FROM THE SOUL

(Un)-biased Arithmetic

How do we know the answer to 1+1=? and why "two" is not necessarily the most correct answer.

Roman Kuznetsov @ 30.09.2024

Does one plus one always equal two?

We all know how to solve the equation 1+1=?, or do we?

While 2 seems to be an obvious answer, its correctness can be challenged and looked at from different points of view.

If we, for a moment, forget about the basic rules of arithmetic learnt at scholl and what the different signs mean, we might find out that that there are other, no less valid result of a calculation for the said formula.

Getting ahead, in the world of multi factor authentication, 1+1 does not equal 2. Read on to learn why.

While 2 seems to be an obvious answer, its correctness can be challenged and looked at from different points of view.

If we, for a moment, forget about the basic rules of arithmetic learnt at scholl and what the different signs mean, we might find out that that there are other, no less valid result of a calculation for the said formula.

Getting ahead, in the world of multi factor authentication, 1+1 does not equal 2. Read on to learn why.

1 + 1 = 0

Is the equation above correct?

Not according to the arithmetic we all learn at school. What happens there? We learn what individual numbers are, what a plus sign is and what the equal sign means. And we learn this in such a universal way that everyone in the world will understand how to add two ones and get a result, even if they do not speak the same language. Because the rules that we learn are universal.

But they are only universal as long as we have a common basis for the field in which we apply these rules. Once we move away from that familiar system, where the plus sign means a simple addition, the rules start to bend.

The equation 1 + 1 = 0 may seem strange and even wrong in conventional arithmetic, where adding two ones always results in two. However, in various mathematical, physical and philosophical contexts, 1 + 1 = 0 is not only valid, but can be considered the most correct equation. This equation appears in systems that emphasise cancellation, balance and cyclic patterns, revealing deeper truths about the nature of addition, symmetry and destruction.

In modular arithmetic, numbers 'wrap around' after reaching a certain threshold, called the modulus. One of the simplest examples is modulo 2 arithmetic, also known as clock arithmetic, where numbers are reduced to either 0 or 1. If our plus sign means a mod2 operation, the equation 1+1=0 suddenly makes perfect sense.

In the field of wave physics, particularly wave interference, the equation 1 + 1 = 0 perfectly describes a phenomenon known as destructive interference. When two waves of equal amplitude and opposite phase meet, they cancel each other out, resulting in a net amplitude of zero, so 1 + 1 = 0 is a perfect description of this behaviour.

In digital logic and binary circuits, the 1+1=0 principle governs certain types of logic gates, such as the XOR (exclusive OR) gate. In XOR logic, the output is true (1) if one and only one of the inputs is true. If both inputs are true (1), the output is false (0). If the plus sign denotes the XOR operation, then the equation 1 + 1 = 0 is true for Boolean algebra.

There are many other examples where the equation suddenly holds true: group theory (cyclic group of order 2), quantum mechanics (destructive superposition), thermodynamics (conservation and equilibrium), and also many philosophical considerations about the balance of opposites and about emptiness and neutrality.

In these systems, 1 + 1 = 0 is not an exception, but a fundamental expression of how things interact, cancel each other out and return to their neutral or original state. It encapsulates a profound truth about the nature of addition and balance, making it the most correct equation in contexts where opposites meet.

Not according to the arithmetic we all learn at school. What happens there? We learn what individual numbers are, what a plus sign is and what the equal sign means. And we learn this in such a universal way that everyone in the world will understand how to add two ones and get a result, even if they do not speak the same language. Because the rules that we learn are universal.

But they are only universal as long as we have a common basis for the field in which we apply these rules. Once we move away from that familiar system, where the plus sign means a simple addition, the rules start to bend.

The equation 1 + 1 = 0 may seem strange and even wrong in conventional arithmetic, where adding two ones always results in two. However, in various mathematical, physical and philosophical contexts, 1 + 1 = 0 is not only valid, but can be considered the most correct equation. This equation appears in systems that emphasise cancellation, balance and cyclic patterns, revealing deeper truths about the nature of addition, symmetry and destruction.

