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From Wikipedia, the free encyclopedia

The senary numeral system (also known as base-6, heximal, or seximal) has six as its base. It has been adopted independently by a small number of cultures. Like decimal, it is a semiprime, though being the product of the only two consecutive numbers that are both prime (2 and 3) it has a high degree of mathematical properties for its size. As six is a superior highly composite number, many of the arguments made in favor of the duodecimal system also apply to base-6.

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  • ✪ The Map of Mathematics
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Transcription

The mathematics we learn in school doesn’t quite do the field of mathematics justice. We only get a glimpse at one corner of it, but the mathematics as a whole is a huge and wonderfully diverse subject. My aim with this video is to show you all that amazing stuff. We’ll start back at the very beginning. The origin of mathematics lies in counting. In fact counting is not just a human trait, other animals are able to count as well and e vidence for human counting goes back to prehistoric times with check marks made in bones. There were several innovations over the years with the Egyptians having the first equation, the ancient Greeks made strides in many areas like geometry and numerology, and negative numbers were invented in China. And zero as a number was first used in India. Then in the Golden Age of Islam Persian mathematicians made further strides and the first book on algebra was written. Then mathematics boomed in the renaissance along with the sciences. Now there is a lot more to the history of mathematics then what I have just said, but I’m gonna jump to the modern age and mathematics as we know it now. Modern mathematics can be broadly be broken down into two areas, pure maths: the study of mathematics for its own sake, and applied maths: when you develop mathematics to help solve some real world problem. But there is a lot of crossover. In fact, many times in history someone’s gone off into the mathematical wilderness motivated purely by curiosity and kind of guided by a sense of aesthetics. And then they have created a whole bunch of new mathematics which was nice and interesting but doesn’t really do anything useful. But then, say a hundred hears later, someone will be working on some problem at the cutting edge of physics or computer science and they’ll discover that this old theory in pure maths is exactly what they need to solve their real world problems! Which is amazing, I think! And this kind of thing has happened so many times over the last few centuries. It is interesting how often something so abstract ends up being really useful. But I should also mention, pure mathematics on its own is still a very valuable thing to do because it can be fascinating and on its own can have a real beauty and elegance that almost becomes like art. Okay enough of this highfalutin, lets get into it. Pure maths is made of several sections. The study of numbers starts with the natural numbers and what you can do with them with arithmetic operations. And then it looks at other kinds of numbers like integers, which contain negative numbers, rational numbers like fractions, real numbers which include numbers like pi which go off to infinite decimal points, and then complex numbers and a whole bunch of others. Some numbers have interesting properties like Prime Numbers, or pi or the exponential. There are also properties of these number systems, for example, even though there is an infinite amount of both integers and real numbers, there are more real numbers than integers. So some infinities are bigger than others. The study of structures is where you start taking numbers and putting them into equations in the form of variables. Algebra contains the rules of how you then manipulate these equations. Here you will also find vectors and matrices which are multi-dimensional numbers, and the rules of how they relate to each other are captured in linear algebra. Number theory studies the features of everything in the last section on numbers like the properties of prime numbers. Combinatorics looks at the properties of certain structures like trees, graphs, and other things that are made of discreet chunks that you can count. Group theory looks at objects that are related to each other in, well, groups. A familiar example is a Rubik’s cube which is an example of a permutation group. And order theory investigates how to arrange objects following certain rules like, how something is a larger quantity than something else. The natural numbers are an example of an ordered set of objects, but anything with any two way relationship can be ordered. Another part of pure mathematics looks at shapes and how they behave in spaces. The origin is in geometry which includes Pythagoras, and is close to trigonometry, which we are all familiar with form school. Also there are fun things like fractal geometry which are mathematical patterns which are scale invariant, which means you can zoom into them forever and the always look kind of the same. Topology looks at different properties of spaces where you are allowed to continuously deform them but not tear or glue them. For example a Möbius strip has only one surface and one edge whatever you do to it. And coffee cups and donuts are the same thing - topologically speaking. Measure theory is a way to assign values to spaces or sets tying together numbers and spaces. And finally, differential geometry looks the properties of shapes on curved surfaces, for example triangles have got different angles on a curved surface, and brings us to the next section, which is changes. The study of changes contains calculus which involves integrals and differentials which looks at area spanned out by functions or the behaviour of gradients of functions. And vector calculus looks at the same things for vectors. Here we also find a bunch of other areas like dynamical systems which looks at systems that evolve in time from one state to another, like fluid flows or things with feedback loops like ecosystems. And chaos theory which studies dynamical systems that are very sensitive to initial conditions. Finally complex analysis looks at the properties of functions with complex numbers. This brings us to applied mathematics. At this point it is worth mentioning that everything here is a lot more interrelated than I have drawn. In reality this map should look like more of a web tying together all the different subjects but you can only do so much on a two dimensional plane so I have laid them out as best I can. Okay we’ll start with physics, which uses just about everything on the left hand side to some degree. Mathematical and theoretical physics has a very close relationship with pure maths. Mathematics is also used in the other natural sciences with mathematical chemistry and biomathematics which look at loads of stuff from modelling molecules to evolutionary biology. Mathematics is also used extensively in engineering, building things has taken a lot of maths since Egyptian and Babylonian times. Very complex electrical systems like aeroplanes or the power grid use methods in dynamical systems called control theory. Numerical analysis is a mathematical tool commonly used in places where the mathematics becomes too complex to solve completely. So instead you use lots of simple approximations and combine them all together to get good approximate answers. For example if you put a circle inside a square, throw darts at it, and then compare the number of darts in the circle and square portions, you can approximate the value of pi. But in the real world numerical analysis is done on huge computers. Game theory looks at what the best choices are given a set of rules and rational players and it’s used in economics when the players can be intelligent, but not always, and other areas like psychology, and biology. Probability is the study of random events like coin tosses or dice or humans, and statistics is the study of large collections of random processes or the organisation and analysis of data. This is obviously related to mathematical finance, where you want model financial systems and get an edge to win all those fat stacks. Related to this is optimisation, where you are trying to calculate the best choice amongst a set of many different options or constraints, which you can normally visualise as trying to find the highest or lowest point of a function. Optimisation problems are second nature to us humans, we do them all the time: trying to get the best value for money, or trying to maximise our happiness in some way. Another area that is very deeply related to pure mathematics is computer science, and the rules of computer science were actually derived in pure maths and is another example of something that was worked out way before programmable computers were built. Machine learning: the creation of intelligent computer systems uses many areas in mathematics like linear algebra, optimisation, dynamical systems and probability. And finally the theory of cryptography is very important to computation and uses a lot of pure maths like combinatorics and number theory. So that covers the main sections of pure and applied mathematics, but I can’t end without looking at the foundations of mathematics. This area tries to work out at the properties of mathematics itself, and asks what the basis of all the rules of mathematics is. Is there a complete set of fundamental rules, called axioms, which all of mathematics comes from? And can we prove that it is all consistent with itself? Mathematical logic, set theory and category theory try to answer this and a famous result in mathematical logic are Gödel’s incompleteness theorems which, for most people, means that Mathematics does not have a complete and consistent set of axioms, which mean that it is all kinda made up by us humans. Which is weird seeing as mathematics explains so much stuff in the Universe so well. Why would a thing made up by humans be able to do that? That is a deep mystery right there. Also we have the theory of computation which looks at different models of computing and how efficiently they can solve problems and contains complexity theory which looks at what is and isn’t computable and how much memory and time you would need, which, for most interesting problems, is an insane amount. Ending So that is the map of mathematics. Now the thing I have loved most about learning maths is that feeling you get where something that seemed so confusing finally clicks in your brain and everything makes sense: like an epiphany moment, kind of like seeing through the matrix. In fact some of my most satisfying intellectual moments have been understanding some part of mathematics and then feeling like I had a glimpse at the fundamental nature of the Universe in all of its symmetrical wonder. It’s great, I love it. Ending Making a map of mathematics was the most popular request I got, which I was really happy about because I love maths and its great to see so much interest in it. So I hope you enjoyed it. Obviously there is only so much I can get into this timeframe, but hopefully I have done the subject justice and you found it useful. So there will be more videos coming from me soon, here’s all the regular things and it was my pleasure se you next time.

