Problem 87 - Algebra 2

Question

Name the property of real numbers illustrated by each equation. $$ 3+(-3)=0 $$

Step-by-Step Solution

Answer

Additive Inverse Property.

Step 1: Identify the operation

The equation \(3 + (-3) = 0\) shows that a number added to its opposite (negative) gives zero.

Step 2: Name the property

This is the \(\textbf{Additive Inverse Property}\) (also called the Inverse Property of Addition): for every real number \(a\), \(a + (-a) = 0\).

Key Concepts

Understanding Real Numbers

The world of mathematics comprises various types of numbers, each with its unique properties. Among them, real numbers are perhaps the most utilized in everyday life. Whether measuring distance, calculating expenses, or even determining temperature, real numbers are always at play.

Real numbers include the set of all rational and irrational numbers. This means that every fraction, decimal that terminates or repeats, and even numbers with non-repeating, non-terminating decimals (such as \(\pi\)) fall under the umbrella of real numbers. In the context of the exercise above, when we say that \(3\) and \( -3\) are real numbers, we acknowledge that they can be found on the number line and are subject to the same rules and properties that govern all real numbers, such as the Additive Inverse Property.

The Role of Additive Identity

A key player in understanding the behavior of real numbers in addition is the additive identity. This concept revolves around the number zero, which is special in the world of mathematics due to its neutral effect in addition. The additive identity property states that any real number added to zero will result in the original number. This is expressed as \( a + 0 = a \) or \( 0 + a = a \), confirming that zero is the only number that, when added to another number, doesn't change its value.

In our original equation \(3+(-3)=0\), the sum of the number and its inverse results in zero, which aligns with the definition of the additive identity. This relationship highlights the power of zero within the realm of real numbers.

Algebraic Properties at Play

Algebra is filled with a variety of properties that govern the operations and interactions of numbers. Algebraic properties are the rules that allow us to manipulate terms and equations in consistent and logical ways. These include distributive property, commutative property, associative property, and of course, the Additive Inverse Property, as seen in our example.

The Additive Inverse Property is particularly profound because it ensures that for every real number, there's a way to return to the additive identity of zero by adding the number's opposite. This property is essential in solving equations, simplifying expressions, and understanding the symmetry of the number line. It's a cornerstone of algebra that allows us to 'undo' addition, thus maintaining the balance in an equation.