When in elementary school there was a chart showing the numbers. But the zero was to the right of the nine. That confused me then. No wonder kids have difficulty grasping the concept of zero.
On Fri, Jun 8, 2018, 8:34 AM Björn Helgason <gos...@gmail.com> wrote: > beenary numbers > > https://m.phys.org/news/2018-06-scientists-bees-concept.html > > On Fri, 8 Jun 2018 07:40 'Bo Jacoby' via Chat, <c...@jsoftware.com> wrote: > > > Cardinal numbers (0, 1, 2, . . .) are including 0 (zero). > > Ordinal numbers (1, 2, 3, . . .) are starting with 1 (first). There is > no > > "zeroth". > > There is arithmetic of cardinal numbers (including the J verbs + * ^ ! ) > , > > but there is no arithmetic of ordinal numbers. > > The codes of the UDC are important numerical objects, but they are > neither > > integers, nor decimal fractions, nor rational numbers, nor real numbers, > > nor complex numbers, nor quaternions, nor vectors, nor matrices, nor > > functions, nor operators. They have been neglected by mathematicians. > > A new kind of numbers must be considered. I dubbed them 'ordinal > > fractions' . > > A cardinal number, such as 'one', counts a set. > > An ordinal number, such as 'the first', identifies an element in a set. > > A cardinal fraction, such as 'one half', measures a part of a totality. > > An ordinal fraction, such as 'the first half', identifies a part of a > > totality. > > Consider for simplicity the binary, rather than the decimal, notation. > > one = 1 = 0001. The digit positions, right to left, indicate ones, twos, > > fours, and eights, and the digit values are one-digit binary cardinal > > numbers, 0 and 1. > > the first = 1 = 0001. This is the cardinal number corresponding to the > > ordinal number in question. > > one half = 0.1 = 0.1000. The digit positions after the binary point > > indicate halfs, fourths, eights, and sixteenths, and the digit values are > > one-digit binary cardinal numbers, 0 and 1. > > the first half = ????? > > My solution to this problem is > > the first half = 1 = 1000 > > the second half = 2 = 2000 > > > > the first fourth = 11 = 1100 > > > > the second fourth = 12 = 1200 > > > > the third fourth = 21 = 2100 > > > > the fourth fourth = 22 = 2200 > > > > the odd fourths = 01 = 0100 > > the even fourths = 02 = 0200 > > the sixteenth sixteenth = 2222 > > Note that the digit positions indicate halfs, fourths, eights, and > > sixteenths, and the digit values are either 1 meaning first, and 2 > meaning > > second, or 0 meaning both. 1000 means: first half, both fourths, both > > eights, both sixteenths. 'both' goes without saying, just as 0 goes > without > > saying. 1000 = 1 = first half. That is one reason for choosing 0 for > > 'both'. > > I did not know the words hyponymy and hypernymy. Thanks! That is just > what > > I need. > > In logic I let 1 and 2 represent True and False. 0 means unknown or > > unimportant. > > > > The Transylvanian problem: > > 0001 Minna is human > > > > 0002 Minna is vampire > > 0010 Minna is sane > > 0020 Minna is insane > > 0100 Lucy is human > > 0200 Lucy is vampire > > 1000 Lucy is sane > > 2000 Lucy is insane > > Check the 2^4 ordinal fractions from 1111 to 2222 against the data. (I > > have not done it) > > Thanks! Bo. . > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > Den 5:58 fredag den 8. juni 2018 skrev Donna Y <dy...@sympatico.ca>: > > > > > > Can we agree on definitions > > > > Ordinal numbers and Cardinal numbers are Natural numbers which do no > > include the number 0. The Natural numbers are well ordered. > > Whole numbers are the Natural numbers and 0. 0 is the least element of > the > > Whole numbers. > > Integers are Whole numbers and Negative signed Natural numbers. (-n + n= > > 0 Zero is not a positive or a negative integer—0 has no sign.) The set > of > > integers has no least element. > > Real numbers are continuous and complete and can be rational or > > irrational—rational numbers are integers and fractions, irrational > numbers > > cannot be expressed as a ratio of two integers. > > Imaginary numbers are not real—they exist in another dimension—root > > negative 1 or i, complex numbers have real and imaginary components. > > > > Tables are two dimensional arrays. > > > > The basic structure of UDC is hierarchy but it could be viewed in other > > ways—as you said yourself. > > > > Are you using tables where Rows are records and Columns are attributes? > Is > > there a primary key? > > > > I am not sure where your Ordinal Fraction comes in but a computer > > application of UDC would need full integration of information retrieval > > (IR) features into a database management system (DBMS). > > > > Right truncation specifies hypernomy (is a—as in number theory is a > subset > > of mathematics)—I copied this table but from my view forms of higher > degree > > is not a subset of diophantine equations. > > > > Table 1: > > > > 5: mathematics and natural sciences > > 51: mathematics > > 511: number theory > > 511.5: diophantine equations > > 511.57: forms of higher degree > > > > What is the precise reason you chose to use 0 as a wild card?