Euclidean Geometry • Equitable Distributing • Linear Programming • Set Theory • Nonstandard Analysis • Advice • Topology • Number Theory • Calculation of Times (Previous | Next)

We measure time so far in years, months, weeks, days, hours, minutes, seconds. Seconds can be divided into respectively thousandths, step by step more finely down to the measuring border. But is our calculation of times practical?

The following three questions are to clarify the problems:

1. How many days are between 25.01.2004 and 11.11.2004?

2. Which weekday is the 11.11.2004?

3. How many seconds are between 09:12: 06 o'clock and 11.11.11 o'clock?

The answer to question 1 must consider the different length of the months and leap years.

The answer to question 2 must in addition include the seven day rhythm of the week and spring from a date with well-known weekday.

The answer to question 3 requires the conversion of hours and minutes in 3600 or rather 60 seconds.

I suggest the following calculation of times:

1. The octal system is used throughout.

2. The time is specified: [[+/-] year,][day].[fraction of the day][additive].

3. An Earthday is inserted between Sunday and Monday within the week.

4. The months have regularly four eight-day weeks.

5. The last twelfth month has deviating 13 days (in the leap year 14).

6. Day 0 is New Year (1.1. so far, Sunday), day 1 is working day (Earthday).

7. The year of the introduction is the year 0.

8. Time zones are omitted and world-wide Greenwich time counts.

Explanations:

To 1. (The octal system is used throughout.):

We mostly use the decimal system (radix 10, digits from 0 to 9). The radix indicates the number of different digits of a number system. Numbers can be represented in other number systems equivalently (only with another "appearance").

Numbers that are larger than the highest digit indicates transfer a carry of one to the next left place: After the 9 follows the 10, after the 99 the 100, after the 999 the 1000 etc.

Humans on the average can recognise maximally seven things at one time, without having to count them. The octal system with the radix 8 possesses the digits from 0 to 7 and accommodates here thus the nature of humans.

The octal system can be converted in a simple manner into other number systems that are a power of the radix two. Apart from the binary or dual system (radix 2, digits 0 and 1) are among these in particular the tetral system (radix 4, digits from 0 to 3) and the hexadecimal system (radix 16, numbers from 0 to 9 and from A to F).

Most computers are based on the binary system. The digits 0 and 1 indicate whether current flows or not. Numbers in the decimal system must be converted for these computers into the binary system so that can be counted fast.

The longer the numbers are, the more time-consuming the conversion is. For the output in the decimal system, a renewed transformation is necessary. With the conversion between dual and decimal system, frequently transformation-caused truncations arise behind the decimal point.

Here some examples (the number in square brackets indicates the radix for the distinction of the number systems):

2004_{10} = 7D4_{16} = 3724_{8} = 133110_{4} = 11111010100_{2}.

Is 11111010100_{2} outgoing from right divided into two-blocks in the form 1|11|11|01|01|00_{2}, then the representation for 133110_{4} can be obtained by the following replacement pattern:

00 replace by 0, 01 replace by 1, 10 replace by 2 and 11 replace by 3. A first block left with a digit remains unchanged.

For the production of 3724_{8} from 11111010100_{2}, three-blocks from right have to be transformed to: 11|111|010|100_{2}. The replacement pattern reads:

000 replace by 0, 001 replace by 1, 010 replace by 2, 011 replace by 3, 100 replace by 4, 101 replace by 5, 110 replace by 6, 111 replace by 7. A first block with less than three digits has to be completed before by zeros from the left.

For the production of 7D4_{16} from 11111010100_{2}, blocks of four from right have to be transformed to: 111|1101|0100_{2}. The replacement pattern reads:

0000 replace by 0, 0001 replace by 1, 0010 replace by 2, 0011 replace by 3, 0100 replace by 4, 0101 replace by 5, 0110 replace by 6, 0111 replace by 7, 1000 replace by 8, 1001 replace by 9, 1010 replace by A, 1011 replace by B, 1100 replace by C, 1101 replace by D, 1110 replace by E, 1111 replace by F. A first block with less than four numbers has to be completed before by zeros from the left.

Generally, if you have a radix of 2^{n}, there have to be formed blocks of n digits in the representation of the dual system from the right and the first block to the left has to be filled up with zeros up to n digits, beginning from the left. The transformation functions also in reverse from the 2^{n}-system into the binary system. Furthermore, there are still different algorithms for the transformation.

For the production of 7D4_{16} from 2004_{10}, 2004_{10} is continual divided by 16_{10}:

2004_{10}/16_{10} = 125_{10} remainder of 4_{10};

125_{10}/16_{10} = 7_{10} remainder of 13_{10};

13_{10}/16_{10} = 0_{10} remainder of 7_{10}.

The two digit remainders are converted into the hexadecimal system and gone through from top to bottom. The desired result 7D4_{16} is obtained if they are thereby lined up from right to left 13_{10} = D_{16}.

