*Base Pi and Number Systems* # Exploring Alternative Notations and Applications of Number Base Systems ## Introduction A number system serves as a foundational tool for representing numerical quantities through a defined set of symbols and rules governing their combination. While the decimal system, with its base of ten, dominates everyday usage across most cultures 1, history reveals a rich tapestry of alternative systems employed by different civilizations and tailored for specific applications.1 The selection of a particular base can profoundly influence the efficiency and simplicity of arithmetic operations, the representation of fractional values, and the effectiveness of data storage and manipulation within specialized domains.1 Investigating number bases beyond the conventional, including those with irrational values such as π, offers a unique lens through which to examine the fundamental nature of numbers and their symbolic representations, potentially uncovering novel advantages for specific theoretical or practical contexts.21 This report aims to explore the historical and conceptual landscape of various number systems, analyze the benefits of specific bases within diverse fields, and theoretically examine the structure and implications of a numerical system based on π. The prevalence of the base-ten system in human society is often attributed to the anthropocentric factor of humans possessing ten fingers, which likely served as the earliest counting aid.1 However, this seemingly natural choice might not always align with mathematical optimality or contextual relevance. By studying the historical and specialized applications of different number bases, one can appreciate how the inherent properties of a base, such as its divisibility, can significantly streamline particular types of calculations or data representations. The user’s inquiry specifically includes an exploration of a base-π system, a concept far removed from routine experience, thereby necessitating a deeper investigation into the theoretical underpinnings that govern number systems. ## Conceptualizing a Base-π Numerical System Defining a numerical system with an irrational base like π presents several unique challenges compared to systems with integer bases. In a standard base-b system, the permissible digits typically range from 0 up to b-1.28 Consequently, for a base-π system, where π is approximately 3.14159, the integer digits available would be 0, 1, 2, and 3.21 This constraint arises from the fundamental requirement that each digit’s value must be a non-negative integer strictly less than the base.28 Representing integer values within a base-π framework introduces further complexity. Unlike integer bases where integers can have finite representations, in a base-π system, most integers beyond 3 would necessitate non-terminating sequences of digits.21 For instance, the integer 4, when expressed in base-π, would require an infinite number of digits.30 This is because π itself is not an integer, and therefore, integer powers of π will generally not combine in a finite manner to yield integer results. The positional values in a base-π system are determined by powers of π, extending both positively and negatively from a central “units” place (..., π², π¹, π⁰, π⁻¹, π⁻²,...).29 This is a departure from integer base systems where these positional values are straightforward integer powers of the base.16 Furthermore, the concept of normalization, which ensures a standard and often unique representation of a number, becomes intricate in the context of non-integer bases.21 A single numerical value can possess multiple valid representations in such systems. For example, the number 1 in base-π can be expressed as 1.0, but also as an infinite series 0.3011....21 This non-uniqueness stems from the fact that the greedy algorithm, typically employed for integer bases to obtain a unique representation, does not always guarantee this property for non-integer bases, allowing for alternative, yet mathematically equivalent, digit sequences.21 Finally, performing arithmetic operations within a base-π system would deviate significantly from the familiar procedures used with integer bases.34 Standard algorithms for addition, subtraction, multiplication, and division are predicated on the properties of integer bases and would need substantial adaptation for a non-integer base like π, potentially introducing considerable complexity.34 The existence of the Golden Ratio base (base-φ ≈ 1.618) provides a compelling parallel, demonstrating the mathematical viability of non-integer number systems.21 In this system, any non-negative real number can be uniquely represented using only the digits 0 and 1, provided the digit sequence “11” is avoided in its standard form.37 This highlights that even with an irrational base, specific rules and constraints on digit usage can lead to standardized representations. Similar to the hypothetical base-π system, integers in base-φ possess unique terminating representations when expressed in their standard form.37 This suggests that while terminating integer representations might not be a universal feature of all non-integer bases, they are achievable in certain cases. Arithmetic operations within the base-φ system, such as addition and multiplication, follow defined rules derived from the algebraic properties of the golden ratio.37 This indicates that arithmetic in non-integer bases is indeed possible, albeit requiring a thorough understanding of the base’s algebraic characteristics. The existence of the base-φ system implies that a base-π system, while presenting its own unique challenges due to π’s transcendental nature, could also theoretically have defined rules for representation and arithmetic. However, it is important to note that the mathematical properties of transcendental numbers are generally more intricate than those of algebraic irrational numbers like the golden ratio.38 In a scenario where Arabic numerals are unavailable, devising alternative symbolic representations for the digits 0, 1, 2, and 3 in a base-π system necessitates drawing inspiration from historical numeral systems.45 These ancient systems offer a diverse range of approaches to representing the initial integers using distinct symbols. For the digit 0, a symbol denoting absence or nothingness would be appropriate, echoing the later Babylonian use of a placeholder 59 or the Greek symbol for zero in astronomical contexts.65 Let us propose a small circle:. For 1, a simple vertical stroke, as found in Egyptian hieroglyphs (I) 51 and Roman numerals (I) 45, would serve well: |. Following this additive principle common in early systems, 2 could be represented by two strokes: ||, and 3 by three strokes: |||, consistent with Roman numerals and Egyptian hieroglyphs for these values.45 Consequently, within this hypothetical base-π system devoid of Arabic numerals, the number π itself, following the convention that any base b is represented as 10 in that base 21, would be symbolized as |. The exercise of conceptualizing a base-π system underscores the fundamental distinction between a number’s inherent mathematical value and its chosen symbolic representation.24 The intrinsic properties of π as an irrational and transcendental number 38 would dictate the essential characteristics of such a system, most notably the infinite and non-repeating nature of the symbolic representations for the majority of numbers. The representation of π as ‘10’ in base-π is a direct consequence of the definition of a number base, where the base value itself is always represented as ‘10’. However, this choice of base has profound implications for the representation of other numbers, particularly those that are simple and finite in more conventional integer-based systems. ## Advantages of Conventional Number Base Systems Different number base systems offer distinct advantages that make them particularly well-suited for specific applications across various domains. Binary (Base-2) stands as the bedrock of modern computing and digital logic. Its inherent simplicity in representing two distinct states–typically denoted as 0 and 1–directly mirrors the fundamental on/off nature of electronic switches (transistors) that form the core of digital circuits. This direct correspondence allows for a straightforward mapping to Boolean logic (True/False, 1/0), making binary the ideal system for implementing logic gates and the fundamental circuits within computers. It is fundamental for representing all forms of data within computer systems, including numbers, text, images, and sound, in memory, storage devices, and during communication. Furthermore, binary search algorithms leverage the ordered nature of binary representations for efficient retrieval of information within large datasets.74 The simplicity of binary signals also contributes to their robustness against noise and allows for flawless copying of digital data, crucial for maintaining data integrity in computer systems.75 Hexadecimal (Base-16) offers a more compact and human-readable alternative for representing binary data, widely used in programming and computer science. Since 16 is a power of 2 (16 = 2⁴), each hexadecimal digit directly corresponds to a group of four binary digits (bits).77 This makes it significantly easier for humans to read and write large binary numbers, as one hexadecimal digit can represent the same information as four binary digits, leading to more concise representations.79 Hexadecimal notation is commonly employed for representing memory addresses, color codes in web design, Media Access Control (MAC) addresses, and Uniform Resource Locators (URLs).79 The straightforward conversion between hexadecimal and