In modular arithmetic, numbers 'wrap around' after reaching a certain threshold, called the modulus. One of the simplest examples is modulo 2 arithmetic, also known as clock arithmetic, where numbers are reduced to either 0 or 1. If our plus sign means a mod2 operation, the equation 1+1=0 suddenly makes perfect sense.

In the field of wave physics, particularly wave interference, the equation 1 + 1 = 0 perfectly describes a phenomenon known as destructive interference. When two waves of equal amplitude and opposite phase meet, they cancel each other out, resulting in a net amplitude of zero, so 1 + 1 = 0 is a perfect description of this behaviour.

In digital logic and binary circuits, the 1+1=0 principle governs certain types of logic gates, such as the XOR (exclusive OR) gate. In XOR logic, the output is true (1) if one and only one of the inputs is true. If both inputs are true (1), the output is false (0). If the plus sign denotes the XOR operation, then the equation 1 + 1 = 0 is true for Boolean algebra.

There are many other examples where the equation suddenly holds true: group theory (cyclic group of order 2), quantum mechanics (destructive superposition), thermodynamics (conservation and equilibrium), and also many philosophical considerations about the balance of opposites and about emptiness and neutrality.

In these systems, 1 + 1 = 0 is not an exception, but a fundamental expression of how things interact, cancel each other out and return to their neutral or original state. It encapsulates a profound truth about the nature of addition and balance, making it the most correct equation in contexts where opposites meet.

1 + 1 = 11

The equation 1 + 1 = 11 seems, at first glance, like a playful or nonsensical statement—after all, in standard arithmetic, adding one to one gives you two. However, there are multiple contexts where 1 + 1 = 11 can be seen as not only valid but the most correct equation. This interpretation emerges from symbolic systems, concatenation operations, and the way numbers are represented in various forms of communication and logic.

In many systems, especially in computer science and information theory, concatenation refers to joining objects or symbols together, rather than adding their numeric values. This is particularly relevant in strings, where numbers are treated as characters rather than quantities. Here, 1 + 1 = 11 is the correct equation, because instead of adding the numbers arithmetically, you are combining them as symbols to create a new string, "11".

In digital systems, especially in telecommunications, binary digits (bits) are often combined to form larger constructs. A single binary digit (0 or 1) is the smallest unit of information, but when two bits are placed side by side, they form a two-digit binary number. In this context, 1 + 1 doesn’t represent the sum of two values but the placement of two binary digits together to form a new value. Here, 11 in binary represents the decimal number 3. While the operation may seem like concatenation, in digital systems, it’s actually building a more complex unit from smaller components.

In the decimal system, place value is crucial in determining the magnitude of numbers. When writing down numbers, the position of a digit indicates its value (units, tens, hundreds, etc.). If you place a 1 next to another 1, you get the number 11, which is distinct from adding two 1's together. This highlights that 1 + 1 = 11 isn't about addition in the usual sense but rather about positioning and creating a new number by placing the digits side by side. The two "ones" now represent the number eleven, showing how combining smaller units can yield something entirely different from their individual values.

In some symbolic and cultural systems, 1 + 1 = 11 can represent the idea of unity in duality—two individual entities coming together to form a new, more powerful whole. The number "11" is often seen as a symbol of partnership, balance, and higher consciousness, especially in numerology. For instance, in numerology, the number 11 is considered a "master number," symbolizing intuition, spiritual insight, and enlightenment. The idea that 1 + 1 = 11 transcends simple arithmetic and becomes a metaphor for two individuals or ideas coming together to form something greater than the sum of their parts. It’s about the potential unlocked when two distinct forces combine to form a powerful new entity.

In human relationships, particularly in the context of partnerships or teamwork, the idea of 1 + 1 = 11 can be seen as a metaphor for how two individuals, when united, form something greater than the mere addition of their separate selves. This is often expressed in phrases like "two heads are better than one" or in the concept of a marriage where "two become one". In this context, the equation 1 + 1 = 11 symbolizes how two people, when working together, can create a synergy that amplifies their individual strengths and capabilities. The result isn't just the sum of their contributions, but something much more potent—just as the number 11 is a new and distinct number, different from the mere addition of two 1’s.