—why not * > or > > # or & or … ? What advantages are derived by using 0? For example when > you > > use boolean logic the 0 and 1 can represent False and True and then the > > result vector of 0 and 1 can be used for selection. > > > > hat does this even mean? > > > > > For example: 0=0 and 0>1 and 1<0 and 10<>01 and 1><2. (meaning that the > > whole is equal to the whole, and the whole comprises the first part, and > > the first part is part of the whole, and the first half is compatible > with > > the odd fourths, and the first part is disjoint with the second part). > > > The notation for ordinal fractions makes ordinal fraction arithmetic > > easy, just as the notation for cardinal numbers makes cardinal number > > arithmetic easy. > > > Keep asking! > > > > > > Maybe you can show how Ordinal Fractions can be applied to the problem > > below: > > > > PUZZLE BREAK > > > > Inspector Craig Visits Transylvania > > Inspector Craig of Scotland Yard was called to Transylvania to solve some > > cases of vampirism. Arriving there, he found the country inhabited both > by > > vampires and humans. Vampires always lie and humans always tell the > truth. > > However, half the inhabitants, both human and vampire, are insane and > > totally deluded in their beliefs: all true propositions they believe > false, > > and all false propositions they believe true. > > The other half of the inhabitants are completely sane: all true > statements > > they know to be true, and all false statements they know to be false. > Thus > > sane humans and insane vampires make only true statements; insane humans > > and sane vampires make only false statements. > > Inspector Craig met two sisters, Lucy and Minna. He knew that one was a > > vampire and one was a human, but knew nothing about the sanity of either. > > Here is the investigation: > > Craig (to Lucy): Tell me about yourselves. > > Lucy: We are both insane. > > Craig (to Minna): Is that true? > > Minna: Of course not! > > From this, Craig was able to prove which of the sisters was the vampire. > > Which one was it? > > — From Logician Raymond Smullyan > > > > > > Donna Y > > dy...@sympatico.ca > > > > > > > On Jun 7, 2018, at 5:55 PM, 'Bo Jacoby' via Chat <c...@jsoftware.com> > > wrote: > > > > > > > > > "You do not show how to access particular rows or columns or elements > > from a table". > > > The table, and the table name, is addressed by 00. > > > The left column, and the left column header, is addressed by 01. > > > The right column, and the right column header, is addressed by 02. > > > The upper row, and the upper row header, is addressed by 10. > > > The upper left table entry, and its data content, is addressed by 11. > > > The upper right table entry, and its data content, is addressed by 12. > > > The lower row, and the lower row header, is addressed by 20. > > > The lower left table entry, and its data content, is addressed by 21. > > > The lower right table entry, and its data content, is addressed by 22. > > > > > > "without knowing the number of rows or columns." > > > In the example there are two rows and two columns. > > > If you need a third row, call it 30. > > > > > > "Why not the concept of arrays with an index?" > > > Arrays may have different shapes. Any ordinal fraction has the shape > > (_$9). > > > Arrays have elements. Ordinal fractions don't. > > > > > > Only array elements contains data. Any ordinal fraction may contain > data. > > > > > > Arrays may have subarrays. Any ordinal fraction has subordinate ordinal > > fractions. > > > Arrays and atoms have names. Ordinal fractions don't. > > > > > > "the set of Natural numbers begin with 1 yet we use base 10 > > representation that uses 0." > > > > > > Cardinal numbers and ordinal fractions have similarities and > > differences. > > > Cardinal number 0 (meaning "nothing") is not the same thing as ordinal > > fraction 0 (meaning "everything"). > > > > > > Cardinal number 1 (meaning "one") is not the same thing as ordinal > > fraction 1 (meaning "first part"). > > > > > > Using "0" for wild card character does not get along with using "0" for > > counting to ten. > > > A cardinal number is represented