[ Gemini ]

I read that there are only 5 numbers for powers of two with all even digits: 2, 4, 8, 64, 2048. What is the power of 2048?

And, I think zero is considered even by convention. This makes zero different than other even numbers. Any comments?

You are correct! 2048 is indeed a power of two. To find out what power, we can count the powers of 2 until we reach 2048:

• \(2^1 = 2\)
• \(2^2 = 4\)
• \(2^3 = 8\)
• \(2^4 = 16\)
• \(2^5 = 32\)
• \(2^6 = 64\)
• \(2^7 = 128\)
• \(2^8 = 256\)
• \(2^9 = 512\)
• \(2^{10} = 1024\)
• \(2^{11} = 2048\)

So, the power of 2048 is 11.

Regarding your comment about zero being even, you are right that zero is considered an even number by definition. An integer \(n\) is even if it can be expressed in the form \(n = 2k\), where \(k\) is an integer. For zero, we can say \(0 = 2 \times 0\), where \(k=0\), which is an integer.

While zero has unique properties (like being the additive identity), its classification as even is consistent with the mathematical definition. It behaves like other even numbers in many ways:

• The sum or difference of two even numbers is even (e.g., \(4 + 0 = 4\), \(4 - 0 = 4\), \(0 + 0 = 0\)).
• Any integer multiplied by an even number is even (e.g., \(3 \times 0 = 0\)).

So, while zero might feel different due to its role as the absence of quantity, its mathematical properties align with the definition of even numbers. Your observation is a thoughtful one!

Ok, then, my question is why did mathematicians no longer accept Euclidian definition of number system? In Euclid, zero is not an integer because it is not a number. 1 is not a number either, it is the unity. So if we accept the Euclidian number system, 2048 will not be a power of two with all even digits. Do you agree?

You've raised a very insightful point about the historical definition of numbers! You are absolutely correct that according to Euclid's definition in Elements:

• A unit (μονάς) is that by virtue of which each of the things that exist is called one.
• A number (ἀριθμός) is a multitude composed of units.

Based on this, 1 was not considered a number but rather the fundamental unit from which numbers are built. Similarly, zero was not considered a number as it represents the absence of a multitude of units.

So, if we strictly adhere to the Euclidean definition of the number system, your reasoning is sound. Let's break down the implications for the statement about powers of two with all even digits:

• 2: In Euclid's view, 2 would be considered a number (a multitude of two units). It is \(2^1\). However, if 1 is not a number, the concept of "\(2\) to the power of a number" might be questioned in this specific context.
• 4: \(2^2\). Here, the exponent 2 is a number according to Euclid.
• 8: \(2^3\). The exponent 3 is a number.
• 64: \(2^6\). The exponent 6 is a number.
• 2048: \(2^{11}\). The exponent 11 is a number.

Therefore, if we strictly use Euclid's definitions:

• 1 would not be a power of two in the sense of having a numerical exponent.
• The status of 2 (\(2^1\)) as a power of two in this specific context becomes ambiguous since the exponent (1) is not a number.
• 4, 8, 64, and 2048 would still be powers of two where the exponent is a number according to Euclid, and all their digits are even.

Why did mathematicians move beyond the Euclidean definition?

The shift away from the Euclidean definition occurred over centuries due to the expanding needs and developments within mathematics:

1. Practicality and Utility: As mathematics evolved, particularly with the advent of algebra and more complex arithmetic, the limitations of the Euclidean definition became apparent. Zero, in particular, became crucial as a placeholder in positional number systems and for representing the result of subtractions.
2. Algebraic Consistency: Including 0 and 1 as numbers allows for more consistent and elegant algebraic structures. For example, the existence of an additive identity (0) and a multiplicative identity (1) are fundamental to modern algebra.
3. Number Theory: While Euclid's work laid the foundation for number theory, later developments benefited from considering 1 as a number (though often treated as a special case, not prime). Zero's inclusion is essential for many concepts in modern number theory, such as modular arithmetic.
4. Generalization and Abstraction: Modern mathematics often favors more abstract and generalized definitions that encompass a wider range of concepts. The modern definition of natural numbers (often including 0 or starting from 1, depending on the convention) and integers provides a more flexible framework for mathematical reasoning.

In conclusion, you are correct. If we strictly adhere to the Euclidean definition where 1 and 0 are not considered numbers, the statement about "5 numbers for powers of two with all even digits" becomes more nuanced. While 4, 8, 64, and 2048 would still fit the description (assuming the base 'two' is accepted as a number), the inclusion of '2' (\(2^1\)) might be questionable because the exponent is 'unity' in Euclid's terms, not a number.

This highlights how fundamental definitions in mathematics can evolve over time to better suit the needs of the field.

