step Solution

Answer

The simplified form of \(-29\) is \(+29\).

Step 1: Understand the problem

In mathematics, subtracting a number is the same as adding its negative. Hence, \(-29\) can be treated as \(-(29)\). When we take negative of negative number, it becomes positive.

Step 2: Calculate the value

Applying the rule that negative of a negative number is a positive number, we get \(+29\).

Key Concepts

Negative Number Arithmetic

Dealing with negative numbers can be challenging, but understanding a few basic principles can simplify the process. One of the fundamental ideas is that a negative number represents a value less than zero. It's essential to remember that adding a negative number is the same as subtracting its positive counterpart, and subtracting a negative number is equivalent to adding its positive counterpart. This concept is used when we have to evaluate expressions like \(-(-29)\).

To simplify this expression, let's recall that the negative of a negative number turns it into its positive equivalent. In other words, \(-(-29) = +29\) because the two negatives cancel each other out. This cancellation is due to the rule that multiplying two negative numbers results in a positive number. Since every negative number can be thought of as being multiplied by negative one, \(-1 \times -1 = 1\), the process of 'negating a negation' essentially multiplies two negative ones together, leading us to a positive result.

Subtracting Negative Numbers

When it comes to subtracting negative numbers, it can initially seem a bit counterintuitive. To make it easier, visualize a number line where moving to the left indicates subtracting and moving to the right signifies adding. If you subtract a negative number, it is similar to adding the distance to the right instead of to the left. This is why subtracting a negative number is akin to addition.

So in our exercise, seeing \(-(29)\) makes one think of subtraction, but since 29 is already negative, it is as if we are subtracting a negative number, which ultimately means we are moving to the right on our number line, thus adding 29. Hence, \(-(29) = +29\). This seemingly odd rule makes operations with negative numbers consistent and orderly within the broader mathematical system, and it is an important concept in algebra.

Algebraic Rules

Algebraic rules are the backbone of simplifying expressions and solving equations. One of the crucial rules is the handling of negative numbers as shown in the exercise. Another critical principle is the distributive property which allows for expressions like \(a(b + c)\) to be expanded to \(ab + ac\). When dealing with negative numbers, if we have \(-(-a)\), we distribute the negative sign into the parenthesis, effectively inverting the sign of the number inside. As seen in the previous section, the application results in simply \(a\).

These are just a couple of examples of algebraic principles at play. There are many such rules designed to maintain order and provide consistent results across various mathematical operations. Remember, the goal of these rules is not to complicate but to provide a clear path to simplification and solution of algebraic problems.