Contents

Mathematical properties

Senary multiplication table
× 1 2 3 4 5 10
1 1 2 3 4 5 10
2 2 4 10 12 14 20
3 3 10 13 20 23 30
4 4 12 20 24 32 40
5 5 14 23 32 41 50
10 10 20 30 40 50 100

Senary may be considered interesting in the study of prime numbers, since all primes other than 2 and 3, when expressed in senary, have 1 or 5 as the final digit. In senary the prime numbers are written

2, 3, 5, 11, 15, 21, 25, 31, 35, 45, 51, 101, 105, 111, 115, 125, 135, 141, 151, 155, 201, 211, 215, 225, 241, 245, 251, 255, 301, 305, 331, 335, 345, 351, 405, 411, 421, 431, 435, 445, 455, 501, 515, 521, 525, 531, 551, ... (sequence A004680 in the OEIS)

That is, for every prime number p greater than 3, one has the modular arithmetic relations that either p ≡ 1 or 5 (mod 6) (that is, 6 divides either p − 1 or p − 5); the final digit is a 1 or a 5. This is proved by contradiction. For any integer n:

  • If n ≡ 0 (mod 6), 6 | n
  • If n ≡ 2 (mod 6), 2 | n
  • If n ≡ 3 (mod 6), 3 | n
  • If n ≡ 4 (mod 6), 2 | n

Additionally, since the smallest four primes (2, 3, 5, 7) are either divisors or neighbors of 6, senary has simple divisibility tests for many numbers.

