Problem 45 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ 20-(-8) $$

Step-by-Step Solution

Answer

Answer: 28

Step 1: Rewrite the expression with double negative

Rewrite the given expression as the addition of the opposite of the negative number. In this case, we have: $$ 20 - (-8) = 20 + 8 $$

Step 2: Perform the addition

Now, we will simply add the two numbers to find the value of the expression: $$ 20 + 8 = 28 $$ So, the value of the given expression $$20 - (-8)$$ is $$28$$.

Key Concepts

Addition of Opposites

When working with integers, a fundamental concept to understand is the addition of opposites, also known as additive inverses. An additive inverse is a number, which when added to a given number, will result in zero. Essentially, it is the 'opposite' of a number. In arithmetic, when facing a double negative such as \( -(-a) \), it simplifies to just \( a \). This is because adding the opposite of a negative number is like adding the positive counterpart of that number.

For instance, in the expression \( 20 - (-8) \), the two negatives effectively cancel each other out, and it changes to \( 20 + 8 \), turning subtraction into addition. Simplifying the double negative makes solving such expressions straightforward.

Arithmetic Operations

Arithmetic operations are the building blocks of math, including addition, subtraction, multiplication, and division. It's integral to master these operations to work with various types of numbers accurately. When processing expressions with negative numbers, it's critical to be familiar with the rules of these operations.

For example, subtraction can be thought of as the addition of a negative number. So when you see an expression like \( 20 - (-8) \), it's helpful to reimagine it as \( 20 + (+8) \), which is a typical addition problem. Understanding how to convert a subtraction into addition by identifying the additive inverse is crucial in simplifying the expression and reaching the correct answer.

Negative Numbers

Negative numbers are a concept that can initially perplex students, but they are simply numbers with a value less than zero, indicated by a minus (-) sign. They arise naturally when discussing debts or temperatures below zero, for example.

Working with negative numbers involves certain rules—particularly when combined with other negative or positive numbers. The rule of thumb is that two negatives make a positive. This is especially important when dealing with double negatives in an expression, such as in \( 20 - (-8) \), where the double negative turns into a positive operation. This transformation is an excellent example of how negative numbers interact within arithmetic operations to alter the outcome of an expression.