The Curious Case of 11 Not Equal to 11

When first encountering the concept that 11 is not equal to 11, many might wonder if it’s a serious mathematical proposition or a playful intellectual exercise. After all, in the simplest context, the equation 1 1 2 is a fundamental truth in arithmetic. However, as we delve into the nuances of mathematical notation and numeral systems, a more nuanced and intriguing picture emerges. This article explores the conditions under which 11 can equal 11, challenging our conventional understanding of mathematics and numeral systems.

Mathematical Expectations and Context

Most people instinctively think of 1 1 as 2 without a second thought. However, this assumption is deeply rooted in conventional expectations. When presented with the expression ‘11’, our brains often interpret it similarly to how we read '1' followed by '1', leading to the conclusion that it equates to '2' in base-10 notation. This, however, is an assumption driven by convention, and not necessarily a universally applicable rule.

Unary Notation: A Case for 11 11

One of the most straightforward contexts where 11 can equal 11 is in unary notation. In unary, a number is represented by a string of identical symbols, where the number of symbols corresponds to the magnitude of the number. For example, the number '1' is represented by a single mark, while the number '2' is represented by '11'. Here, if we consider '1' as the symbol for 1, then '11' is simply the representation of 2. Therefore, in this notation, 11 and 2 are indeed the same. This means that 11 (interpreted as '1' followed by '1') is equal to 2 (interpreted as '11'), reaffirming that 11 11 in this specific context.

Arbitrary Semigroups: A Mathematical Construction

Mathematically, we can construct a very simple semigroup where 11 11 by defining our operation and symbols arbitrarily. A semigroup is a set equipped with an associative binary operation. If we define a set ( S {1, 11} ) and declare that the operation (otimes) is such that for all ( a, b in S ), ( a otimes b 11 ), then clearly, in this semigroup, 11 11 is always true. This construction is purely abstract and serves to illustrate the flexibility of mathematical definitions, provided they adhere to certain axioms.

Place-Value Systems: The Intricacy of Numerals

Our familiar Indo-Arabic numeral system, which operates on a place-value basis, is far from the only system. In place-value systems, the value of a digit depends on its position within the numeral. For instance, in the numeral '11', the first '1' represents ten, and the second '1' represents one, resulting in the sum of 11. This is why '11' is read as eleven. However, in some numeral systems, '11' might simply denote 2. This is particularly evident in ancient systems where numerals were represented differently.

Ancient Egyptian Numerals: A Diverse Approach

In ancient Egyptian numerals, the use of symbols to denote numbers was quite different from our modern system. The symbol for one vertical stroke was ' | ', and two such strokes were ' || '. Ninine was represented by nine vertical strokes ' || || || || || || || | '. Ten was symbolized by a figure resembling an upside-down 'U'. In their numeral system, ' | ' meant one, ' || ' meant two, and so forth. This system used additive notation, where a string of identical symbols represented a number equal to the count of the symbols. For instance, ' || | ' represented three.

Consider the image that follows, which shows several numerals:- The first row has ' || | ' (three), ' | ' (one), and ' || || ' (two).- The third row starts with ' || || || || || ' (six) and ends with ' || || || || || | | | | | ' (twenty-eight).- The second row shows ' || || || | | ' (forty-eight) and ' || || || | | | | | | ' (eighty-eight), and ends with ' || || || || || | | | ' (one hundred twenty-eight).

The Importance of Context

The significance of the context in which we interpret numbers cannot be overstated. While 1 1 2 is a bedrock of arithmetic, the interpretation of 11 as 11 or 2 can vary widely depending on the numeral system and the conventions in place. Understanding these variations enriches our knowledge of mathematics and numeral systems, highlighting the importance of always specifying the context in which a mathematical statement is made.

Conclusion

The concept that 11 is not equal to 11 may seem paradoxical and intriguing. Whether 11 equals 11 depends on the framework of the system and the context in which the symbol is used. Unary notation, arbitrary semigroups, and place-value systems all play a role in shaping our understanding of how numbers and symbols interact. By exploring these concepts, we gain a deeper appreciation for the versatile and nuanced nature of mathematics, and the importance of context in mathematical interpretation.