https://claude.ai/share/5066ea8b-eba5-4c03-831b-a31f2283e792 hi claude i often think of adding 1/0 to math, one idea is on an "invertive zero" or iz numberline representing all the values that can recover from multiplication with zero, other approaches exist, but basically it breaks every rule of math and doesn't work. can you help me comprehend that by showing things clearly close together so i can see the parts right next to what they imply. or are there normal ways to define special rules for preserving accuracy through division by zero in normal equations? also are "invertive numbers" an existing term, analogous for negative numbers, that anybody's ever thought up? Edit Hello! Division by zero is indeed a fascinating mathematical concept that challenges our fundamental rules. Let me help you explore this. When we think about division by zero, we're essentially asking: "What number, when multiplied by 0, gives us another specific number?" The problem is that any number multiplied by 0 always gives 0, so there's no way to "undo" this operation to get back to our original number. Let's look at what happens as we approach division by zero: 1/1 = 1 1/0.1 = 10 1/0.01 = 100 1/0.001 = 1000 ... 1/ε → ∞ as ε → 0 ... 1/0 = ??? Similarly, approaching from the negative side: 1/(-0.1) = -10 1/(-0.01) = -100 1/(-0.001) = -1000 ... 1/(-ε) → -∞ as ε → 0 This shows why 1/0 is problematic - it seems to approach both positive and negative infinity depending on how you approach it. Regarding your idea of an "invertive zero" or "iz" - it's an interesting concept. What you're looking for is essentially a number that, when multiplied by 0, doesn't result in 0. This fundamentally breaks the field axioms of real numbers, particularly the property that 0 × a = 0 for all a. Some approaches to handle division by zero include: Projective geometry: Adding a single "point at infinity" to the real number line. Extended real number line: Adding both +∞ and -∞, but division by zero remains undefined. Riemann sphere: In complex analysis, we can think of 1/0 as representing infinity in a geometric sense. Wheel theory: A formal system that defines division by zero, but sacrifices other algebraic properties. IEEE 754 floating-point standard: In computing, division by zero yields special values like ±Infinity or NaN (Not a Number). As for "invertive numbers," the term isn't standard in mainstream mathematics, though the concept of multiplicative inverses (reciprocals) is fundamental. Your usage seems novel - drawing an analogy between additive inverses (negatives) and potential multiplicative behaviors around zero. Would you like me to elaborate on any particular aspect of this topic? Perhaps I could explore one of the formal systems that attempts to define division by zero in more detail? could we define the iz numbers with special rules such that things are consistent? have existing theories done this, such that the specialness only comes into play when a value is divided by 0? Edit Yes, there have been attempts to create consistent mathematical systems that allow division by zero. The most notable approach is "wheel theory," developed by mathematicians like David Carlson in the 1970s and further refined by others. In wheel theory, we define a new algebraic structure that extends the real numbers by adding a special element often denoted as "∞" or "⊥" that represents the result of division by zero. However, this comes with some significant trade-offs: The system sacrifices some familiar algebraic properties. For instance, in wheels, we lose the property that a·(b+c) = a·b + a·c for all values. Instead of complete distributivity, we get more limited properties that hold except in special cases involving division by zero. The element representing 1/0 behaves as a kind of "absorbing element" in many operations. There's also the "transreal" and "transcomplex" number systems proposed by James Anderson, which define division by zero in ways that attempt to be consistent. In this system: 1/0 = ∞ (positive infinity) -1/0 = -∞ (negative infinity) 0/0 = Φ (a special undefined value) The key insight from these theories is that you can define a system where division by zero has meaning, but you must accept modifications to the standard rules of arithmetic that we're accustomed to. The "specialness" does indeed only come into play during operations involving zero divisors. Another approach is the "Riemann sphere" in complex analysis, which adds a single point at infinity. This works well for many purposes in complex analysis, though it doesn't completely solve all issues with division by zero. Your concept of "iz numbers" could potentially be developed along similar lines - the challenge would be defining operations consistently so that most standard mathematical properties hold except in clearly defined special cases involving your invertive zeros. Would you like me to elaborate on how one of these systems handles specific algebraic properties, or perhaps explore how your "iz" concept might