Like addition, multiplication is also a binary operation requiring two operands. Multiplication of units requires two distinct numbers with different roles The first number is an adjective that describes a count of noun units called a multiplicand. The second number is an adverb that counts either additions of the given number of units in the multiplicand to zero, or a count of subtractions of the number of units from zero. So we have a number of units to the left of the multiplication sign and the number of additions or subtractions of the multiplicand to or from zero (depending on the sign of the multiplier) to the right of the multiplication sign. Thus we say multiplication is a binary operation requiring two quantities. Yet unlike addition these numbers count different things. This, a number cannot be multiplied by itself as you cannot multiply a number of things by a number of things. You cannot multiply three eggs by three eggs. You can only multiply a number of units by a number. So you can multiply three eggs by three. That three multiplier means three eggs are added to an empty basket three times in succession which results in nine eggs in the basket. This you cannot multiply a number of units by itself for several reasons. First as with addition, multiplication is binary requires two numbers. Secondly a number of units cannot be multiplied by a number of units as the unit of the product will differ from the unit of the multiplicand. A number cannot be multiplied by itself as it cannot be both itself (a count of units) and another a count of actions. A single number can only refer so a single concept otherwise we would never know what unit a number counts. Prove you can't multiply a number of units by itself.

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Multiplication is a binary operation that requires exactly two numbers with distinct roles. The first number, called the multiplicand, represents a count of units or things. The second number, called the multiplier, represents how many times we perform an action on those units. Unlike addition where both numbers can represent the same type of quantity, multiplication requires these two different conceptual roles. For example, three eggs times four means we have three eggs as our base unit, and we add this group four times. The multiplicand and multiplier have fundamentally different roles. The multiplicand is an adjective that describes a count of noun units - actual things we can count. The multiplier is an adverb that describes how many addition operations we perform. In five apples times three, we have five apples as our multiplicand - this is what we have. The multiplier three tells us to add this group three times: five plus five plus five equals fifteen apples. The multiplier counts actions, not things. Unit analysis reveals a fundamental problem with self-multiplication. When we multiply correctly, like three meters times four, we get twelve meters. The units are preserved because the multiplier four is dimensionless - it's just a count of actions. However, if we try to multiply three meters by four meters, we create dimensional confusion. This would give us square meters, which is a completely different type of measurement representing area, not basic multiplication. For basic multiplication to work, the multiplier must always be dimensionless to preserve the multiplicand's units. Here is the formal proof that self-multiplication is impossible. First, multiplication is a binary operation requiring exactly two distinct operands. Second, a single number cannot simultaneously represent both a count of units and a count of actions - these are fundamentally different concepts. When we write three eggs times three eggs, we create a logical contradiction because the number three cannot be both the multiplicand describing what we have and the multiplier describing how many times we act. The correct form is three eggs times three, where three eggs is what we have and three is how many times we add it. This proves that self-multiplication violates the basic principles of binary operations. Let's address common counterarguments. People often point to examples like three times three equals nine in pure arithmetic, but this involves abstract numbers with no physical units. Area calculations like three meters times three meters equal nine square meters represent completely different geometric operations, not basic multiplication of counted units. The key distinction is context: abstract mathematics deals with pure numbers, while unit counting represents real physical quantities. Mathematical notation can be misleading because the same symbols can represent different conceptual operations. In conclusion, you cannot multiply a number of units by itself in basic multiplication because it violates the fundamental requirement that multiplication needs two operands with distinct roles.