binary is a key advantage, allowing programmers to work with binary data in a more manageable format. Base-12 (Duodecimal) offers practical advantages, particularly in calculations involving division and everyday measurements. The number 12 boasts a higher number of divisors (1, 2, 3, 4, 6) compared to base-10 (1, 2, 5), which simplifies the representation of many common fractions. For instance, fractions like 1/3, 1/4, and 1/6 have short, terminating representations in base-12 (0.4, 0.3, 0.2 respectively), whereas 1/3 becomes a repeating decimal in base-10 (0.333...). Historically, base-12 has been used in various units of measurement, such as inches in a foot, months in a year, and in traditional currency systems like the British pound, shilling, and pence system.12 Some proponents also suggest that duodecimal multiplication tables exhibit more easily recognizable patterns compared to decimal tables.10 Base-60 (Sexagesimal) holds historical significance, originating in ancient Mesopotamia, and continues to be used in specific measurement domains. The number 60 is a highly composite number with twelve divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), including the prime numbers 2, 3, and 5. This high divisibility simplifies many fractional calculations. Its historical use in ancient Mesopotamia has left a lasting legacy in our modern systems for measuring time (hours, minutes, seconds) and angles (degrees, minutes, seconds). ## Exploring Niche Applications of Other Number Bases Beyond the commonly used binary, hexadecimal, base-12, and base-60 systems, other number bases find specialized applications in various fields, often leveraging unique properties suited to particular problems. Ternary (base-3) and balanced ternary systems have garnered interest in computation and logic.84 The standard ternary system employs digits 0, 1, and 2, while the balanced ternary system uses -1, 0, and 1, often represented as T, 0, and 1.85 Balanced ternary offers an elegant way to represent negative numbers without the need for a separate minus sign, simplifying certain arithmetic operations and potentially reducing the carry rate in multi-digit multiplication.87 Experimental computers, such as the Soviet Setun, explored the use of balanced ternary.87 Furthermore, ternary systems provide an efficient framework for representing self-similar fractal structures like the Sierpinski triangle and the Cantor set.84 Vigesimal (base-20) systems have a rich history, having been utilized by various cultures across the globe, including the Maya and Aztec civilizations.94 Remnants of vigesimal counting can still be found in the number naming conventions of some modern languages, such as French and Danish.94 The prevalence of base-20 systems in different cultures is often attributed to the natural method of counting using both fingers and toes.94 Quaternary (base-4) systems find specific utility due to their close relationship with the binary system.95 Since 4 is a power of 2 (4 = 2²), each quaternary digit maps directly to two binary digits, facilitating efficient conversion between the two systems.95 This property can be advantageous in certain computational contexts. Quaternary numbers are also employed in representing Hilbert curves, a type of space-filling curve.95 Interestingly, parallels exist between quaternary numerals and the four nucleotides (A, C, G, T) that form the genetic code of DNA, allowing for DNA sequences to be represented as quaternary numbers.95 Beyond these examples, other number bases have found niche applications. Base-8 (octal) is sometimes used in computing, particularly in representing file permissions in Unix-based systems, as it provides a more human-readable format than binary while still having a close relationship with powers of 2. Base-36, utilizing digits 0-9 and letters A-Z, is employed as a binary-to-text encoding scheme, offering a more compact representation than pure binary. These examples illustrate that the choice of number base is not merely a matter of convention but can be strategically selected to optimize representation, calculation, or mapping to specific structures within a given domain. ## The Unique Case of Base-π Fundamental mathematical constants like π, when represented in different number bases, exhibit unique characteristics dictated by their inherent mathematical properties. In a base-π system, the number π itself is represented simply as 10.21 Following this convention, powers of π would be represented as 1 followed by a corresponding number of zeros; for instance, π² would be 100, and π³ would be 1000.21 However, other fundamental mathematical constants, such as e (Euler’s number) and √2 (the square root of two), would have infinite, non-repeating representations in a base-π system.21 This is because e is a transcendental number, meaning it is not a root of any non-zero polynomial equation with rational coefficients 38, and √2 is an irrational number, meaning it cannot be expressed as a ratio of two integers.38 Due to π’s transcendental nature, rational numbers, which have terminating or repeating representations in integer bases, would generally also have infinite, non-repeating representations in base-π.23 Theoretically, a base-π system might offer certain advantages, particularly in relation to geometry and cyclical phenomena, given the fundamental role of π in these areas.109 Such a system could potentially simplify mathematical formulas involving circles and spheres, as the relationships between diameter, circumference, radius, and area would become more directly expressed in terms of powers of the base.121 For instance, in base-π, a circle with a diameter of 1 would have a circumference of 10.121 Similarly, modeling periodic phenomena, which are often mathematically described using π (e.g., sine and cosine functions) 38, might find a more natural and concise representation in a base-π system. Some even speculate that fundamental constants and patterns observed in the natural world might exhibit more elegant expressions when viewed through the lens of a base-π numerical system.112 Despite these potential theoretical advantages, a base-π system would face significant challenges and limitations, especially concerning the representation of rational numbers.22 Simple rational numbers like 1, 2, and 3, which have straightforward finite representations in our familiar base-10 system, would generally require infinite, non-repeating “digits” in base-π, making even basic arithmetic operations exceptionally cumbersome.21 The comparison and ordering of numbers could also become more complex due to the infinite nature of their representations.28 Furthermore, given the widespread adoption and utility of base-10 for everyday calculations and base-2 for computational applications, there is no clear practical advantage that would necessitate or even suggest the adoption of a base-π system.96 The concept of “normalcy,” which refers to the statistical distribution of digits in an infinite sequence, is also more complex to define and analyze for irrational bases compared to integer bases.38 In essence, while a base-π system offers an intriguing perspective centered around the mathematical constant π, it introduces substantial complexities for representing other numbers, particularly those within the set of rational numbers that form the foundation of most practical and scientific computations. The theoretical elegance it might provide for formulas directly involving circles and cyclical phenomena likely does not outweigh the significant practical difficulties that would arise in basic arithmetic and general numerical representation. ## Conclusion The exploration of alternative number base systems reveals that the strategic choice of a base is often dictated by the specific requirements and priorities of a given domain. The binary system, with its simplicity and direct correspondence to electronic states, remains the cornerstone of modern computing. Hexadecimal provides a more concise and human-friendly representation of binary data. The duodecimal system simplifies common fractional calculations due to the higher number of divisors of 12. The sexagesimal system, with its base of 60, continues to be used for measurements of time and angles, a testament to its high divisibility and historical precedent. While a base-π system offers a unique perspective on the number π, simplifying its representation, it introduces significant complexities for representing other numbers, particularly rational numbers, which form the bedrock of most everyday and scientific computations. The theoretical elegance for formulas directly involving circles might not outweigh the practical difficulties in arithmetic and general representation. Ultimately, the strategic choice of number base depends heavily on the specific needs and priorities of the domain of application, balancing factors such as efficiency of representation, simplicity of arithmetic, and ease of conversion with existing systems. While base-10 serves as a convenient system for everyday human use, and binary is indispensable for the functioning of computers, the exploration of bases like base-π underscores the rich mathematical landscape of number systems and the potential for tailored solutions in specialized contexts. The choice of base is ultimately a tool, and its efficacy is intrinsically linked to the specific task at hand. (Table 1: Comparison of Number Base Systems) | | | | | | | | ---- | ----------- | ------------------- | ---------------------------------------------------------------------- | ---------------------------------------------------------------------- | ----------------------------------------------------------------------------- | | Base | Name | Digits Used | Primary Advantages | Primary Disadvantages | Common Applications | | 2 | Binary | 0, 1 | Simple implementation in electronics, direct Boolean logic mapping | Lengthy representation for large numbers | Computer science, digital electronics, data storage | | 10 | Decimal | 0-9 | Intuitive for humans, widely adopted | Not optimal for all divisions (e.g., 1/3) | Everyday calculations, general mathematics | | 12 | Duodecimal | 0-9, T (10), E (11) | Easier division by 2, 3, 4, 6 | Not as widely adopted, requires new symbols | Historical measurements, some proposals for wider adoption | | 16 | Hexadecimal | 0-9, A-F | Compact representation of binary, easy conversion to binary | Less intuitive for non-programmers | Programming, memory addressing, color codes | | π | Base-π | 0, 1, 2, 3 | π represented as 10, potentially simplifies formulas involving circles | Infinite representations for most rational numbers, complex arithmetic | Theoretical exploration, potential niche applications in geometry and physics | (Table 2: Representation of Integers 1-10 in Base-π (Hypothetical)) | | | |---|---| |Decimal Number|Base-π Representation| |1|1.0| |2|2.0| |3|3.0| |4|10.2201...