be formalized? do the wheel theory rules upset normal arithmetic, or only come into play around 1/0 ? my approach was to say that 1/0 cannot distribute into or out of a grouping operator Edit In wheel theory, the special rules mostly come into play only when dealing with division by zero scenarios, which is exactly what you're looking for. Regular arithmetic remains unchanged for all other operations. Here's how wheel theory works in relation to normal arithmetic: All standard arithmetic operations with non-zero divisors behave exactly as expected. The special element (often denoted as ∞ or ⊥) representing division by zero only affects calculations when it appears. When this special element enters a calculation, specific rules apply that preserve as much consistency as possible. The most significant departure from standard algebra is exactly what you suggested - restrictions on distributivity. In wheel theory: The distributive property a(b+c) = ab+ac holds EXCEPT when a = 0 and either b or c involves division by zero. Similarly, factoring expressions works normally except in cases involving division by zero. For example, in wheel theory: 2 + 3 = 5 (normal arithmetic) 7 × 6 = 42 (normal arithmetic) 1/0 = ∞ (special element) 0 × (1/0) = 0 (not 1, as distribution doesn't apply here) Your intuition about restricting distribution with 1/0 aligns perfectly with how wheel theory handles these situations. The key insight of wheel theory is that it isolates the "unusual" behavior to only those situations where division by zero actually occurs, which preserves most of our familiar arithmetic in everyday calculations. Would you like me to explain some specific examples of how wheel theory handles particular calculations involving division by zero? is wheel theory strictly an extension of normal algebra that could in theory be united into it? Edit Yes, wheel theory is indeed an extension of normal algebra that could theoretically be united with it. The relationship is similar to how complex numbers extend real numbers, or how real numbers extend rational numbers. In a formal sense, wheel theory creates a larger algebraic structure that contains the field of real numbers as a substructure. Within this larger structure: All operations among regular numbers behave exactly as they do in standard algebra The special elements and rules only come into play when division by zero occurs The standard field of real numbers is embedded as a substructure where all the familiar properties hold What makes this unification possible is that wheel theory is designed to be conservative - it only modifies the algebraic structure in the minimal way necessary to accommodate division by zero. The standard numbers form a "reduced wheel" within the larger structure. This means you could work entirely within the standard algebraic system until you encounter division by zero, at which point you'd use the extended rules. This is conceptually similar to how we use real numbers for most calculations but can switch to complex numbers when we need to take the square root of a negative number. The main challenge with this unification is that some of the algebraic properties we take for granted (like universal distributivity) need to be slightly restricted, but these restrictions only matter in very specific cases involving division by zero. what is 1+1/0 Edit In wheel theory, operations involving the element representing 1/0 follow special rules. For the expression 1+1/0: When we have 1/0, this is typically represented by the special element (often denoted as ∞ or ⊥) in wheel theory. In a wheel, the sum of a regular number and the element representing division by zero is defined to be equal to the division-by-zero element itself. This is because the division-by-zero element acts like an "absorbing element" for addition in many formulations of wheel theory. Therefore: 1 + 1/0 = 1/0 = ∞ This reflects the intuition that adding any finite number to infinity doesn't change the result - it remains infinite. This property helps maintain some level of consistency in the system while accommodating division by zero. This is one of the special rules that only comes into play when the division-by-zero element is involved in calculations, while regular arithmetic remains unchanged for all other operations. that seems strange. does wheel theory provide for recovering coefficients of zero via 1/0? Edit You're right to question this aspect. What you're touching on is one of the key limitations of wheel theory. In wheel theory, division by zero does not allow you to "recover" coefficients of zero in the way you might hope. That is, if we have an equation like 0 × x = 0, wheel theory doesn't provide a way to determine what x is by multiplying both sides by 1/0. The reason is that wheel theory preserves the property that 0 × a = 0 for all regular values of a. So while 1/0 exists in the system, it doesn't work as a true multiplicative inverse of zero. In particular: 0 × (1/0) = 0 (not 1) This is necessary to