Furthermore, all even perfect numbers besides 6 have 44 as the final two digits when expressed in senary, which is proven by the fact that every even perfect number is of the form 2p−1(2p−1), where 2p−1 is prime.

Senary is also the largest number base r that has no totatives other than 1 and r − 1, making its multiplication table highly regular for its size, minimizing the amount of effort required to memorize its table. This property maximizes the probability that the result of an integer multiplication will end in zero, given that neither of its factors do.

Fractions

Because six is the product of the first two prime numbers and is adjacent to the next two prime numbers, many senary fractions have simple representations:

Decimal base
Prime factors of the base: 2, 5
Prime factors of one below the base: 3
Prime factors of one above the base: 11
Senary base
Prime factors of the base: 2, 3
Prime factors of one below the base: 5
Prime factors of one above the base: 11
Fraction Prime factors
of the denominator
Positional representation Positional representation Prime factors
of the denominator
Fraction
1/2 2 0.5 0.3 2 1/2
1/3 3 0.3333... = 0.3 0.2 3 1/3
1/4 2 0.25 0.13 2 1/4
1/5 5 0.2 0.1111... = 0.1 5 1/5
1/6 2, 3 0.16 0.1 2, 3 1/10
1/7 7 0.142857 0.05 11 1/11
1/8 2 0.125 0.043 2 1/12
1/9 3 0.1 0.04 3 1/13
1/10 2, 5 0.1 0.03 2, 5 1/14
1/11 11 0.09 0.0313452421 15 1/15
1/12 2, 3 0.083 0.03 2, 3 1/20
1/13 13 0.076923 0.024340531215 21 1/21
1/14 2, 7 0.0714285 0.023 2, 11 1/22
1/15 3, 5 0.06 0.02 3, 5 1/23
1/16 2 0.0625 0.0213 2 1/24
1/17 17 0.0588235294117647 0.0204122453514331 25 1/25
1/18 2, 3 0.05 0.02 2, 3 1/30
1/19 19 0.052631578947368421 0.015211325 31 1/31
1/20 2, 5 0.05 0.014 2, 5 1/32
1/21 3, 7 0.047619 0.014 3, 11 1/33
1/22 2, 11 0.045 0.01345242103 2, 15 1/34
1/23 23 0.0434782608695652173913 0.01322030441 35 1/35
1/24 2, 3 0.0416 0.013 2, 3 1/40
1/25 5 0.04 0.01235 5 1/41
1/26 2, 13 0.0384615 0.0121502434053 2, 21 1/42
1/27 3 0.037 0.012 3 1/43
1/28 2, 7 0.03571428 0.0114 2, 11 1/44
1/29 29 0.0344827586206896551724137931 0.01124045443151 45 1/45
1/30 2, 3, 5 0.03 0.01 2, 3, 5 1/50
1/31 31 0.032258064516129 0.010545 51 1/51
1/32 2 0.03125 0.01043 2 1/52
1/33 3, 11 0.03 0.01031345242 3, 15 1/53
1/34 2, 17 0.02941176470588235 0.01020412245351433 2, 25 1/54
1/35 5, 7 0.0285714 0.01 5, 11 1/55
1/36 2, 3 0.027 0.01 2, 3 1/100

Finger counting

34senary = 22decimal, in senary finger counting

Each regular human hand may be said to have six unambiguous positions; a fist, one finger (or thumb) extended, two, three, four and then all five extended.

If the right hand is used to represent a unit, and the left to represent the 'sixes', it becomes possible for one person to represent the values from zero to 55senary (35decimal) with their fingers, rather than the usual ten obtained in standard finger counting. e.g. if three fingers are extended on the left hand and four on the right, 34senary is represented. This is equivalent to 3 × 6 + 4 which is 22decimal.

Additionally, this method is the least abstract way to count using two hands that reflects the concept of positional notation, as the movement from one position to the next is done by switching from one hand to another. While most developed cultures count by fingers up to 5 in very similar ways, beyond 5 non-Western cultures deviate from Western methods, such as with Chinese number gestures. As senary finger counting also deviates only beyond 5, this counting method rivals the simplicity of traditional counting methods, a fact which may have implications for the teaching of positional notation to young students.