| |5|11.2201...| |6|12.2201...| |7|20.2021...| |8|21.2021...| |9|22.2021...| |10|100.0102...| (Table 3: Divisibility of Base-10 vs. Base-12) | | | | |---|---|---| |Number (n)|1/n (Base-10)|1/n (Base-12)| |1|1|1| |2|0.5|0.6| |3|0.333...|0.4| |4|0.25|0.3| |5|0.2|0.2497...| |6|0.1666...|0.2| |7|0.1428...|0.186A...| |8|0.125|0.16| |9|0.111...|0.14| |10|0.1|0.1249...| |11|0.0909...|0.10E0...| |12|0.0833...|0.1| (Table 4: Comparison of Precision in Different Numerical Bases (Approximate)) | | | | |---|---|---| |Base|Digits/Bits to Represent 0-255|Approximate Decimal Precision| |2|8 bits|2.4 digits| |10|3 digits|3 digits| |16|2 digits|2.4 digits| |12|3 digits|2.8 digits| |π|~2.something digits|N/A (complex due to irrationality)| ### Works Cited 1. math.libretexts.org, accessed April 7, 2025, [https://math.libretexts.org/Courses/Mount_Royal_University/Higher_Arithmetic/7%3A_Numeration_Systems/7.2%3A_Number_Bases#:~:text=A%20number%20base%20is%20the,commonly%20known%20as%20base%2010.](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://math.libretexts.org/Courses/Mount_Royal_University/Higher_Arithmetic/7%25253A_Numeration_Systems/7.2%25253A_Number_Bases%2523:~:text%253DA%252520number%252520base%252520is%252520the,commonly%252520known%252520as%252520base%25252010.%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184297192%26amp;usg%3DAOvVaw3UDkhk3RqC5Kd1PxF3or8x&sa=D&source=docs&ust=1744019184402439&usg=AOvVaw2Gk9Ef7PJQ_iiOxD7VT5TW) 2. Understand Base in Math - Definition, Examples, Different Numeral Systems - Cuemath, accessed April 7, 2025, [https://www.cuemath.com/numbers/base/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.cuemath.com/numbers/base/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184297769%26amp;usg%3DAOvVaw2yZUG4dgf7FDUJWF6pB1No&sa=D&source=docs&ust=1744019184402715&usg=AOvVaw2Oy8_ptBVuoCdu4cWWHvdr) 3. brilliant.org, accessed April 7, 2025, [https://brilliant.org/wiki/number-base/#:~:text=A%20number%20base%20(or%20base,also%20known%20as%20base%2010.](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://brilliant.org/wiki/number-base/%2523:~:text%253DA%252520number%252520base%252520\(or%252520base,also%252520known%252520as%252520base%25252010.%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184298345%26amp;usg%3DAOvVaw3Qd3PcSVsUMWAUIt1rMjeX&sa=D&source=docs&ust=1744019184402805&usg=AOvVaw2W_wn8xFVSrwWG-U2xmMzw) 4. Number Bases - (AP Computer Science Principles) - Vocab, Definition, Explanations | Fiveable, accessed April 7, 2025, [https://library.fiveable.me/key-terms/ap-comp-sci-p/number-bases](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://library.fiveable.me/key-terms/ap-comp-sci-p/number-bases%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184298995%26amp;usg%3DAOvVaw2OwwFcU5DG7ZE6lDTB26IO&sa=D&source=docs&ust=1744019184402917&usg=AOvVaw39aabiLY-LCN-tvtny6vsp) 5. Number Systems / Number Bases - TRCCompSci - AQA Computer Science, accessed April 7, 2025, [https://www.trccompsci.online/mediawiki/index.php/Number_Systems_/_Number_Bases](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.trccompsci.online/mediawiki/index.php/Number_Systems_/_Number_Bases%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184299610%26amp;usg%3DAOvVaw13j0Z-9_y9R77YQnmxRmpR&sa=D&source=docs&ust=1744019184403022&usg=AOvVaw24Xz4N_dGZj_UuLUBJz4EF) 6. NUMBER BASES - Base 10, Base 2, Base 8 & Base 16 - YouTube, accessed April 7, 2025, [https://www.youtube.com/watch?v=j_pK0QOpN2I](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.youtube.com/watch?v%253Dj_pK0QOpN2I%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184300062%26amp;usg%3DAOvVaw2qYLPc7ax29O0Ccmv9NnXx&sa=D&source=docs&ust=1744019184403100&usg=AOvVaw3PNmQilZE-ncLDjMxaBJlv) 7. Lesson Note: Application of Number Base Systems in Computer Programming - FCT EMIS, accessed April 7, 2025, [https://fctemis.org/notes/17057_Application%20of%20Number%20Base%20system%20in%20computer%20programming%20.pdf](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://fctemis.org/notes/17057_Application%252520of%252520Number%252520Base%252520system%252520in%252520computer%252520programming%252520.pdf%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184300816%26amp;usg%3DAOvVaw3-K32gst7Tv9Ndw5B2byuw&sa=D&source=docs&ust=1744019184403201&usg=AOvVaw0O6VNE7lx66xxYU1f2ismT) 8. www.reddit.com, accessed April 7, 2025, [https://www.reddit.com/r/askscience/comments/1wx004/what_are_the_advantages_for_a_base_12_system/#:~:text=One%20major%20advantage%20is%20that,10%20does%20not%20quarter%20easily.](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.reddit.com/r/askscience/comments/1wx004/what_are_the_advantages_for_a_base_12_system/%2523:~:text%253DOne%252520major%252520advantage%252520is%252520that,10%252520does%252520not%252520quarter%252520easily.%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184301561%26amp;usg%3DAOvVaw1Ci7acVptR2FyFE-6HnLnp&sa=D&source=docs&ust=1744019184403321&usg=AOvVaw2_r-G6CrXVd9bYuGbo-nmk) 9. What are the advantages for a base 12 system?: r/askscience - Reddit, accessed April 7, 2025, [https://www.reddit.com/r/askscience/comments/1wx004/what_are_the_advantages_for_a_base_12_system/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.reddit.com/r/askscience/comments/1wx004/what_are_the_advantages_for_a_base_12_system/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184302285%26amp;usg%3DAOvVaw39jGjSGDRU3n0YqnUbO5qA&sa=D&source=docs&ust=1744019184403433&usg=AOvVaw2IxewN0zVOBnFSe84QtDow) 10. The Curious Case For Base 12 (Why Dozens Are Easier For Everyday Maths Than Tens), accessed April 7, 2025, [https://steemit.com/mathematics/@rocking-dave/the-curios-case-for-base-12-why-dozens-are-easier-for-everyday-maths-than-tens](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://steemit.com/mathematics/@rocking-dave/the-curios-case-for-base-12-why-dozens-are-easier-for-everyday-maths-than-tens%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184303077%26amp;usg%3DAOvVaw1WxUK8G-wZLKeRaaSxj094&sa=D&source=docs&ust=1744019184403532&usg=AOvVaw0WJhDVZDKDaggZSNFTftfD) 11. Base 12 - Why Counting In Twelves Would Make Life Easier - YouTube, accessed April 7, 2025, [https://www.youtube.com/watch?v=y_QBDrBlbds](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.youtube.com/watch?v%253Dy_QBDrBlbds%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184303590%26amp;usg%3DAOvVaw2e4-jhB3ApDOKky2Ct1KaH&sa=D&source=docs&ust=1744019184403641&usg=AOvVaw2JDxwjDaCZQc0y2mVjfXBM) 12. Base 12: An Introduction | Built In, accessed April 7, 2025, [https://builtin.com/data-science/base-12](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://builtin.com/data-science/base-12%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184304083%26amp;usg%3DAOvVaw1v-gKAJ9UMwWB_vYBzIDlp&sa=D&source=docs&ust=1744019184403724&usg=AOvVaw2z8xnU2_XX1dZ_vVv2SAMq) 13. Sexagesimal - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Sexagesimal](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Sexagesimal%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184304467%26amp;usg%3DAOvVaw0shJL5YewetoqYcS0GEBuy&sa=D&source=docs&ust=1744019184403815&usg=AOvVaw15YiDemfP2_hN0Mxr5rKrE) 14. Babylonian Mathematics and the Base 60 System - ThoughtCo, accessed April 7, 2025, [https://www.thoughtco.com/why-we-still-use-babylonian-mathematics-116679](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.thoughtco.com/why-we-still-use-babylonian-mathematics-116679%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184305026%26amp;usg%3DAOvVaw1OKXONSUR7eK_osi4BHfyz&sa=D&source=docs&ust=1744019184403901&usg=AOvVaw35XSUatXBz_2PnzVAy1Rxx) 15. Why is base 60 more precise for trigonometry, can you give an example?, accessed April 7, 2025, [https://math.stackexchange.com/questions/2405480/why-is-base-60-more-precise-for-trigonometry-can-you-give-an-example](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://math.stackexchange.com/questions/2405480/why-is-base-60-more-precise-for-trigonometry-can-you-give-an-example%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184305808%26amp;usg%3DAOvVaw0kHboWRPXtgspHTWDNKprJ&sa=D&source=docs&ust=1744019184403992&usg=AOvVaw0mmSlro6JmiuzbJm4HpEd-) 16. 