avoid contradictions in the system. If we had 0 × (1/0) = 1, we could create paradoxes. For instance, using that property would allow us to "prove" that any two numbers are equal: Let a and b be any two numbers a - b = a - b (a - b) × 0 = (a - b) × 0 0 = 0 0 × (1/0) = 0 × (1/0) 1 × (a - b) = 1 × 0 a - b = 0 a = b So wheel theory deliberately designs the system so that 1/0 exists but doesn't allow you to recover coefficients of zero. It's not a true "invertive zero" in the sense you were describing earlier. This reflects a fundamental mathematical principle: there's no way to create a system that both allows division by zero and maintains all the familiar algebraic properties we're used to. Something has to give, and in wheel theory, it's the ability to use 1/0 to recover coefficients in expressions like 0 × x. what happens if we relax 0*x=0 and instead keep 0x as a symbolic construction until solving is over ... is there any pursuit along lines like that? Edit That's an insightful approach! Yes, there are mathematical systems that treat expressions like 0×x as symbolic constructions rather than immediately evaluating them to 0. Symbolic computation systems (like computer algebra systems) often work this way. For example, in symbolic mathematical software like Mathematica or MAPLE, expressions can be maintained in unevaluated form until simplification is explicitly requested. In formal mathematics, there are several approaches similar to what you're suggesting: Differential algebra: This treats derivatives as formal operators rather than limits, allowing expressions with "infinitesimals" to be manipulated symbolically before evaluation. Non-standard analysis: Developed by Abraham Robinson, this provides a rigorous foundation for infinitesimals and infinite numbers, allowing for symbolic manipulation of expressions that would be problematic in standard analysis. Computer algebra systems: These maintain expressions symbolically, evaluating only when necessary, which can avoid division-by-zero issues in intermediate steps. Formal power series: These allow operations with expressions that might not converge in standard analysis. Residue calculus: In complex analysis, integrals around singularities (including division by zero) can be evaluated by examining the behavior near the singularity without directly dealing with division by zero. Your idea of keeping 0×x as a symbolic construction has merit in certain contexts. For example, in limit calculations, we often work with expressions like 0×∞ by rewriting them in forms that avoid direct evaluation until the final step. This approach moves away from the standard algebraic structure of fields but can be useful for solving specific types of problems where maintaining the symbolic form provides insight that immediate evaluation would obscure. Would you like to explore any of these approaches in more detail? i mean as a set of rules to provide for consistent manipulation through division by zero. but it kind of seems like creating a symbol for sqrt(-1) simply worked out better than creating constructs for 1/0; information is lost when scaling by 0 ... of course a little bit of information is lost when multiplying negatives too ... Edit You're making a profound observation. The comparison between introducing √(-1) and attempting to define 1/0 is illuminating. When we introduced i = √(-1), we expanded our number system in a way that preserved most algebraic properties while solving a specific problem (finding roots of negative numbers). The key difference is that multiplication by i doesn't lose information - it's reversible. In contrast, multiplication by 0 fundamentally loses information. When we compute 0×a, we can't recover what "a" was, regardless of what system we're working in. This irreversible information loss is what makes division by zero so much more problematic than square roots of negative numbers. You could create a symbolic system where: 0×x is kept as a symbolic entity Division by zero produces symbols that carry information about what was divided Operations with these symbols follow special rules But such a system would need to sacrifice some basic algebraic properties. For instance, if (0×a)÷0 = a, then we'd lose the property that equal values divided by equal values give equal results, since (0×a) = (0×b) = 0 for any a and b. This fundamental difference explains why complex numbers became a standard part of mathematics, while systems handling division by zero remain specialized tools rather than general extensions of our number system. In a sense, multiplying by zero is more like a projection operation that collapses an entire dimension of information, while multiplying by i is merely a rotation that preserves information. Creating an inverse for projection is mathematically much more challenging than creating an inverse for rotation. you'd define 0x and izx as distinct, inequal values as x changed Edit That's a fascinating approach! By treating 0*x as distinct values that depend on x (rather