Let's consider the statement given in the question:
"The ____ Identity Element defines [tex]\( 0 \)[/tex] as such that [tex]\( a + 0 = 0 + a = a \)[/tex] for all real numbers."
To understand and solve this problem, let's break it down step-by-step:
1. Understanding Identity Elements:
- An identity element in mathematics is an element that, when combined with another given element in an operation, leaves that given element unchanged.
2. Types of Identity Elements:
- Additive Identity: In the context of addition, the identity element is [tex]\( 0 \)[/tex]. This means that when you add [tex]\( 0 \)[/tex] to any number [tex]\( a \)[/tex], the result is [tex]\( a \)[/tex]. Mathematically, it is expressed as:
[tex]\[
a + 0 = 0 + a = a
\][/tex]
- Multiplicative Identity: In the context of multiplication, the identity element is [tex]\( 1 \)[/tex]. When you multiply any number [tex]\( a \)[/tex] by [tex]\( 1 \)[/tex], the result is [tex]\( a \)[/tex]. Mathematically, it is expressed as:
[tex]\[
a \times 1 = 1 \times a = a
\][/tex]
3. Analyzing the Given Statement:
- The statement specifies the operation of addition and mentions that [tex]\( 0 \)[/tex] is the identity element that satisfies the equation [tex]\( a + 0 = 0 + a = a \)[/tex].
4. Identifying the Relevant Type of Identity:
- Since the operation in the question is addition and we want to find the type of identity element, we look at the definition of the additive identity. For addition, the identity element [tex]\( 0 \)[/tex] fulfills the given equation.
5. Options Provided:
- Additive
- Inverse
- Associative
6. Determination:
- The option "Additive" correctly describes the identity element for the operation of addition.
Therefore, the best answer for the given question is:
Additive