7.2: Number Bases - Mathematics LibreTexts, accessed April 7, 2025, [https://math.libretexts.org/Courses/Mount_Royal_University/Higher_Arithmetic/7%3A_Numeration_Systems/7.2%3A_Number_Bases](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://math.libretexts.org/Courses/Mount_Royal_University/Higher_Arithmetic/7%25253A_Numeration_Systems/7.2%25253A_Number_Bases%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184306459%26amp;usg%3DAOvVaw3FPMVL9I39Ah2Wh7r7hQK9&sa=D&source=docs&ust=1744019184404091&usg=AOvVaw0-3i9FiVaVxnN1dEPZNdNu) 17. Number Base | Brilliant Math & Science Wiki, accessed April 7, 2025, [https://brilliant.org/wiki/number-base/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://brilliant.org/wiki/number-base/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184307018%26amp;usg%3DAOvVaw0H_88XB0sr146HdhL1sDJ7&sa=D&source=docs&ust=1744019184404204&usg=AOvVaw2r_K-uB0DiHHTK5USvt1cx) 18. Intro to number bases & How to do binary numbers - Purplemath, accessed April 7, 2025, [https://www.purplemath.com/modules/numbbase.htm](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.purplemath.com/modules/numbbase.htm%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184307547%26amp;usg%3DAOvVaw0XK9GVDZbEkPTdx-904W8l&sa=D&source=docs&ust=1744019184404292&usg=AOvVaw3QWrI5tEX4e3rDU9EPFKCj) 19. Base (mathematics) - Simple English Wikipedia, the free encyclopedia, accessed April 7, 2025, [https://simple.wikipedia.org/wiki/Base_(mathematics)](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://simple.wikipedia.org/wiki/Base_\(mathematics\)%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184308133%26amp;usg%3DAOvVaw2QU1hN_7ZVocThAXLM500w&sa=D&source=docs&ust=1744019184404384&usg=AOvVaw1TOWbvSIhLW2OlHSmHmUgh) 20. Introduction to number systems and binary (video) - Khan Academy, accessed April 7, 2025, [https://www.khanacademy.org/math/algebra-home/alg-intro-to-algebra/algebra-alternate-number-bases/v/number-systems-introduction](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.khanacademy.org/math/algebra-home/alg-intro-to-algebra/algebra-alternate-number-bases/v/number-systems-introduction%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184308932%26amp;usg%3DAOvVaw0LGr6h2x3B1TnIb7NfGb4g&sa=D&source=docs&ust=1744019184404489&usg=AOvVaw0Bz4UzkKwEd2Rnc7djIaWe) 21. Numbers in base φ, π, and so on | Adam Scherlis, accessed April 7, 2025, [https://adam.scherlis.com/2022/06/30/numbers-in-base-%CF%86-%CF%80-and-so-on/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://adam.scherlis.com/2022/06/30/numbers-in-base-%2525CF%252586-%2525CF%252580-and-so-on/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184309507%26amp;usg%3DAOvVaw3jwKmLejODCzk2w4qrk2sq&sa=D&source=docs&ust=1744019184404591&usg=AOvVaw23A_4DxSJ0TswbW2iX51xv) 22. Are there any significant advantages or funny problems with using base pi number system?, accessed April 7, 2025, [https://www.reddit.com/r/mathematics/comments/176l2qj/are_there_any_significant_advantages_or_funny/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.reddit.com/r/mathematics/comments/176l2qj/are_there_any_significant_advantages_or_funny/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184310184%26amp;usg%3DAOvVaw0nIWe9Ppkav72Mu5ooYZa3&sa=D&source=docs&ust=1744019184404698&usg=AOvVaw0Cdd7uTeW_oeTayYa6cjRY) 23. Can pi be expressed exactly by using a system other than decimal?: r/math - Reddit, accessed April 7, 2025, [https://www.reddit.com/r/math/comments/dtksf/can_pi_be_expressed_exactly_by_using_a_system/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.reddit.com/r/math/comments/dtksf/can_pi_be_expressed_exactly_by_using_a_system/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184310874%26amp;usg%3DAOvVaw0KrLVL-Aa6fc5k1y7WE8HB&sa=D&source=docs&ust=1744019184404802&usg=AOvVaw1XLAMV3ip0gciGYqAyuBJD) 24. Can pi be expressed rationally in base pi? I still don’t understand. - Reddit, accessed April 7, 2025, [https://www.reddit.com/r/learnmath/comments/1bngym5/can_pi_be_expressed_rationally_in_base_pi_i_still/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.reddit.com/r/learnmath/comments/1bngym5/can_pi_be_expressed_rationally_in_base_pi_i_still/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184311511%26amp;usg%3DAOvVaw0eQaA2wLLepWtL11npCG6v&sa=D&source=docs&ust=1744019184404978&usg=AOvVaw25pIDCAiPnzr0mMpq1DTn2) 25. Base Systems, accessed April 7, 2025, [https://www.ucd.ie/mathstat/t4media/3.%20Base%20systems.pdf](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.ucd.ie/mathstat/t4media/3.%252520Base%252520systems.pdf%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184311971%26amp;usg%3DAOvVaw042CMbW4LCQAK1UHmRb9Z2&sa=D&source=docs&ust=1744019184405087&usg=AOvVaw2_MSKhu5yUOkW55Rct2XWA) 26. What would be the changes and advantages/disadvantages using numbers systems other than base 10?: r/askmath - Reddit, accessed April 7, 2025, [https://www.reddit.com/r/askmath/comments/18jpkfs/what_would_be_the_changes_and/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.reddit.com/r/askmath/comments/18jpkfs/what_would_be_the_changes_and/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184312680%26amp;usg%3DAOvVaw1nb-NJd8fZFtXVdTIngAq-&sa=D&source=docs&ust=1744019184405194&usg=AOvVaw2CEFNJ20k_oreMnou5vC1N) 27. Big Idea: The Base Ten System | Strategies for Teaching Elementary Mathematics, accessed April 7, 2025, [https://mathteachingstrategies.wordpress.com/2009/04/02/big-idea-the-base-ten-system/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://mathteachingstrategies.wordpress.com/2009/04/02/big-idea-the-base-ten-system/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184313311%26amp;usg%3DAOvVaw3FoeYavMyGCtLW3RisI5EN&sa=D&source=docs&ust=1744019184405303&usg=AOvVaw1FlTuZmTjqGqI8YCXupo6l) 28. Is it possible to write any rational number using base pi - Math Stack Exchange, accessed April 7, 2025, [https://math.stackexchange.com/questions/2385022/is-it-possible-to-write-any-rational-number-using-base-pi](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://math.stackexchange.com/questions/2385022/is-it-possible-to-write-any-rational-number-using-base-pi%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184314013%26amp;usg%3DAOvVaw3HnOvDa5AYdox3M0kBwjP3&sa=D&source=docs&ust=1744019184405418&usg=AOvVaw0wG2qkQAZspICNQHYmLhyD) 29. Integers in Base Pi - Code Golf Stack Exchange, accessed April 7, 2025, [https://codegolf.stackexchange.com/questions/161206/integers-in-base-pi](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://codegolf.stackexchange.com/questions/161206/integers-in-base-pi%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184314545%26amp;usg%3DAOvVaw2FkXKcctw7NCQXxVmA-Nc0&sa=D&source=docs&ust=1744019184405529&usg=AOvVaw3M1uZhVbsWqOnSZNDsF6SU) 30. Way Off Base (by David A. Wheeler), accessed April 7, 2025, [https://dwheeler.com/essays/bases.html](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://dwheeler.com/essays/bases.html%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184314985%26amp;usg%3DAOvVaw36b_oAf76jnuJHYH2MuTv2&sa=D&source=docs&ust=1744019184405629&usg=AOvVaw2-8xsxeliQTmdpFL3nmJvv) 31. What is the Base-10 Number System? - ThoughtCo, accessed April 7, 2025, [https://www.thoughtco.com/definition-of-base-10-2312365](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.thoughtco.com/definition-of-base-10-2312365%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184315489%26amp;usg%3DAOvVaw1nnRnwwdmyNbbV7FCLGwGo&sa=D&source=docs&ust=1744019184405720&usg=AOvVaw1JXLTia4Uca2HkarKZEsVK) 32. Number Bases - Discrete Math - Codecademy, accessed April 7, 2025, [https://www.codecademy.com/resources/docs/discrete-math/number-bases](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.codecademy.com/resources/docs/discrete-math/number-bases%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184316045%26amp;usg%3DAOvVaw0fmOPXsWQgzGNH0JGZsr_5&sa=D&source=docs&ust=1744019184405814&usg=AOvVaw1ySgmlTN5O7dG8QLFje9W5) 33. www.techtarget.com, accessed April 7, 2025, [https://www.techtarget.com/whatis/definition/decimal#:~:text=The%20decimal%20system%20consists%20of,the%20right%20goes%20beyond%209.](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.techtarget.com/whatis/definition/decimal%2523:~:text%253DThe%252520decimal%252520system%252520consists%252520of,the%252520right%252520goes%252520beyond%2525209.%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184316720%26amp;usg%3DAOvVaw2Q_3s-P3T1EB5VHcSyEgrn&sa=D&source=docs&ust=1744019184405919&usg=AOvVaw1XdiGwt16lPNSghvzcOc-f) 34. 5.3: Calculating in Other Bases - Mathematics LibreTexts, accessed April 7, 2025, [https://math.libretexts.org/Courses/Las_Positas_College/Math_for_Liberal_Arts/05%3A_Numeration_Systems/5.03%3A_Calculating_in_Other_Bases](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://math.libretexts.org/Courses/Las_Positas_College/Math_for_Liberal_Arts/05%25253A_Numeration_Systems/5.03%25253A_Calculating_in_Other_Bases%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184317502%26amp;usg%3DAOvVaw1J_tIb9oaZmWLIGLVToipN&sa=D&source=docs&ust=1744019184406039&usg=AOvVaw2K_iQVdeAsuC4fohW3b7Wq) 35. Number Base Calculator, accessed April 7, 2025, [http://www.cleavebooks.co.uk/scol/calnumba.htm](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttp://www.cleavebooks.co.uk/scol/calnumba.htm%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184317941%26amp;usg%3DAOvVaw1SeRlVqjuP4Q_ot42cCxHO&sa=D&source=docs&ust=1744019184406146&usg=AOvVaw2P5FczyOQxxl7FSSRYtRqN) 36. Equations Involving Number Bases | Likely Examination Questions | Maths Center, accessed April 7, 2025, [https://m.youtube.com/watch?v=sb-4l6dMOHA&pp=ygUGIzEzMnh3](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://m.youtube.com/watch?v%253Dsb-4l6dMOHA%2526pp%253DygUGIzEzMnh3%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184318547%26amp;usg%3DAOvVaw0r-t8TfiyeLg4FwCU2x6sQ&sa=D&source=docs&ust=1744019184406249&usg=AOvVaw3HLVzIplGfdQjLRFRYNX6M) 37. Golden ratio base - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Golden_ratio_base](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Golden_ratio_base%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184318966%26amp;usg%3DAOvVaw00DnczJpoXUyEmyOaKzHJ-&sa=D&source=docs&ust=1744019184406356&usg=AOvVaw1tA5CzybWIy0hJreeaQNJM) 38. Pi - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Pi](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Pi%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184319330%26amp;usg%3DAOvVaw28Ut4yxOH-buTpt7XEfiav&sa=D&source=docs&ust=1744019184406451&usg=AOvVaw1cApGolmk76QN7GfKp820u) 39. How do we know pi is an irrational number? - Live Science, accessed April 7, 2025, [https://www.livescience.com/physics-mathematics/mathematics/how-do-we-know-pi-is-an-irrational-number](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.livescience.com/physics-mathematics/mathematics/how-do-we-know-pi-is-an-irrational-number%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184319976%26amp;usg%3DAOvVaw1LFV0IUCNgLJIWLEenJ_gS&sa=D&source=docs&ust=1744019184406549&usg=AOvVaw19Srn2KEa1kxbwy3gpqrqv) 40. Knowing PI better - Atractor, accessed April 7, 2025, [https://www.atractor.pt/mat/fromPI/piintro-_en.html](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.atractor.pt/mat/fromPI/piintro-_en.html%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184320411%26amp;usg%3DAOvVaw0ebzuZzl11__qhgkwBDetX&sa=D&source=docs&ust=1744019184406716&usg=AOvVaw3OLO7DfV4hQGw6eSCUrtCd) 41. Secrets of Pi: Unlocking the Mystry of Mathematical Constant, accessed April 7, 2025, [https://mathematicalexplorations.co.in/secrets-of-pi-unlocking-the-mystry/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://mathematicalexplorations.co.in/secrets-of-pi-unlocking-the-mystry/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184320988%26amp;usg%3DAOvVaw3uABABCwTQm0ly4aERIwRy&sa=D&source=docs&ust=1744019184406818&usg=AOvVaw2Xk0mWiIJmjpZ3RIa74bg-) 42. Deciphering π: A Journey from 22/7 to Transcendental Truth. | by Srinivasa Raghava K, accessed April 7, 2025, [https://medium.com/@srinivasaraghavak/deciphering-%CF%80-a-journey-from-22-7-to-transcendental-truth-19c73488d506](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://medium.com/@srinivasaraghavak/deciphering-%2525CF%252580-a-journey-from-22-7-to-transcendental-truth-19c73488d506%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184321767%26amp;usg%3DAOvVaw30VT9A86OwBdtQLHFHxdWp&sa=D&source=docs&ust=1744019184406928&usg=AOvVaw31-R02QtJFrpuxOfyqFZ7W) 43. Irrational Numbers: Examples & Properties - Vaia, accessed April 7, 2025, [https://www.vaia.com/en-us/explanations/math/pure-maths/irrational-numbers/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.vaia.com/en-us/explanations/math/pure-maths/irrational-numbers/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184322316%26amp;usg%3DAOvVaw1ycRglDgfIJIOlbssxoYNV&sa=D&source=docs&ust=1744019184407037&usg=AOvVaw0bP1dszcOPfqZegZa8w5hk) 44. en.wikipedia.org, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Pi#:~:text=It%20is%20a%20transcendental%20number,with%20a%20compass%20and%20straightedge.](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Pi%2523:~:text%253DIt%252520is%252520a%252520transcendental%252520number,with%252520a%252520compass%252520and%252520straightedge.%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184322903%26amp;usg%3DAOvVaw0PtPicnUdg83F8-tZtNm15&sa=D&source=docs&ust=1744019184407147&usg=AOvVaw2A78dCxzO8KH9J9DA4Zkeg) 45. Exploring Different Number Systems: From Roman Numerals to Mayan Numbers, accessed April 7, 2025, [https://www.mathnasium.com/math-centers/pointloma/news/exploring-different-number-systems-roman-numerals-mayan-numbers-pl](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.mathnasium.com/math-centers/pointloma/news/exploring-different-number-systems-roman-numerals-mayan-numbers-pl%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184323684%26amp;usg%3DAOvVaw1s9WajFhd1y_ILeoeuln-N&sa=D&source=docs&ust=1744019184407270&usg=AOvVaw2yokyXGJKfhOSRm_Q0_QIF) 46. byjus.com, accessed April 7, 2025, [https://byjus.com/maths/roman-numerals/#:~:text=What%20are%20Roman%20Numerals%3F,8%2C%209%20and%2010%20respectively.](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://byjus.com/maths/roman-numerals/%2523:~:text%253DWhat%252520are%252520Roman%252520Numerals%25253F,8%25252C%2525209%252520and%25252010%252520respectively.%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184324261%26amp;usg%3DAOvVaw0xQkZGl3KAHh8T21PXhliV&sa=D&source=docs&ust=1744019184407381&usg=AOvVaw0iMFWKFo3rJdA6X2hMtPqn) 47. Roman numerals - Kids | Britannica Kids | Homework Help, accessed April 7, 2025, [https://kids.britannica.com/kids/article/Roman-numerals/390051](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://kids.britannica.com/kids/article/Roman-numerals/390051%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184324842%26amp;usg%3DAOvVaw39X3Shi2FfKF1-XX4SdtnR&sa=D&source=docs&ust=1744019184407497&usg=AOvVaw1x3OAB2S-VIRwmZGqaOd3g) 48. How To Use Roman Numerals - Dictionary.com, accessed April 7, 2025, [https://www.dictionary.com/e/roman-numerals/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.dictionary.com/e/roman-numerals/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184325291%26amp;usg%3DAOvVaw2OIYXAOzP7gE1-Pq_-GXm5&sa=D&source=docs&ust=1744019184407581&usg=AOvVaw2crFoLE4FKcZJDySy_ODD0) 49. Roman numeral | Chart, LIX, & Facts - Britannica, accessed April 7, 2025, [https://www.britannica.com/topic/Roman-numeral](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.britannica.com/topic/Roman-numeral%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184325745%26amp;usg%3DAOvVaw1bwTBOeqrQAE-CZHl9MVip&sa=D&source=docs&ust=1744019184407688&usg=AOvVaw10fIKxp26q4ZXiu-mF3gw7) 50. Roman Numerals - Definition, Chart, Conversion Rules, Examples - BYJU’S, accessed April 7, 2025, [https://byjus.com/maths/roman-numerals/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://byjus.com/maths/roman-numerals/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184326226%26amp;usg%3DAOvVaw1demcLkMDbPJ74QpczqRNK&sa=D&source=docs&ust=1744019184407775&usg=AOvVaw0CjDHmbPcYD8De7yNrI9gV) 51. Egyptian Mathematics Numbers Hieroglyphs - Discovering Egypt, accessed April 7, 2025, [https://discoveringegypt.com/egyptian-hieroglyphic-writing/egyptian-mathematics-numbers-hieroglyphs/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://discoveringegypt.com/egyptian-hieroglyphic-writing/egyptian-mathematics-numbers-hieroglyphs/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184326878%26amp;usg%3DAOvVaw1pjWE_se3REajJ7SsWW6JV&sa=D&source=docs&ust=1744019184407856&usg=AOvVaw22F8Me_UtJXtqBZHx7pZVZ) 52. History of Ancient Egyptian Numbers - Historicaleve -, accessed April 7, 2025, [https://historicaleve.com/history-of-ancient-egyptian-numbers/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://historicaleve.com/history-of-ancient-egyptian-numbers/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184327363%26amp;usg%3DAOvVaw33Mmn9GX0cfiHzrRsPXywp&sa=D&source=docs&ust=1744019184407959&usg=AOvVaw16oTeYRw93xHegUaLA0sZ5) 53. An Ancient Egyptian Mathematical Photo Album–Hieroglyph Numerals and More, accessed April 7, 2025, [https://old.maa.org/press/periodicals/convergence/an-ancient-egyptian-mathematical-photo-album-hieroglyph-numerals-and-more](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://old.maa.org/press/periodicals/convergence/an-ancient-egyptian-mathematical-photo-album-hieroglyph-numerals-and-more%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184328159%26amp;usg%3DAOvVaw0lojXXRrxm9ZtRTeUZ87X_&sa=D&source=docs&ust=1744019184408058&usg=AOvVaw3Hq6zTYOGsdKi-9hUjJWbE) 54. Ancient Egyptian Numerals - Teaching London Computing, accessed April 7, 2025, [https://teachinglondoncomputing.org/egyptians/ancient-egyptian-numerals/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://teachinglondoncomputing.org/egyptians/ancient-egyptian-numerals/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184328702%26amp;usg%3DAOvVaw3CBui0Bx5TwfL7vRJ-Tu2Q&sa=D&source=docs&ust=1744019184408160&usg=AOvVaw1PMUaqAvfbxTj41LnVM7el) 55. OER Ancient Egyptian Numerals and Arithmetic Activity - Digital Commons, accessed April 7, 2025, [https://digitalcommons.pittstate.edu/cgi/viewcontent.cgi?article=1007&context=math_faculty](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://digitalcommons.pittstate.edu/cgi/viewcontent.cgi?article%253D1007%2526context%253Dmath_faculty%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184329309%26amp;usg%3DAOvVaw0opqYFxo02HZIzb81zDwY7&sa=D&source=docs&ust=1744019184408260&usg=AOvVaw1imV_0IpCgejRY2jWIdcdD) 56. Babylonian Mathematics - CDLI Wiki, accessed April 7, 2025, [https://cdli.ox.ac.uk/wiki/doku.php?id=babylonian_mathematics](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://cdli.ox.ac.uk/wiki/doku.php?id%253Dbabylonian_mathematics%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184329813%26amp;usg%3DAOvVaw2afZvKuYVWxYbFJ5F-IAEM&sa=D&source=docs&ust=1744019184408357&usg=AOvVaw3adpU8pZ4BCsQ8ZBWTt4NS) 57. Babylonian Numeration System - YouTube, accessed April 7, 2025, [https://www.youtube.com/watch?v=WN3Dh5SvpA0](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.youtube.com/watch?v%253DWN3Dh5SvpA0%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184330260%26amp;usg%3DAOvVaw2N_U0z0PRtsTTxXOWBTnaz&sa=D&source=docs&ust=1744019184408465&usg=AOvVaw1rLEJbq5bBBWubQwEJv1z_) 58. Babylonian Numeration System, accessed April 7, 2025, [https://www.esu.edu/tutoring/documents/21-22/numeration/babylonian_va.pdf](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.esu.edu/tutoring/documents/21-22/numeration/babylonian_va.pdf%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184330769%26amp;usg%3DAOvVaw099UpeRYyBD5IZNoEAvUGL&sa=D&source=docs&ust=1744019184408552&usg=AOvVaw2jB9UgqcqFchbOefYLlH-8) 59. Babylonian cuneiform numerals - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Babylonian_cuneiform_numerals](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Babylonian_cuneiform_numerals%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184331248%26amp;usg%3DAOvVaw0F-jQ3zhUX2H_fRAyJr3-T&sa=D&source=docs&ust=1744019184408636&usg=AOvVaw3sTHCNy_SXWU02xjq8orgW) 60. Babylonian Cuneiform & Sexagesimal Numbers–Robert Hatch - University of Florida, accessed April 7, 2025, [https://people.clas.ufl.edu/ufhatch/2022/07/28/babylonian-cuneiform-and-sexagesimal-numbers/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://people.clas.ufl.edu/ufhatch/2022/07/28/babylonian-cuneiform-and-sexagesimal-numbers/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184331905%26amp;usg%3DAOvVaw3eGKe3e3k2VDKFIg8svqz1&sa=D&source=docs&ust=1744019184408722&usg=AOvVaw34Co8TrU0fKCm-zWhY5EKg) 61. simple.wikipedia.org, accessed April 7, 2025, [https://simple.wikipedia.org/wiki/Greek_numerals#:~:text=The%20earliest%20alphabet%2Drelated%20system,%3D%2010000%20and%20%CE%93%CE%9C%20%3D%2050000.](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://simple.wikipedia.org/wiki/Greek_numerals%2523:~:text%253DThe%252520earliest%252520alphabet%25252Drelated%252520system,%25253D%25252010000%252520and%252520%2525CE%252593%2525CE%25259C%252520%25253D%25252050000.