than all collapsing to 0), you could potentially create a system where division by zero recovers the original value. In such a system: 0a ≠ 0b when a ≠ b (these are distinct symbolic entities) iz0a = a (where iz represents your "invertive zero") 0*iz = 1 (the multiplicative identity) This would mean 0*x retains information about x, effectively "tagging" the zero with information about what it was multiplied by. Division by zero (or multiplication by iz) would then retrieve this tag. This system would need to handle operations like: (0a) + (0b) = 0*(a+b)? Or something else? (0a) × c = 0(a×c)? Or does this need special handling? What you're proposing has some conceptual similarities to dual numbers in automatic differentiation, where we track both a value and its derivative symbolically, or to approaches in non-standard analysis where infinitesimals carry information. The challenge is maintaining consistency while defining all operations in this extended system. You'd need to carefully work through the implications for addition, multiplication, and other operations to ensure you don't create contradictions. This approach differs from wheel theory by preserving the information lost during multiplication by zero, rather than just adding a division-by-zero element with special properties. Is this the direction you were thinking of with your iz concept? yeah Edit Your idea has genuine mathematical merit. Creating a system where 0*x preserves information about x would let iz (your invertive zero) act as a true multiplicative inverse of zero. In such a system: 0a and 0b are distinct objects for different values of a and b iz0a = a (recovering the original value) 0*iz = 1 (making iz a true multiplicative inverse) This approach differs fundamentally from wheel theory because you're changing the nature of multiplication by zero itself, rather than just adding special elements to handle division by zero. Some natural questions about this system: How would addition work? Would (0a) + (0b) = 0*(a+b)? What is (0a) × (0b)? Perhaps 0*(a×b)? Would associativity hold? That is, would 0*(ab) = (0a)*b? This reminds me somewhat of how we handle nilpotent elements in certain algebraic structures or how we use "tagged zeros" in some computer arithmetic systems, though your approach is more ambitious in creating a full inverse. While I'm not aware of an established theory that does exactly what you're proposing, your idea connects to several advanced mathematical concepts like hyperreal numbers, transfinite arithmetic, and certain areas of abstract algebra where multiplication can have non-traditional behavior. Would developing the formal rules for such a system interest you? It could potentially lead to interesting mathematical insights, even if it ultimately serves more as a thought experiment than a practical extension of standard arithmetic. it's just a small idea and i'm pretty tired and only have this 1 message remaining. can you consider the idea and point out where a contradiction would arise if we kept all the normal rules, then start adding restrictions like wheel theory does Edit Let me analyze your idea of preserving information in 0*x expressions and identify potential contradictions: If we try to maintain all normal algebraic rules while allowing division by zero through your "invertive zero" (iz) concept, here are the key points where contradictions would emerge: Distributivity conflict: If 0a ≠ 0b for a ≠ b, and 0*(a+b) = (0a) + (0b), then: 0*(a+b) = 0a + 0b But also 0*(a+b) = 0*(a+b) (a single "tagged" zero) This creates two different representations for the same expression. Equality problem: If 0*a ≠ 0 for all a, then: 0a = 0a (true by reflexivity) But 0a = 01a = 0a*1 (by associativity and commutativity) So 01 = 0, which contradicts our assumption that 0a retains information about a. Identity element issue: If iz*0 = 1, then: iz0a = a (your recovery property) But also iz0a = iz*(0a) = (iz0)a = 1a = a This works fine, but requires 0*a to be "tagged" while preserving associativity. Transitivity contradiction: If 0a = 0b = 0 (standard arithmetic) but we also require 0a ≠ 0b when a ≠ b, we've created a direct contradiction. To avoid these contradictions, we would need restrictions similar to wheel theory: Limit distributivity: (a+b)c = ac + b*c except when c = 0 Define special addition rules for expressions like (0a) + (0b) Restrict associativity for expressions involving 0 and iz Create clear separation between standard zeros and "tagged" zeros Your system would need to sacrifice some algebraic properties we take for granted, but could potentially be developed into a consistent framework that extends standard arithmetic while providing a pathway to recover information lost in multiplication by zero. The fundamental issue remains that multiplication by zero inherently loses information in standard arithmetic, so any system that allows recovery must alter some basic properties we usually expect to hold universally. You are out of free messages until 3:00 AM