Which hand is used for the 'sixes' and which the units is down to preference on the part of the counter, however when viewed from the counter's perspective, using the left hand as the most significant digit correlates with the written representation of the same senary number. Flipping the 'sixes' hand around to its backside may help to further disambiguate which hand represents the 'sixes' and which represents the units. The downside to senary counting, however, is that without prior agreement two parties would be unable to utilize this system, being unsure which hand represents sixes and which hand represents ones, whereas decimal-based counting (with numbers beyond 5 being expressed by an open palm and additional fingers) being essentially a unary system only requires the other party to count the number of extended fingers.

In NCAA basketball, the players' uniform numbers are restricted to be senary numbers of at most two digits, so that the referees can signal which player committed an infraction by using this finger-counting system.[1]

More abstract finger counting systems, such as chisanbop or finger binary, allow counting to 99, 1,023, or even higher depending on the method (though not necessarily senary in nature). The English monk and historian Bede, described in the first chapter of his work De temporum ratione, (725), titled "Tractatus de computo, vel loquela per gestum digitorum," a system which allowed counting up to 9,999 on two hands.[2][3]

Natural languages

Despite the rarity of cultures that group large quantities by 6, a review of the development of numeral systems suggests a threshold of numerosity at 6 (possibly being conceptualized as "whole", "fist", or "beyond five fingers"[4]), with 1–6 often being pure forms, and numerals thereafter being constructed or borrowed.[5]

The Ndom language of Papua New Guinea is reported to have senary numerals.[6] Mer means 6, mer an thef means 6 × 2 = 12, nif means 36, and nif thef means 36 × 2 = 72.

Another example from Papua New Guinea are the Morehead-Maro languages. In these languages, counting is connected to ritualized yam-counting. These languages count from a base six, employing words for the powers of six; running up to 66 for some of the languages. One example is Kómnzo with the following numerals: nimbo (61), féta (62), tarumba (63), ntamno (64), wärämäkä (65), wi (66).

Some Niger-Congo languages have been reported to use a senary number system, usually in addition to another, such as decimal or vigesimal.[5]

Proto-Uralic has also been suspected to have had senary numerals, with a numeral for 7 being borrowed later, though evidence for constructing larger numerals (8 and 9) subtractively from ten suggests that this may not be so.[5]

Base 36 as senary compression

For some purposes, base 6 might be too small a base for convenience. This can be worked around by using its square, base 36 (hexatrigesimal), as then conversion is facilitated by simply making the following replacements:

Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Base 6 0 1 2 3 4 5 10 11 12 13 14 15 20 21 22 23 24 25
Base 36 0 1 2 3 4 5 6 7 8 9 A B C D E F G H
 
Decimal 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Base 6 30 31 32 33 34 35 40 41 42 43 44 45 50 51 52 53 54 55
Base 36 I J K L M N O P Q R S T U V W X Y Z

Thus, the base-36 number WIKIPEDIA36 is equal to the senary number 5230323041222130146. In decimal, it is 91,730,738,691,298.

The choice of 36 as a radix is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z: this choice is the basis of the base36 encoding scheme. The compression effect of 36 being the square of 6 causes a lot of patterns and representations to be shorter in base 36:

1/910 = 0.046 = 0.436

1/1610 = 0.02136 = 0.2936

1/510 = 0.16 = 0.736

1/710 = 0.056 = 0.536

See also

  • Diceware method to encode base-6 values into pronounceable passwords.
  • Base36 encoding scheme

Related number systems

References

  1. ^ Schonbrun, Zach (March 31, 2015), "Crunching the Numbers: College Basketball Players Can't Wear 6, 7, 8 or 9", The New York Times, archived from the original on February 3, 2016.
  2. ^ Bloom, Jonathan M. (2001). "Hand sums: The ancient art of counting with your fingers". Yale University Press. Archived from the original on August 13, 2011. Retrieved May 12, 2012.
  3. ^ "Dactylonomy". Laputan Logic. 16 November 2006. Archived from the original on 23 March 2012. Retrieved May 12, 2012.
  4. ^ Blevins, Juliette (3 May 2018). "Origins of Northern Costanoan ʃak:en 'six':A Reconsideration of Senary Counting in Utian". International Journal of American Linguistics. 71 (1): 87–101. doi:10.1086/430579. JSTOR 10.1086/430579.
  5. ^ a b c "Archived copy" (PDF). Archived (PDF) from the original on 2016-04-06. Retrieved 2014-08-27.CS1 maint: Archived copy as title (link)
  6. ^ Owens, Kay (2001), "The Work of Glendon Lean on the Counting Systems of Papua New Guinea and Oceania", Mathematics Education Research Journal, 13 (1): 47–71, doi:10.1007/BF03217098, archived from the original on 2015-09-26

External links

This page was last edited on 4 July 2019, at 12:34
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