%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184332562%26amp;usg%3DAOvVaw3oKkNp03tzWZADzyZWLCqg&sa=D&source=docs&ust=1744019184408821&usg=AOvVaw1m-daEKvboeg93Mo5aoU6L) 62. Greek numerals - FutureLearn, accessed April 7, 2025, [https://www.futurelearn.com/info/courses/maths-puzzles/0/steps/13997](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.futurelearn.com/info/courses/maths-puzzles/0/steps/13997%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184333083%26amp;usg%3DAOvVaw0DWPezA_WoStRqdviSDS3Z&sa=D&source=docs&ust=1744019184408933&usg=AOvVaw2l95lM8SPd6qEJSWwzKU6u) 63. Your Ultimate Guide to Counting in Greek: Numbers 1 to 1,000, accessed April 7, 2025, [https://www.theonlinegreektutor.com/counting-in-greek/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.theonlinegreektutor.com/counting-in-greek/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184333620%26amp;usg%3DAOvVaw3EltTsrbzJ8-C2UmNR2oHl&sa=D&source=docs&ust=1744019184409022&usg=AOvVaw3cF6OOlvxBcNI-NznNvkKE) 64. Greek numerals - Simple English Wikipedia, the free encyclopedia, accessed April 7, 2025, [https://simple.wikipedia.org/wiki/Greek_numerals](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://simple.wikipedia.org/wiki/Greek_numerals%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184334127%26amp;usg%3DAOvVaw3ZbdoUrccpVWlOGMseiQlP&sa=D&source=docs&ust=1744019184409104&usg=AOvVaw2-nzzzwAuuoBfF0LqsAxv3) 65. Greek numerals - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Greek_numerals](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Greek_numerals%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184334554%26amp;usg%3DAOvVaw0ljj367iS2u25Z7cah3AW_&sa=D&source=docs&ust=1744019184409178&usg=AOvVaw0WI62t52UolYz6yCnCrJEs) 66. Greek Numeration - YouTube, accessed April 7, 2025, [https://www.youtube.com/watch?v=pgAVmM8wswE](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.youtube.com/watch?v%253DpgAVmM8wswE%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184334981%26amp;usg%3DAOvVaw2CdmN7PYv3M7vGgMq3Hulk&sa=D&source=docs&ust=1744019184409262&usg=AOvVaw1se4Juo_w7hmUg5OyzbhDU) 67. Babylonian cuneiform numerals - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Babylonian_numerals](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Babylonian_numerals%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184335431%26amp;usg%3DAOvVaw0s_YsE8YupYmqXh2ck7lq6&sa=D&source=docs&ust=1744019184409334&usg=AOvVaw0y9fh_7TBzuNv2BiqQmBZD) 68. Egyptian numerals - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Egyptian_numerals](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Egyptian_numerals%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184335891%26amp;usg%3DAOvVaw2EF5fNE2qRfDq5gc7GaJzI&sa=D&source=docs&ust=1744019184409412&usg=AOvVaw1GBWm627jBbe1n3sFSzrqr) 69. Roman numerals - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Roman_numerals](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Roman_numerals%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184336309%26amp;usg%3DAOvVaw07-_B2TvUAZ1ekVkR9k_yO&sa=D&source=docs&ust=1744019184409484&usg=AOvVaw33OrESctbxgoegGGtuJ-Fl) 70. The Beginning of the Number Pi, in Binary Through Hexadecimal, etc. | RobertLovesPi.net, accessed April 7, 2025, [https://robertlovespi.net/2014/06/09/the-beginning-of-the-number-pi-in-binary-through-hexadecimal-etc/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://robertlovespi.net/2014/06/09/the-beginning-of-the-number-pi-in-binary-through-hexadecimal-etc/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184337010%26amp;usg%3DAOvVaw0niJvXEUQgO0C8UL1Alde_&sa=D&source=docs&ust=1744019184409561&usg=AOvVaw3lFYJwndZTkkicmWwJelSQ) 71. pi - Is π normal in base π? - Mathematics Stack Exchange, accessed April 7, 2025, [https://math.stackexchange.com/questions/216926/is-%CF%80-normal-in-base-%CF%80](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://math.stackexchange.com/questions/216926/is-%2525CF%252580-normal-in-base-%2525CF%252580%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184337620%26amp;usg%3DAOvVaw1dbREEOu4Ro92mMPlOmsX7&sa=D&source=docs&ust=1744019184409661&usg=AOvVaw1EhswHVgcF93xYGLOt2tS0) 72. Can pi be expressed rationally in a non base 10 number system?: r/askscience - Reddit, accessed April 7, 2025, [https://www.reddit.com/r/askscience/comments/57wvdw/can_pi_be_expressed_rationally_in_a_non_base_10/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.reddit.com/r/askscience/comments/57wvdw/can_pi_be_expressed_rationally_in_a_non_base_10/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184338340%26amp;usg%3DAOvVaw0ebx8yfoUM2zhqqpWgJ7yG&sa=D&source=docs&ust=1744019184409761&usg=AOvVaw3daw42ifWqww1l9xmmOKVC) 73. The Natural Base? - Science & Spaceflight - Kerbal Space Program Forums, accessed April 7, 2025, [https://forum.kerbalspaceprogram.com/topic/147685-the-natural-base/?do=getLastComment](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://forum.kerbalspaceprogram.com/topic/147685-the-natural-base/?do%253DgetLastComment%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184338963%26amp;usg%3DAOvVaw3mohvrCAX5Q_ZH0glBHQON&sa=D&source=docs&ust=1744019184409845&usg=AOvVaw3QRtGt0j0RtVfk1s7OpIxF) 74. Introduction to Binary: Basics and Importance | Lenovo US, accessed April 7, 2025, [https://www.lenovo.com/us/en/glossary/what-is-binary/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.lenovo.com/us/en/glossary/what-is-binary/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184339541%26amp;usg%3DAOvVaw3Yoz1r7o-BN6O1r3Vzr4DY&sa=D&source=docs&ust=1744019184409939&usg=AOvVaw0ZC67MWY0w_zn2qb65Ajnh) 75. 3.3 Binary and Its Advantages - Runestone Academy, accessed April 7, 2025, [https://runestone.academy/ns/books/published/welcomecs2/binary_binary-and-its-advantages.html](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://runestone.academy/ns/books/published/welcomecs2/binary_binary-and-its-advantages.html%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184340207%26amp;usg%3DAOvVaw16ulqUDtwe6Dmw6_gkvX_9&sa=D&source=docs&ust=1744019184410017&usg=AOvVaw1xr2lb9lzCwULA6_u5whbN) 76. 3.3. Binary and Its Advantages—CS160 Reader - Chemeketa CS, accessed April 7, 2025, [https://computerscience.chemeketa.edu/cs160Reader/Binary/Binary.html](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://computerscience.chemeketa.edu/cs160Reader/Binary/Binary.html%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184340782%26amp;usg%3DAOvVaw1XdVz4dCx4ny90LOyBtcfn&sa=D&source=docs&ust=1744019184410098&usg=AOvVaw0uss2I3iHV06jO8xB-wryF) 77. Uses of Hexadecimal in Computer Science - IGCSE Revision - Save My Exams, accessed April 7, 2025, [https://www.savemyexams.com/igcse/computer-science/cie/23/revision-notes/1-data-representation/number-systems/uses-of-hexadecimal/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.savemyexams.com/igcse/computer-science/cie/23/revision-notes/1-data-representation/number-systems/uses-of-hexadecimal/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184341661%26amp;usg%3DAOvVaw08I8UELFpPoc1i5KDIphhl&sa=D&source=docs&ust=1744019184410180&usg=AOvVaw230Kei_ixmYuKh8TmCXNfr) 78. Hexadecimal - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Hexadecimal](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Hexadecimal%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184342086%26amp;usg%3DAOvVaw2aMPvQ3EKvvutizAvg7dv3&sa=D&source=docs&ust=1744019184410263&usg=AOvVaw2FhDt_2JlbVH4Hd-qsYT1H) 79. Hexadecimal number system (video) - Khan Academy, accessed April 7, 2025, [https://www.khanacademy.org/math/algebra-home/alg-intro-to-algebra/algebra-alternate-number-bases/v/hexadecimal-number-system](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.khanacademy.org/math/algebra-home/alg-intro-to-algebra/algebra-alternate-number-bases/v/hexadecimal-number-system%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184342839%26amp;usg%3DAOvVaw1NAu-Rz4Xtb3v7Y7_gNb5L&sa=D&source=docs&ust=1744019184410368&usg=AOvVaw1LpqrCTnMWHqJ7snA4eqZq) 80. www.techtarget.com, accessed April 7, 2025, [https://www.techtarget.com/whatis/definition/hexadecimal#:~:text=The%20format%20of%20hex%20numbers,strings%20of%201s%20and%200s.](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.techtarget.com/whatis/definition/hexadecimal%2523:~:text%253DThe%252520format%252520of%252520hex%252520numbers,strings%252520of%2525201s%252520and%2525200s.