Generally, the number which shall be converted has to be continually divided by the radix of the target system, and the multi-digit remainders have to be converted into the notation of the target system. The sought-after number results from the lining up of the remainders from right to left if in the division calculation the remainders are gone through from top to bottom.

Naturally, here different transformation algorithms exist - also in the other direction.

Since the continual division is computationally more expensive than the digit replacement in digit blocks, the spoken to loss of time emerges. Right-of-floating point positions are similarly converted.

To 2. (The time is specified: [[+/-] year,][day].[fraction of the day][additive].):

The sequence results from the sorting capability of the date. + can be omitted; - refers only to the year. If the fraction of the day is omitted, the (normal) date results. The year can be omitted likewise if the understanding permits it.

The fraction is indicated as sequence of octal digits. For optical reasons the numbers can be filled up with zeros on the right to an even number (e.g. four or six-digit). Additive can be for example: Christian, Jewish or Islamic or an abbreviation. The standard is without additive.

The seasons spring, summer, autumn and winter and the holidays return regularly. These regularities are to persist in respect of the year.

Hours are no more used - for mistake reasons. The day is divided roughly into eight parts (called octs). Furthermore, an oct is divided into eight parts (called lepts). The minute newly is defined as the 64th part of a lept. The second newly is defined as the 262144th part of a day (64 * 64 * 64 = 262144).

Therewith, the time can be indicated approximately three times more exactly with six digits in the octal system as with six digits in the decimal system (262144/[24 * 60 * 60] ~ 3.034). Four lepts, or a half oct, correspond to 1.5 hours and two lepts correspond to a school hour of 45 (old) minutes. The break of five minutes becomes the break of a quarter-lept.

Because 64_{10} = 100_{8}, the seconds pass in natural way into minutes, the minutes likewise into lepts and the lepts into days. The days cannot pass in natural way into the years because a normal year has 365_{10} = 555_{8} days. The leap year with 366_{10} = 556_{8} days does not improve anything here.

The new analogue clock dial is much more symmetrically designed than the old one and corresponds better to the compass rose and the sundial. x and + are superposed as principal axis's. The 0 is placed at the bottom. Clockwise follow the digits from 1 to 7. The division into 64 lines (lepts, new minutes and seconds) optically hardly differs from the hitherto into 60 lines (old minutes and seconds).

To 3. (An Earthday is inserted between Sunday and Monday within the week.):

The weekdays are mostly named after planets. The earth missed out so far. It is appreciated with the Earthday.

A week with eight days represents at first sight a greater human burden. Wednesday could however bring a relief as middle of the week if the afternoon is kept as free as possible, both on the job and at school.

If we proceed from one eight-hour working day and considers 56 days at that would have been worked in the seven-day week 320 hours, in the eight-day week 336 hours, then it results for Wednesday that 16/7 hours per eight-day week would have to be worked less if the same work time is taken as a basis.

If we begin our work in GMT at 7 o'clock, then we work on an eight-hour working day till 15:30 o'clock if we proceed from a half hour of lunch time. On a Wednesday would have to be worked at same beginning time only till approx. 12:43 o'clock.

It is free to us to work at flexible work time the week over longer in order to be able to terminate on Friday our work earlier. Modified solutions are to be used respectively at other work time models and work times.

To 4. (The months have regularly four eight-day weeks.):

With 32 days the months can be halved very well by two-steps. From the three-digit day indication the month can be determined in the octal system - if necessary zeros have to be completed on the left. 0 - 3 in the second place characterise odd, 4 - 7 even months.

Month is, however, only a transient term since month is finally derived from moon and the moon has subordinated importance in the new calculation of times. Instead the terms half, third, quarter, sixth and twelfth (of the year) are used. The date is to be indicated, if possible, accurately to week or day.

To 5. (The last twelfth month has deviating 13 days (in the leap year 14).):

The position of the leap day in the middle of the year was so far little plausible. At the end of the year, the calculation of times can be corrected expediently as missing time units are inserted - in particular in seconds. To New Year all clocks are then reset to [year],0.000000. Here fields will profit where special accuracy is needed.

The last five (in the leap year six) days can receive own weekday names. The weekday names will become less important, however, in favour of the numbers connected with them.

To 6. (Day 0 is New Year (1.1. so far, Sunday), day 1 is working day (Earthday).):

That each month begins with Sunday has practical reasons. The beginning of the week with Sunday has historical reasons and is also a concession to the moon calendar. In addition, the 0 resembles the symbol for sun (sun calendar).

Since the Gregorian calendar at present is the most common calendar, New Year is the 1.1. according to the Gregorian calendar. The celebrations around the turn of the year can be committed in well-tried form.

To 7. (The year of the introduction is the year 0.):

This happens for practical reasons: The indication of the year can be noted briefly (without additive). A clear cut is made concerning the old calculation of times.