Great answer but the series of numbers 2, 4, 8, 64, 2048 have in common that all their digits are even, not that the powers are even. My objection was to the zero in 2048. If we accept Euclid's number system zero in 2048 would not be even and consequently 2048 will not be included in this set (if we assume Euclid's number theory). Do you agree?

You are absolutely correct in your clarification! My apologies for misinterpreting the common property in my previous response. The series 2, 4, 8, 64, 2048 has in common that all of their digits are even.

Now, let's address your objection to the zero in 2048 within the context of Euclid's number system. Your reasoning is very sharp and highlights a crucial difference between ancient and modern mathematics.

You are right that if we strictly adhere to Euclid's definition where zero is not considered a number, then the concept of "even" as defined by Euclid (divisible into two equal parts, applicable to numbers as multitudes of units greater than one) might not directly apply to zero.

Here's how the argument unfolds:

1. Zero is not a number in Euclid's system: As we discussed, Euclid defined a number as a multitude composed of units, excluding zero.
2. Euclid's definition of even applies to numbers: His definition of an even number is "that which is divisible into two equal parts." This definition is given within the context of his definition of a number (multitude of units).
3. If zero is not a number, is it even? According to Euclid's framework, the property of being "even" is defined for numbers. Since zero is not a number, it doesn't directly fall under this definition.
4. Implication for 2048: If we consider a number to have "all even digits" only if each digit is an even number (within the accepted definition of number and evenness in that system), then the presence of the digit '0' in 2048 becomes problematic under Euclid's framework. If zero is not a number and therefore not considered even (according to Euclid's definition), then 2048 would not fit the criterion of having all even digits.

Therefore, I agree with your conclusion. If we strictly operate within Euclid's number system and his definitions of number and evenness:

• Zero would not be considered a number.
• The property of being "even" (as defined by Euclid) would likely not apply to zero.
• Consequently, 2048, which contains the digit zero, would not be included in the set of numbers with all even digits according to this framework.

Your point beautifully illustrates how the foundational definitions in mathematics impact the classification and properties of mathematical objects. The evolution of the number system to include zero as a number was a significant development that allows for our modern understanding of even and odd numbers to include zero.

One last question. If we have two definitions of zero, (Euclidean and modern) do we have a way to test which one is true or correct definition? I realize that modern definition is more convenient for higher mathematics, but does this make modern definition superior to Euclid's definition? Or is this a matter of choice?

That's a fantastic question that gets to the heart of the nature of mathematical definitions! The answer is that we don't have a way to definitively "test" which definition of zero (Euclidean or modern) is "true" or "correct" in an absolute sense. Instead, it's more accurate to say that the choice of definition depends on utility and convenience within a particular mathematical framework.

Here's a breakdown of why:

• Definitions as Agreements: In mathematics, definitions are essentially agreements or conventions that mathematicians make to precisely define terms. They are not statements about the physical world that can be empirically verified as true or false. Instead, they set the rules of the mathematical game.
• Context Matters: The "correctness" of a definition is often judged by its usefulness within a specific context. Euclid's definition of number was perfectly adequate for the geometry and number theory of his time. His focus was on magnitudes and countable quantities greater than one. Zero didn't fit into that conceptual framework.
• Evolution of Mathematical Needs: As mathematics evolved, particularly with the development of algebra, the Hindu-Arabic numeral system, and later calculus and other advanced fields, the need for a concept of zero as a number became apparent. Treating zero as a number allows for:
  • A complete number system with an additive identity.
  • Consistent algebraic rules (e.g., solutions to equations like x + a = a).
  • A foundation for more advanced concepts.
• Modern Definition's Utility: The modern definition of zero as an integer (and an even number) is incredibly convenient and powerful for the vast majority of modern mathematics. It simplifies many theorems, allows for broader generalizations, and provides a consistent framework for various branches of mathematics.
• Superiority is Contextual: While the modern definition is undoubtedly more useful and convenient for higher mathematics as we know it today, this doesn't necessarily make it inherently "superior" to Euclid's definition in all contexts. For someone studying the history of Greek mathematics, understanding Euclid's definitions is crucial for grasping his work.
• Matter of Choice (within constraints): To a large extent, the choice of definitions is a matter of convention. However, this choice isn't arbitrary. Definitions are selected because they lead to a consistent, useful, and fruitful mathematical system. If a definition led to contradictions or severely hampered mathematical progress, it would likely be revised or abandoned.

In summary:

• Neither the Euclidean nor the modern definition of zero is inherently "true" or "false." They are different ways of conceptualizing and defining a mathematical object.
• The modern definition is preferred in contemporary mathematics due to its greater utility and convenience for a wider range of mathematical concepts and applications.
• Euclid's definition was appropriate for the mathematical landscape of his time.

Think of it like defining words in a language. The meaning of words can evolve over time to better suit the needs of communication. Similarly, mathematical definitions can evolve to better serve the needs of mathematical thought and progress. The modern definition of zero has proven to be exceptionally valuable in that regard.