%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184343448%26amp;usg%3DAOvVaw3jBhmT7lwv0CYKQl9JZqIK&sa=D&source=docs&ust=1744019184410472&usg=AOvVaw3NldUwdFtseiUcORougfCL) 81. Uses of Hexadecimal - Teach Computer Science, accessed April 7, 2025, [https://teachcomputerscience.com/uses-of-hexadecimal/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://teachcomputerscience.com/uses-of-hexadecimal/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184343952%26amp;usg%3DAOvVaw0f4uH5a7xnOts7dhl5JwJ8&sa=D&source=docs&ust=1744019184410577&usg=AOvVaw340F9EeHwr7NJ1D6GTyAbH) 82. Why do we use hexadecimal? - Medium, accessed April 7, 2025, [https://medium.com/@savas/why-do-we-use-hexadecimal-d6d80b56f026](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://medium.com/@savas/why-do-we-use-hexadecimal-d6d80b56f026%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184344449%26amp;usg%3DAOvVaw0NMmAbvOm6yyQVNw1fDxIf&sa=D&source=docs&ust=1744019184410654&usg=AOvVaw3f4nYlSuG6DDO8agSSMzht) 83. Duodecimal - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Duodecimal](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Duodecimal%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184344869%26amp;usg%3DAOvVaw2oFHEKug0KUx5ttVaZeHi3&sa=D&source=docs&ust=1744019184410736&usg=AOvVaw0S59-UIVH7P1rhY27b0gKf) 84. en.wikipedia.org, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Ternary_numeral_system#:~:text=Ternary%20numbers%20can%20be%20used,the%20Cantor%20set%20is%20constructed.](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Ternary_numeral_system%2523:~:text%253DTernary%252520numbers%252520can%252520be%252520used,the%252520Cantor%252520set%252520is%252520constructed.%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184345506%26amp;usg%3DAOvVaw1ovSKY1dCiLzilqpCv62LE&sa=D&source=docs&ust=1744019184410823&usg=AOvVaw3DtlSJ4uMZaqejuVmiNx6P) 85. Ternary numeral system - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Ternary_numeral_system](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Ternary_numeral_system%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184345982%26amp;usg%3DAOvVaw1ZFkMPJn_1PtH7cjNwH4S6&sa=D&source=docs&ust=1744019184410912&usg=AOvVaw0Ym40XXZqWiworhPdw07dq) 86. Ternary -- from Wolfram MathWorld, accessed April 7, 2025, [https://mathworld.wolfram.com/Ternary.html](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://mathworld.wolfram.com/Ternary.html%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184346426%26amp;usg%3DAOvVaw2Mufejcx8UNuvV5hdAVvwP&sa=D&source=docs&ust=1744019184410983&usg=AOvVaw0mLaVc4S1JLt5MUQb9KEaf) 87. Balanced ternary - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Balanced_ternary](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Balanced_ternary%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184346874%26amp;usg%3DAOvVaw2W_-WI74rbHwWb4xxdC0aa&sa=D&source=docs&ust=1744019184411052&usg=AOvVaw2QpwTcNVJuKw3ujJvshLSE) 88. Ternary Digital Systems - Oregon State University, accessed April 7, 2025, [https://ir.library.oregonstate.edu/downloads/wh246w135](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://ir.library.oregonstate.edu/downloads/wh246w135%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184347394%26amp;usg%3DAOvVaw3sMlEpLGCH1YKNEresX_ir&sa=D&source=docs&ust=1744019184411140&usg=AOvVaw234BtY2MLpWpz4OroWxyWW) 89. Base 3: Ternary Numbers - Expii, accessed April 7, 2025, [https://www.expii.com/t/base-ternary-numbers-9194](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.expii.com/t/base-ternary-numbers-9194%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184347817%26amp;usg%3DAOvVaw1ag4m84bpHsqTkYZ3x5anD&sa=D&source=docs&ust=1744019184411212&usg=AOvVaw3umYP7zrP3wt8nkzLslQY2) 90. Balanced Number System, accessed April 7, 2025, [https://www.ias.ac.in/article/fulltext/reso/023/12/1395-1410](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.ias.ac.in/article/fulltext/reso/023/12/1395-1410%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184348261%26amp;usg%3DAOvVaw2cp5_CbcOeuGst2y0tXkgj&sa=D&source=docs&ust=1744019184411286&usg=AOvVaw1KlpLWdzXIT_jmGoETVjNU) 91. Unlocking the Secrets of Balanced Ternary: A Base 3 Number System - SchoolTube, accessed April 7, 2025, [https://www.schooltube.com/unlocking-the-secrets-of-balanced-ternary-a-base-3-number-system/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.schooltube.com/unlocking-the-secrets-of-balanced-ternary-a-base-3-number-system/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184348954%26amp;usg%3DAOvVaw1XvjCzlU5CuIpPjFDjESSA&sa=D&source=docs&ust=1744019184411367&usg=AOvVaw0AuzDTxWGZ-ccbyf5K5g1v) 92. Balanced ternary - Rosetta Code, accessed April 7, 2025, [https://rosettacode.org/wiki/Balanced_ternary](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://rosettacode.org/wiki/Balanced_ternary%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184349394%26amp;usg%3DAOvVaw3SAs7w85vTbPlVNvM-THPC&sa=D&source=docs&ust=1744019184411448&usg=AOvVaw2hz7r4e04yGfs5u1zUqmuq) 93. Ternary numeral system - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Ternary_numeral_system#Applications](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Ternary_numeral_system%2523Applications%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184349896%26amp;usg%3DAOvVaw2DBhYzoP6D3Wuf0HCcJVU2&sa=D&source=docs&ust=1744019184411528&usg=AOvVaw0NSqTZcA99pAmW7lq8-MHl) 94. Vigesimal - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Vigesimal](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Vigesimal%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184350284%26amp;usg%3DAOvVaw07Ylqo0i2AqsVbvZ57IOtM&sa=D&source=docs&ust=1744019184411628&usg=AOvVaw3hqwZwlAHwRBiku9bj-RZE) 95. Quaternary numeral system - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Quaternary_numeral_system](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Quaternary_numeral_system%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184350754%26amp;usg%3DAOvVaw2ziexQSkwRSkFkjlJOsNd9&sa=D&source=docs&ust=1744019184411723&usg=AOvVaw2fl4Kdia9nALwywpifTW_J) 96. What would a base $\pi$number system look like? - Mathematics Stack Exchange, accessed April 7, 2025, [https://math.stackexchange.com/questions/1320248/what-would-a-base-pi-number-system-look-like](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://math.stackexchange.com/questions/1320248/what-would-a-base-pi-number-system-look-like%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184351429%26amp;usg%3DAOvVaw3vpkIvRbQWoLUQH-FXNvQE&sa=D&source=docs&ust=1744019184411801&usg=AOvVaw30e0Fr5rTLmTCI5929lQq1) 97. Constants - Scistats, accessed April 7, 2025, [https://alandefreitas.github.io/scistats/mathematics/constants.html](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://alandefreitas.github.io/scistats/mathematics/constants.html%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184351912%26amp;usg%3DAOvVaw1k2rNh_ri6KTJHR-h51xu5&sa=D&source=docs&ust=1744019184411882&usg=AOvVaw1Iv0EDKIOtcUGCEAVtKhIc) 98. Irrational number - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Irrational_number](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Irrational_number%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184352347%26amp;usg%3DAOvVaw1k0FROde_JZWFkMwtR5ScD&sa=D&source=docs&ust=1744019184411967&usg=AOvVaw3wGkVVNmm2XojRdwttQ6OI) 99. Rational and Irrational Numbers (Definition & Examples) - BYJU’S, accessed April 7, 2025, [https://byjus.com/maths/rational-and-irrational-numbers/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://byjus.com/maths/rational-and-irrational-numbers/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184352861%26amp;usg%3DAOvVaw2QPwoEBiBuvVisfLMJ3ghl&sa=D&source=docs&ust=1744019184412051&usg=AOvVaw3D8Z7s4shC4VN4EUATQmxK) 100. 1.04 Rational and irrational numbers | Level 8 Maths - Mathspace, accessed April 7, 2025, [https://mathspace.co/textbooks/syllabuses/Syllabus-999/topics/Topic-19889/subtopics/Subtopic-263668/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://mathspace.co/textbooks/syllabuses/Syllabus-999/topics/Topic-19889/subtopics/Subtopic-263668/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184353552%26amp;usg%3DAOvVaw3OCp0ATIba1VpFuSu9hhO_&sa=D&source=docs&ust=1744019184412145&usg=AOvVaw2yWAc4mzt3On6NsYgwnmoW) 101. Discover the difference between Rational and Irrational Numbers, accessed April 7, 2025, [https://www.teachtutti.co.uk/blog/irrational-numbers-guide/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.teachtutti.co.uk/blog/irrational-numbers-guide/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184354058%26amp;usg%3DAOvVaw17LnyTkDvV7m65JBjo47X6&sa=D&source=docs&ust=1744019184412238&usg=AOvVaw2gA2KoLRuzP6esYDqSoCmw) 102. What are all the irrational numbers? - Tutorax, accessed April 7, 2025, [https://tutorax.com/blogue/en/what-are-all-the-irrational-numbers/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://tutorax.com/blogue/en/what-are-all-the-irrational-numbers/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184354551%26amp;usg%3DAOvVaw37PWiHOfrCZLQHA65tPa9B&sa=D&source=docs&ust=1744019184412335&usg=AOvVaw1YW6f4rgSBOhDfexzodBlg) 103. Irrational Numbers - Definition, List, Properties, Examples, Symbol - BYJU’S, accessed April 7, 2025, [https://byjus.com/maths/irrational-numbers/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://byjus.com/maths/irrational-numbers/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184355097%26amp;usg%3DAOvVaw1OU2q-aZ9Oqlm9E3wxCErR&sa=D&source=docs&ust=1744019184412436&usg=AOvVaw1ALXk-t4Zvy9yNnmEC_3tp) 104. Rational and irrational numbers | EBSCO Research Starters, accessed April 7, 