by a right-justified sequence of > digits > > with a finite number of nonzero digits. The cardinal number 1 may be > > written 00001. > > > > > > An ordinal fraction is represented by a left-justified sequence of > > digits with a finite number of nonzero digits. The ordinal fraction 1 may > > be written 10000. > > > > > > "However this was not an easy system in which to do arithmetic so I > > cannot see how your base 9 system could be either." > > > Cardinal numbers, A and B, are ordered such that either A=B or A<B or > > A>B. For example: 0=0 and 0<1 and 1>0 (meaning "zero is equal to zero" , > > and "zero is fewer than one", and "one is more than zero"). > > > > > > Ordinal fractions, A and B, are ordered such that either A=B or A<B or > > A>B or A<>B or A><B. For example: 0=0 and 0>1 and 1<0 and 10<>01 and > 1><2. > > (meaning that the whole is equal to the whole, and the whole comprises > the > > first part, and the first part is part of the whole, and the first half > is > > compatible with the odd fourths, and the first part is disjoint with the > > second part). > > > The notation for ordinal fractions makes ordinal fraction arithmetic > > easy, just as the notation for cardinal numbers makes cardinal number > > arithmetic easy. > > > Keep asking! > > > Bo. > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > Den 18:35 torsdag den 7. juni 2018 skrev Donna Y < > dy...@sympatico.ca > > >: > > > > > > > > > UDC is a library classification system akin to the Dewey decimal > > system—the creators collaborated with Dewy. UTC is also used for data. > > > > > >> Every number is thought of as a decimal fraction with the initial > > decimal point omitted, which determines the filing order. > > > > > > I am not sure why you say it cannot handle tables. > > > > > >> Concepts are organized in two kinds of tables in UDC:[27] < > > > https://en.wikipedia.org/wiki/Universal_Decimal_Classification#cite_note-UDC_Structure-27 > > > > > >> Common auxiliary tables (including certain auxiliary signs). These > > tables contain facets of concepts representing, general recurrent > > characteristics, applicable over a range of subjects throughout the main > > tables, including notions such as place, language of the text and > physical > > form of the document, which may occur in almost any subject. UDC numbers > > from these tables, called common auxiliaries are simply added at the end > of > > the number for the subject taken from the main tables. There are over > > 15,000 of common auxiliaries in UDC. > > >> The main tables or main schedules containing the various disciplines > > and branches of knowledge, arranged in 9 main classes, numbered from 0 > to 9 > > (with class 4 being vacant). At the beginning of each class there are > also > > series of special auxiliaries, which express aspects that are recurrent > > within this specific class. Main tables in UDC contain more than 60,000 > > subdivisions. > > > > > > Can you show exactly what is gained by your “ordinal fraction”? > > > > > > Also—the set of Natural numbers begin with 1 yet we use base 10 > > representation that uses 0. > > > > > > However in the 4 BCE the Greeks used their letters to represent numbers > > from one to nine with no zero: > > > > > > alpha beta gamma delta epsilon digamma zeta eta theta > > > > > > They didn’t need to use 0 for higher values because much like Romans > > they had symbols for higher values. > > > > > > However this was not an easy system in which to do arithmetic so I > > cannot see how your base 9 system could be either. > > > > > > You suddenly use 0 when you illustrate tables. You do not show how to > > access particular rows or columns or elements from a table but instead > use > > vague > > > terms upper, lower, left, right without knowing the number of rows or > > columns.. > > > > > > Why not the concept of arrays with an index?. > > > > > > Maybe you could explain what I am missing. > > > > > > Donna Y > > > dy...@sympatico.ca > > > > > > > > >> On Jun 7, 2018, at 3:47 AM, 'Bo Jacoby' via Chat <c...@jsoftware.com> > > wrote: > > >> > > >> Ordinal Fractions is an improvement to the idea behind the Universal > > Decimal Classification (UDC). > > > > > > ---------------------------------------------------------------------- > > > For information about J forums see http://www.jsoftware.com/forums.htm > > > > > > > > > ---------------------------------------------------------------------- > > > For information about J forums see http://www.jsoftware.com/forums.htm > > > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > > > > > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