In all these contexts — concatenation, digital systems, place value, symbolism, information theory, and relationships — the equation 1 + 1 = 11 emerges as the most correct representation of how things can be combined. It reflects the power of joining elements, not to merely sum them, but to create something new and greater. 1 + 1 = 11 is about extending, amplifying, and transforming simple units into a unified and expanded whole, where the combination produces something far more potent than what ordinary addition would suggest.

In many systems, especially in computer science and information theory, concatenation refers to joining objects or symbols together, rather than adding their numeric values. This is particularly relevant in strings, where numbers are treated as characters rather than quantities. Here, 1 + 1 = 11 is the correct equation, because instead of adding the numbers arithmetically, you are combining them as symbols to create a new string, "11".

In digital systems, especially in telecommunications, binary digits (bits) are often combined to form larger constructs. A single binary digit (0 or 1) is the smallest unit of information, but when two bits are placed side by side, they form a two-digit binary number. In this context, 1 + 1 doesn’t represent the sum of two values but the placement of two binary digits together to form a new value. Here, 11 in binary represents the decimal number 3. While the operation may seem like concatenation, in digital systems, it’s actually building a more complex unit from smaller components.

In the decimal system, place value is crucial in determining the magnitude of numbers. When writing down numbers, the position of a digit indicates its value (units, tens, hundreds, etc.). If you place a 1 next to another 1, you get the number 11, which is distinct from adding two 1's together. This highlights that 1 + 1 = 11 isn't about addition in the usual sense but rather about positioning and creating a new number by placing the digits side by side. The two "ones" now represent the number eleven, showing how combining smaller units can yield something entirely different from their individual values.

In some symbolic and cultural systems, 1 + 1 = 11 can represent the idea of unity in duality—two individual entities coming together to form a new, more powerful whole. The number "11" is often seen as a symbol of partnership, balance, and higher consciousness, especially in numerology. For instance, in numerology, the number 11 is considered a "master number," symbolizing intuition, spiritual insight, and enlightenment. The idea that 1 + 1 = 11 transcends simple arithmetic and becomes a metaphor for two individuals or ideas coming together to form something greater than the sum of their parts. It’s about the potential unlocked when two distinct forces combine to form a powerful new entity.

In human relationships, particularly in the context of partnerships or teamwork, the idea of 1 + 1 = 11 can be seen as a metaphor for how two individuals, when united, form something greater than the mere addition of their separate selves. This is often expressed in phrases like "two heads are better than one" or in the concept of a marriage where "two become one". In this context, the equation 1 + 1 = 11 symbolizes how two people, when working together, can create a synergy that amplifies their individual strengths and capabilities. The result isn't just the sum of their contributions, but something much more potent—just as the number 11 is a new and distinct number, different from the mere addition of two 1’s.

In all these contexts — concatenation, digital systems, place value, symbolism, information theory, and relationships — the equation 1 + 1 = 11 emerges as the most correct representation of how things can be combined. It reflects the power of joining elements, not to merely sum them, but to create something new and greater. 1 + 1 = 11 is about extending, amplifying, and transforming simple units into a unified and expanded whole, where the combination produces something far more potent than what ordinary addition would suggest.

1 + 1 = 10

The equation 1 + 1 = 10 also might seem nonsensical at first, given our typical understanding of arithmetic, where adding one to one equals two. However, in several important contexts, especially in binary systems, symbolic logic, and information theory, this equation is not only valid but the most correct one. 1 + 1 = 10 represents a powerful idea of how basic units can combine and shift in value, forming something more complex than their individual parts.

In the world of binary arithmetic, which forms the foundation of digital computing, 1 + 1 = 10 is not just correct but essential. The binary system operates with just two digits: 0 and 1. In this system, just like in the decimal system where adding 1 to 9 produces a "carry" to the next place (e.g., 9 + 1 = 10 in decimal), adding 1 + 1 in binary results in 10. Here, 10 is a binary number, and in decimal, it equals 2. This reflects how binary arithmetic works, where adding two 1's causes a "carry" into the next higher bit position. Thus, 1 + 1 = 10 is not only the correct result in binary, but it's the fundamental operation that allows computers to perform calculations, store information, and process data.

