Problem 115 - Elementary Algebra

Question

Simplify. $$ -(-118911) $$

Step-by-Step Solution

Answer

118911

Step 1: Understanding Negative of a Negative

In mathematics, the negative of a negative number becomes positive. This is because the operation of negation reverses the sign of a number. Thus, we have \[-(-a) = a\] for any number \(a\).

Step 2: Applying the Concept

Apply the rule to the given expression, which is \[-(-118911)\]. According to the rule we stated, the negative of \(-118911\) becomes \[118911\].

Key Concepts

Negation

Negation is a fundamental concept in mathematics, particularly when dealing with numbers. In simple terms, negation involves reversing the sign of a number. For instance, if you start with a positive number and apply negation, the result is a negative number. Conversely, if you begin with a negative number and negate it, you end up with a positive number. This can be thought of as a sort of 'flipping' of the number's sign.
To further illustrate:

• If you negate +5, you get -5.
• If you negate -3, you end up with +3.

The principle behind negation is that it 'cancels out' or reverses the existing sign of the number in question.

Sign Reversal

Sign reversal is closely related to negation and is a crucial operation when working with negative numbers. When a sign reversal occurs, the numerical value remains unchanged, but its sign switches from positive to negative, or vice versa. This concept helps in simplifying expressions, especially those that contain multiple negative signs.
Consider the expression \(-(-a)\). Applying a sign reversal twice will bring you back to the original sign, making \(-(-a) = a\). This is because each negative sign acts like an inversion, or a flip in the direction of the number's sign. In more straightforward terms:

• The first negative sign turns the number opposite of its original.
• The second negative sign flips it back to the original sign.

By understanding sign reversal, you can more easily simplify complex mathematical expressions that involve multiple layers of negation.

Positive Numbers

Positive numbers are numbers that are greater than zero and can be found to the right of zero on a number line. They are as natural to math as breathing is to living. Positive numbers can be written either without a sign or with a '+' sign, and they indicate an increase, gain, or simply a counting number. In math problems, when you simplify expressions, outcomes often resolve to positive numbers, especially when negative signs cancel each other out.
For instance:

• Adding two positive numbers always results in a positive sum.
• Subtracting a smaller positive number from a larger one keeps the result positive.

In practical scenarios, positive numbers depict a variety of things like profits, distances, heights, and temperatures above zero, essentially representing anything that can increase or grow. Understanding basic operations and manipulations involving positive numbers is foundational to mathematics.