2025, [https://www.ebsco.com/research-starters/mathematics/rational-and-irrational-numbers](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.ebsco.com/research-starters/mathematics/rational-and-irrational-numbers%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184355669%26amp;usg%3DAOvVaw2MZmwawVW4OACZScvZzB1S&sa=D&source=docs&ust=1744019184412534&usg=AOvVaw1YiHpCDsB48zl2HVmNTWLu) 105. Non-Terminating, Non-Repeating Decimal - LearnAlberta.ca, accessed April 7, 2025, [https://www.learnalberta.ca/content/memg/division03/non-terminating,%20non-repeating%20decimal/index.html](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.learnalberta.ca/content/memg/division03/non-terminating,%252520non-repeating%252520decimal/index.html%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184356305%26amp;usg%3DAOvVaw14ZBaSikBe7cnV8V8GRpcG&sa=D&source=docs&ust=1744019184412647&usg=AOvVaw19QMKgyy138Q5Yrp68n-g1) 106. Irrational Number: Definitions and Examples - Club Z! Tutoring, accessed April 7, 2025, [http://clubztutoring.com/ed-resources/math/irrational-number-definitions-examples-6-7-5-5/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttp://clubztutoring.com/ed-resources/math/irrational-number-definitions-examples-6-7-5-5/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184356871%26amp;usg%3DAOvVaw10iBZvcAuE852jVDP1DGDW&sa=D&source=docs&ust=1744019184412758&usg=AOvVaw20-RcSPhlTDTFrjrRrNImv) 107. www.learner.com, accessed April 7, 2025, [https://www.learner.com/blog/what-is-an-irrational-number-in-math#:~:text=Irrational%20numbers%20are%20essentially%20non,(1.618033...).](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.learner.com/blog/what-is-an-irrational-number-in-math%2523:~:text%253DIrrational%252520numbers%252520are%252520essentially%252520non,\(1.618033...\).%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184357592%26amp;usg%3DAOvVaw3NYrbCgGkqm5MaX5xWCBaw&sa=D&source=docs&ust=1744019184412867&usg=AOvVaw1Mp6lUPRaHO8KamDNM6V5c) 108. Is Pi irrational in all bases?: r/math - Reddit, accessed April 7, 2025, [https://www.reddit.com/r/math/comments/2xldg4/is_pi_irrational_in_all_bases/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.reddit.com/r/math/comments/2xldg4/is_pi_irrational_in_all_bases/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184358193%26amp;usg%3DAOvVaw1x_agkFx6a1Fz5XNA-mVE5&sa=D&source=docs&ust=1744019184412981&usg=AOvVaw3p_nF6BttoVBNttXPaKMaI) 109. Base Twelve Pi and 360 Degrees - YouTube, accessed April 7, 2025, [https://www.youtube.com/watch?v=PNuLvWGeihY](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.youtube.com/watch?v%253DPNuLvWGeihY%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184358646%26amp;usg%3DAOvVaw2ADCNLBDAewVhiKBZDKuiO&sa=D&source=docs&ust=1744019184413076&usg=AOvVaw2ZyRi5XGplncomw2Zb2bYP) 110. Why is pi here? And why is it squared? A geometric answer to the Basel problem - YouTube, accessed April 7, 2025, [https://www.youtube.com/watch?v=d-o3eB9sfls](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.youtube.com/watch?v%253Dd-o3eB9sfls%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184359151%26amp;usg%3DAOvVaw3MI3_vw_zTWhdcF-gUju6-&sa=D&source=docs&ust=1744019184413169&usg=AOvVaw0m4DulT6CMUhZyCL-gm_O4) 111. The mystique of mathematics: 5 beautiful math phenomena - Mathnasium, accessed April 7, 2025, [https://www.mathnasium.com/math-centers/kirkwood/news/the-mystique-of-mathematics-5-beautiful-math-phenomena-1864784637](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.mathnasium.com/math-centers/kirkwood/news/the-mystique-of-mathematics-5-beautiful-math-phenomena-1864784637%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184359905%26amp;usg%3DAOvVaw1sWi147942Nd-0KbBCGyMA&sa=D&source=docs&ust=1744019184413270&usg=AOvVaw0t0QN01WoU7dU5EVZHk7zi) 112. Pi Is Encoded in the Patterns of Life - Biophysical Society, accessed April 7, 2025, [https://www.biophysics.org/blog/pi-is-encoded-in-the-patterns-of-life](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.biophysics.org/blog/pi-is-encoded-in-the-patterns-of-life%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184360511%26amp;usg%3DAOvVaw1g50SPDl81TF7vJKlOUivA&sa=D&source=docs&ust=1744019184413384&usg=AOvVaw1Zs77UWATI_KZ9PGd5MT5C) 113. en.wikipedia.org, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Pi#:~:text=Because%20it%20relates%20to%20a,thermodynamics%2C%20mechanics%2C%20and%20electromagnetism.](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Pi%2523:~:text%253DBecause%252520it%252520relates%252520to%252520a,thermodynamics%25252C%252520mechanics%25252C%252520and%252520electromagnetism.%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184361161%26amp;usg%3DAOvVaw12N7Z_ddePi3VSyKRzTeEL&sa=D&source=docs&ust=1744019184413491&usg=AOvVaw1Fa7WwXIJpI3FBWWAuD_IT) 114. What is Base Twelve Pi? - YouTube, accessed April 7, 2025, [https://www.youtube.com/watch?v=gmlVe9lbUq8](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.youtube.com/watch?v%253DgmlVe9lbUq8%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184361580%26amp;usg%3DAOvVaw3dnCbF_AceBPywExY3cWSK&sa=D&source=docs&ust=1744019184413603&usg=AOvVaw1PigK5op-VObuUwDrc_FBE) 115. Why does pi appear so much in physics?: r/askscience - Reddit, accessed April 7, 2025, [https://www.reddit.com/r/askscience/comments/1sh50y/why_does_pi_appear_so_much_in_physics/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.reddit.com/r/askscience/comments/1sh50y/why_does_pi_appear_so_much_in_physics/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184362217%26amp;usg%3DAOvVaw3XjcRAUZJwh2L8ovUTWAuD&sa=D&source=docs&ust=1744019184413716&usg=AOvVaw1hnipaiEMOA82xPxZKK8pF) 116. Ultimate Relativity: π Theory - Scientific & Academic Publishing, accessed April 7, 2025, [http://article.sapub.org/10.5923.j.ijtmp.20221202.02.html](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttp://article.sapub.org/10.5923.j.ijtmp.20221202.02.html%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184362743%26amp;usg%3DAOvVaw1oZ0bEXfpJmyckKp_Kdv6z&sa=D&source=docs&ust=1744019184413813&usg=AOvVaw3JSqoGUrP377f5RpuLyGyW) 117. AP Precalculus Unit 3 Topic 3.1A Periodic Phenomena - YouTube, accessed April 7, 2025, [https://www.youtube.com/watch?v=Na23KdObyLM](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.youtube.com/watch?v%253DNa23KdObyLM%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184363253%26amp;usg%3DAOvVaw0yJdUgTP7PCsD_kU6WWly-&sa=D&source=docs&ust=1744019184413909&usg=AOvVaw3q9L9UDQ9QlfD88WQj2vFZ) 118. AP Precalculus - Intro to Periodic Phenomena - Passwater 3.1 - YouTube, accessed April 7, 2025, [https://www.youtube.com/watch?v=UJBO3gikkus](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.youtube.com/watch?v%253DUJBO3gikkus%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184363755%26amp;usg%3DAOvVaw3KuEOKqxCwLwRwWtnS_Fh1&sa=D&source=docs&ust=1744019184414003&usg=AOvVaw1ALUCQU0iPB7exKnWMDHEP) 119. 3-1 Periodic Phenomena - YouTube, accessed April 7, 2025, [https://www.youtube.com/watch?v=hwTrqaJa-Oc](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.youtube.com/watch?v%253DhwTrqaJa-Oc%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184364197%26amp;usg%3DAOvVaw2DltxyPPDzxy8_SRUs_vnl&sa=D&source=docs&ust=1744019184414095&usg=AOvVaw14Zqcmyc2VuOeCk3Mk4qst) 120. Examples of Modeling Periodic Phenomena for AP Precalculus - YouTube, accessed April 7, 2025, [https://www.youtube.com/watch?v=OHzY0X2lq_I](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.youtube.com/watch?v%253DOHzY0X2lq_I%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184364667%26amp;usg%3DAOvVaw3lfIvsyVDcyQ5AjF1ajFlF&sa=D&source=docs&ust=1744019184414197&usg=AOvVaw1XE-hkYvkG6-JuLEoHDodC) 121. Non-integer base of numeration - Wikipedia, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Non-integer_base_of_numeration](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Non-integer_base_of_numeration%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184365152%26amp;usg%3DAOvVaw32tYovMnctwqlwZ_Q45WNW&sa=D&source=docs&ust=1744019184414294&usg=AOvVaw0fartueb8BySOIQSiyhKun) 122. en.wikipedia.org, accessed April 7, 2025, [https://en.wikipedia.org/wiki/Non-integer_base_of_numeration#:~:text=Base%20%CF%80%20can%20be%20used,circumference%20of%20100%CF%80%2C%20etc.](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://en.wikipedia.org/wiki/Non-integer_base_of_numeration%2523:~:text%253DBase%252520%2525CF%252580%252520can%252520be%252520used,circumference%252520of%252520100%2525CF%252580%25252C%252520etc.%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184365779%26amp;usg%3DAOvVaw2cKSDSl40U43-pdg1vpAXo&sa=D&source=docs&ust=1744019184414403&usg=AOvVaw2oG1wUXGDyvvr5Ln-Rwb3j) 123. ELI5 why Pi is important?: r/explainlikeimfive - Reddit, accessed April 7, 2025, [https://www.reddit.com/r/explainlikeimfive/comments/11r4okq/eli5_why_pi_is_important/](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.reddit.com/r/explainlikeimfive/comments/11r4okq/eli5_why_pi_is_important/%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184366424%26amp;usg%3DAOvVaw3vEvY7Pm9tpKjOBXxxq_6x&sa=D&source=docs&ust=1744019184414509&usg=AOvVaw1fV4LwJQgmrXAUlU2N2yly) 124. Midline, amplitude, and period review (article) - Khan Academy, accessed April 7, 2025, [https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:trig/x2ec2f6f830c9fb89:amp-mid-period/a/midline-amplitude-and-period-review](https://www.google.com/url?q=https://www.google.com/url?q%3Dhttps://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:trig/x2ec2f6f830c9fb89:amp-mid-period/a/midline-amplitude-and-period-review%26amp;sa%3DD%26amp;source%3Deditors%26amp;ust%3D1744019184367230%26amp;usg%3DAOvVaw1dJu-KKUXS-MK3xTk5bp2M&sa=D&source=docs&ust=1744019184414602&usg=AOvVaw2918O5VbOTzLvlPJGng2wI)