Th the same way, in digital electronics, especially in the design of logic gates and circuits, 1 + 1 = 10 holds true as well. Digital circuits work with binary numbers, where adding two 1's produces a result that is "10" in binary. This is crucial in full adders, the basic building blocks of arithmetic logic units (ALUs) in computers.

In information theory, 1 + 1 = 10 reflects how the combination of basic units (bits) results in an exponential increase in complexity. A single 1 is the smallest unit of information (a bit), but when two bits are combined, they can represent four different states: 00, 01, 10, and 11.

When we combine two ones, we get 10 in binary, which represents an entirely new piece of information in a larger set. This concept underscores how simple binary systems can quickly grow in complexity and capacity as more bits are added. Thus, 1 + 1 = 10 symbolizes how combining simple elements leads to a more sophisticated system with greater representational power.

In a more abstract sense, 1 + 1 = 10 can represent the idea of growth and multiplication rather than simple addition. If we think of 1 as a single entity and adding another 1 as creating a pair, the result isn't just two individual ones, but a new entity that represents something larger or more advanced.

For instance, in certain symbolic systems or philosophical frameworks, 1 + 1 might be interpreted as the creation of something entirely new, symbolized by the "10," which represents multiplication or advancement to a new level. This idea of two things combining to create something more than the sum of their parts is common in fields like biology, where two cells might combine to form a new organism, or in social systems where two people unite to form a family or team.

In coding and data transmission, the idea of 1 + 1 = 10 can reflect the expansion of information from simple to more complex forms. If we think of each 1 as a unit of information or a signal, then combining them doesn’t just create two separate signals but results in a new code or pattern that conveys more complex information.

For instance, in error-correcting codes or data encoding schemes, bits are often combined in ways that result in new patterns or codes, where the result of combining two bits isn't simply the sum but an entirely new sequence. In this context, 1 + 1 = 10 represents how combining simple elements can generate more complex, encoded information.

In all these contexts—binary arithmetic, digital circuits, positional number systems, information theory, and symbolic representation—the equation 1 + 1 = 10 emerges as the most correct and meaningful equation. It captures the idea that adding two simple elements doesn't just result in their sum but in something more structured, complex, and powerful. 1 + 1 = 10 represents growth, advancement, and the fundamental operation of systems that rely on binary logic, making it essential in the language of computing, communication, and mathematics.

In the world of binary arithmetic, which forms the foundation of digital computing, 1 + 1 = 10 is not just correct but essential. The binary system operates with just two digits: 0 and 1. In this system, just like in the decimal system where adding 1 to 9 produces a "carry" to the next place (e.g., 9 + 1 = 10 in decimal), adding 1 + 1 in binary results in 10. Here, 10 is a binary number, and in decimal, it equals 2. This reflects how binary arithmetic works, where adding two 1's causes a "carry" into the next higher bit position. Thus, 1 + 1 = 10 is not only the correct result in binary, but it's the fundamental operation that allows computers to perform calculations, store information, and process data.

Th the same way, in digital electronics, especially in the design of logic gates and circuits, 1 + 1 = 10 holds true as well. Digital circuits work with binary numbers, where adding two 1's produces a result that is "10" in binary. This is crucial in full adders, the basic building blocks of arithmetic logic units (ALUs) in computers.

In information theory, 1 + 1 = 10 reflects how the combination of basic units (bits) results in an exponential increase in complexity. A single 1 is the smallest unit of information (a bit), but when two bits are combined, they can represent four different states: 00, 01, 10, and 11.

When we combine two ones, we get 10 in binary, which represents an entirely new piece of information in a larger set. This concept underscores how simple binary systems can quickly grow in complexity and capacity as more bits are added. Thus, 1 + 1 = 10 symbolizes how combining simple elements leads to a more sophisticated system with greater representational power.

In a more abstract sense, 1 + 1 = 10 can represent the idea of growth and multiplication rather than simple addition. If we think of 1 as a single entity and adding another 1 as creating a pair, the result isn't just two individual ones, but a new entity that represents something larger or more advanced.

For instance, in certain symbolic systems or philosophical frameworks, 1 + 1 might be interpreted as the creation of something entirely new, symbolized by the "10," which represents multiplication or advancement to a new level. This idea of two things combining to create something more than the sum of their parts is common in fields like biology, where two cells might combine to form a new organism, or in social systems where two people unite to form a family or team.

