Addition of units requires two counts of units, the number of units to the left of the addition sign and the number of units to the right of the addition sign. This we say addition is a binary operation requiring two quantities. A single number can only be added to another number. It cannot be added to itself because it cannot be both itself and another. For example you can add six eggs to another six eggs and get 12 eggs altogether. However you cannot add six eggs to itself. Prove you can't add a number of units to itself.

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Addition is fundamentally a binary operation, which means it requires exactly two operands to function. The term binary comes from the Latin word meaning two. For any addition operation to be valid, we need two distinct quantities that serve as our operands. Consider the simple example of three plus five. Here we have our first operand, the number three, and our second operand, the number five. These are two separate, distinct quantities that can be combined through addition to produce the result eight. The key principle here is that both operands must be present and they must be distinct from each other. To understand why self-addition is impossible, we must examine the concepts of mathematical identity and distinctness. Mathematical identity states that everything equals itself - a equals a. However, distinctness is equally important: if a is not equal to b, then a and b are distinct entities. For addition to work, we need two distinct operands. Consider six red balls and six blue balls. Even though both groups contain six objects, they are distinct quantities because they are separate entities. The red balls are one group, and the blue balls are another group. This distinctness allows us to add them together. But a single entity cannot be both itself and another entity simultaneously - this creates a logical impossibility that prevents true self-addition. The self-addition paradox reveals the fundamental logical contradiction in trying to add a number to itself. For any addition operation a plus b, we require that a and b are distinct entities. However, self-addition would imply adding a to a, where both operands refer to the exact same entity. This violates the distinctness requirement that makes addition possible. Consider the egg example: when we say six eggs plus six eggs equals twelve eggs, we're actually adding two distinct groups of six eggs each, not adding one group of six eggs to itself. The first group and the second group are separate entities, which is why the addition is valid. But a single group of six eggs cannot be added to itself because it cannot simultaneously be both the first operand and the second operand. This creates a logical impossibility that proves true self-addition cannot exist. Now let's examine the formal mathematical proof that demonstrates why self-addition is impossible. The proof follows a clear logical structure. First, we establish the premise that addition requires two operands a and b where a is not equal to b - this is the distinctness requirement. Second, we consider the assumption of self-addition, which would require adding a plus a, where both operands are identical. Third, we identify the logical contradiction: this violates the distinctness requirement from our initial premise. From set theory perspective, addition combines two separate sets, but a single set cannot be combined with itself because it cannot simultaneously exist as two different sets. This creates a fundamental contradiction. Therefore, we conclude that self-addition is undefined and mathematically impossible. The formal proof demonstrates that the very definition of addition prevents any number from being added to itself. It's crucial to clarify common misconceptions about self-addition. When we write six plus six equals twelve, we are not adding six to itself. Instead, we are adding one instance of six to another instance of six. These are two distinct entities that happen to have the same numerical value. The key distinction is between invalid self-addition, where one group tries to add to itself, and valid addition of identical quantities, where two separate groups of the same size are combined. Think of it as having two different boxes, each containing six items. We can add the contents of box one to the contents of box two because they are separate entities. Additionally, it's important to note that multiplication, such as two times six equals twelve, is conceptually different from addition, even though it may yield the same numerical result. Mathematical operations have precise definitions that prevent logical contradictions and ensure consistency in our mathematical system.