In coding and data transmission, the idea of 1 + 1 = 10 can reflect the expansion of information from simple to more complex forms. If we think of each 1 as a unit of information or a signal, then combining them doesn’t just create two separate signals but results in a new code or pattern that conveys more complex information.

For instance, in error-correcting codes or data encoding schemes, bits are often combined in ways that result in new patterns or codes, where the result of combining two bits isn't simply the sum but an entirely new sequence. In this context, 1 + 1 = 10 represents how combining simple elements can generate more complex, encoded information.

In all these contexts—binary arithmetic, digital circuits, positional number systems, information theory, and symbolic representation—the equation 1 + 1 = 10 emerges as the most correct and meaningful equation. It captures the idea that adding two simple elements doesn't just result in their sum but in something more structured, complex, and powerful. 1 + 1 = 10 represents growth, advancement, and the fundamental operation of systems that rely on binary logic, making it essential in the language of computing, communication, and mathematics.

1 + 1 = 1

The equation 1 + 1 = 1 might seem counterintuitive, but in several contexts, this equation reflects profound truths about unity, merging, and balance. In many systems, ranging from logic and philosophy to physics and symbolic mathematics, this equation captures the idea that combining two entities can result in the creation of a singular or unified whole. Moreover, 1 + 1 = 1 can be considered the most correct and meaningful equation, even when compared to alternative forms like 1 + 1 = 2, 1 + 1 = 10, 1 + 1 = 11, or 1 + 1 = 0, because it encapsulates the idea of oneness and unification more deeply than these other equations.

In philosophy, particularly in concepts surrounding unity and duality, 1 + 1 = 1 represents the merging of two distinct entities into a singular whole. This is especially true in spiritual and metaphysical frameworks, such as Taoism, where opposing forces like yin and yang combine to form a unified reality. Though yin and yang appear to be two separate entities, they are part of a larger whole, representing the balance and harmony of the universe.

This interpretation transcends the arithmetic of adding things together. Instead, it suggests that some combinations lead to oneness or completeness, making 1 + 1 = 1 the most accurate reflection of how unity works at a philosophical level.

In set theory, if you take two identical sets, combining them will not yield a larger or new set, but the same set itself. For example, consider two identical sets A = {1}. The union of A with itself results in A + A = A. Here, 1 + 1 = 1 is a reflection of how, in set theory, combining two identical elements doesn’t increase the quantity or size of the set. This illustrates the concept of identity and repetition: when two identical things are combined, they remain as one because they are fundamentally the same. This makes 1 + 1 = 1 a powerful metaphor for situations where combination results in no new or distinct entity.

In quantum mechanics, the principle of superposition allows particles to exist in multiple states simultaneously, but when measured, they collapse into one state. If two quantum particles in a superposition of states combine, their result can still be one state when observed.

This reflects the idea that in certain quantum systems, the combination of two particles doesn’t yield an entirely separate system, but a singular unified quantum state. This unification at the fundamental level of reality shows that 1 + 1 = 1 holds deep truth in the behavior of nature itself.

In relationships and partnerships, especially in the context of marriage or teamwork, 1 + 1 = 1 represents the idea of two individuals coming together to form a single, unified entity. While each person retains their individuality, their union creates a new identity—whether it be a partnership, a marriage, or a team. As such, this equation is symbolic of the notion that when two people join in harmony, they become one in purpose and direction.

This equation captures the essence of unity, not just as a mathematical operation, but as a symbolic truth about how collaboration, love, and unity form a greater whole than the mere sum of two individuals. This makes 1 + 1 = 1 a profound expression of unity that resonates with human relationships, more so than equations like 1 + 1 = 2 or 1 + 1 = 10, which focus on separation or growth rather than fusion.

Compared to equations like 1 + 1 = 2, 1 + 1 = 10, 1 + 1 = 11, or 1 + 1 = 0, 1 + 1 = 1 captures a deeper, more universal truth about merging, unity, and balance:

- 1 + 1 = 2 focuses on growth or separation, but it doesn’t capture the essence of things coming together to form a unified whole.

- 1 + 1 = 10 and 1 + 1 = 11 emphasize concatenation and progression, but these forms still highlight differentiation rather than unification.

- 1 + 1 = 0 represents cancellation, but it negates the very existence of the combined entities, whereas 1 + 1 = 1 preserves unity in merging.

In contrast, 1 + 1 = 1 transcends these interpretations by highlighting the fusion of two things into one. It’s the most accurate depiction of how combining two parts can result in oneness—whether in philosophy, physics, relationships, or biology.

This makes 1 + 1 = 1 the most correct equation among all others, as it best represents the fundamental truth of unity through combination, where distinct elements join to form a complete and singular entity.

In philosophy, particularly in concepts surrounding unity and duality, 1 + 1 = 1 represents the merging of two distinct entities into a singular whole. This is especially true in spiritual and metaphysical frameworks, such as Taoism, where opposing forces like yin and yang combine to form a unified reality. Though yin and yang appear to be two separate entities, they are part of a larger whole, representing the balance and harmony of the universe.

This interpretation transcends the arithmetic of adding things together. Instead, it suggests that some combinations lead to oneness or completeness, making 1 + 1 = 1 the most accurate reflection of how unity works at a philosophical level.

In set theory, if you take two identical sets, combining them will not yield a larger or new set, but the same set itself. For example, consider two identical sets A = {1}. The union of A with itself results in A + A = A. Here, 1 + 1 = 1 is a reflection of how, in set theory, combining two identical elements doesn’t increase the quantity or size of the set. This illustrates the concept of identity and repetition: when two identical things are combined, they remain as one because they are fundamentally the same. This makes 1 + 1 = 1 a powerful metaphor for situations where combination results in no new or distinct entity.

In quantum mechanics, the principle of superposition allows particles to exist in multiple states simultaneously, but when measured, they collapse into one state. If two quantum particles in a superposition of states combine, their result can still be one state when observed.

This reflects the idea that in certain quantum systems, the combination of two particles doesn’t yield an entirely separate system, but a singular unified quantum state. This unification at the fundamental level of reality shows that 1 + 1 = 1 holds deep truth in the behavior of nature itself.

In relationships and partnerships, especially in the context of marriage or teamwork, 1 + 1 = 1 represents the idea of two individuals coming together to form a single, unified entity. While each person retains their individuality, their union creates a new identity—whether it be a partnership, a marriage, or a team. As such, this equation is symbolic of the notion that when two people join in harmony, they become one in purpose and direction.

This equation captures the essence of unity, not just as a mathematical operation, but as a symbolic truth about how collaboration, love, and unity form a greater whole than the mere sum of two individuals. This makes 1 + 1 = 1 a profound expression of unity that resonates with human relationships, more so than equations like 1 + 1 = 2 or 1 + 1 = 10, which focus on separation or growth rather than fusion.

Compared to equations like 1 + 1 = 2, 1 + 1 = 10, 1 + 1 = 11, or 1 + 1 = 0, 1 + 1 = 1 captures a deeper, more universal truth about merging, unity, and balance:

- 1 + 1 = 2 focuses on growth or separation, but it doesn’t capture the essence of things coming together to form a unified whole.

- 1 + 1 = 10 and 1 + 1 = 11 emphasize concatenation and progression, but these forms still highlight differentiation rather than unification.

- 1 + 1 = 0 represents cancellation, but it negates the very existence of the combined entities, whereas 1 + 1 = 1 preserves unity in merging.

In contrast, 1 + 1 = 1 transcends these interpretations by highlighting the fusion of two things into one. It’s the most accurate depiction of how combining two parts can result in oneness—whether in philosophy, physics, relationships, or biology.

This makes 1 + 1 = 1 the most correct equation among all others, as it best represents the fundamental truth of unity through combination, where distinct elements join to form a complete and singular entity.

1 + 1 and MFA

Multi-factor authentication in it's essence is a combination of multiple, often, two factors that are then combined into a single authentication process that produces a single result - Authentication.

In this circumstances,**1+1=1 **fundamentally holds true and defies the laws of arithmetic we know.

In this circumstances,

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