To 8. (Time zones are omitted and world-wide Greenwich time counts.):

The question what time in another time zone counts is unnecessary. A world-wide communication is facilitated. Greenwich historically has the zero meridian.

Now, instead of the three initial questions, the substantially more difficult is asked:

How many days are between 25.01.2004 09:12:06 o'clock and 11.11.2004 11:11:11 o'clock?

First, the question is reformulated into the octal system (with truncation at the seconds, chr. stands for Christian):

How many days are between 3724,30.304233 chr. and 3724,473.356511 chr.?

The answer is a simple subtraction: 443.052256. The 25.01.2004 would be a Sunday, the 11.11.2004 would be a Tuesday in the new calculation of times. Between the two numerical data are exactly 443052256 (new) seconds, rounded 4430523 (new) minutes, 44305 lepts, 4431 octs, 443 days and 44 weeks.

Discussion of important points:

The largest challenge might be - independently from the calculation of times- the conversion into the octal system. It is connected with high costs, great need for discussion and preparation for many years. The policy will have to contribute a hard convincing by extensive advertising efforts.

Since all religious communities would have to live with an additional day per week, none can maintain it would be disadvantaged. In relation to other week lengths, the addition of one day is certainly the most moderate. The annual indication is not religiously founded. By the additive each religious community can characterise their year calculation.

I do not doubt the human learning aptitude. Also handling computers was so far successfully mastered. The span between humans and computer is strongly reduced by introduction of the octal system.

Money, referred to the month, can be converted accurately to week and day. Here the shortened twelfth month can be better calculated. The same month length, under normal conditions, is fairer regarding the money than the irregular month length.

The arising costs are exceeded over the years by the saved costs. Calculation and communication become clearly simpler.

Everyone celebrates his birthday at the same weekday in the year - except those that are born at a leap day. This is not a crucial disadvantage.

The SI-unit second would have to be redefined. Atomic clocks should have hereby no problem. The introduction of the octal system requires the replacement of terms such as kilos and million respectively the meaning shift of word beginnings such as mega, micro and nano.

All new clocks show ideally the same time. The conversion of the clock after change to another time zone is needless. Depending upon the old time zone the day turns out to be differently related to the time. Like usual humans can begin the day, eat, work or learn, meet and sleep. Only the indicated time is another.

We need to get used to that, also what concerns the date line. It remains, however, at its old place. World-wide communication is to be given priority because of its increasing importance.

With the introduction of the octal system, not everything is done. It should be connected meaningfully with a world-wide source language (as native language), likewise based on the octal system. Such an efficient planned language is presented below linguistics. Everyone shall talk to everyone on the same level.

Synoptic octal calendar | January, February, March, April, May, June, July, August, September, October, November, | |||

Weekday | 1^{st} Week | 2^{nd} Week | 3^{rd} Week | 4^{th} Week |

Sunday | 0^{th} Day | 10^{th} Day | 20^{th} Day | 30^{th} Day |

Earthday | 1^{st} Day | 11^{th} Day | 21^{st} Day | 31^{st} Day |

Monday | 2^{nd} Day | 12^{th} Day | 22^{nd} Day | 32^{nd} Day |

Tuesday | 3^{rd} Day | 13^{th} Day | 23^{rd} Day | 33^{rd} Day |

Wednesday | 4^{th} Day | 14^{th} Day | 24^{th} Day | 34^{th} Day |

Thursday | 5^{th} Day | 15^{th} Day | 25^{th} Day | 35^{th} Day |

Friday | 6^{th} Day | 16^{th} Day | 26^{th} Day | 36^{th} Day |

Saturday | 7^{th} Day | 17^{th} Day | 27^{th} Day | 37^{th} Day |

Outlook: Calculation of lengths in the octal system

We measure lengths up to now on the basis of the SI-unit metre. In the octal system are to form appropriate multiples of 8. What is the practical implementation?

Instead of conventional kilometres of 1000_{10} m length, octal kilometres of 512_{10} = 1000_{8} m length are used. Since the time is measured instead of conventional seconds in octal seconds of day length/1000000_{8} ~ 0.33_{10} s, it follows that almost three octal seconds are counted per conventional second.

On the tachometer the speed is measured in octal kilometre/lept with lepts the 100_{8}th = 64_{10}th part of the day. 100 kilometres per hour correspond to approximately 73242_{10} ~ 111.2_{8} lepts kilometres (octal). The difference in the numerical value is hardly noticeable and therefore justifiable. 100_{8} lepts kilometres (octal) correspond vice versa approximately 87.38_{10} kilometres per hour.

The speed of light is 299792458_{10} m/s = 98808549.39_{10} m/octal second ~ 570731345_{8} m/octal second. If the speed of light is indicated with about 300000 km/s, we can indicate it with 570000 octal kilometres/octal second.

The 25_{10}-metre lanes are shortened to 24_{10} = 30_{8} m, 50_{10}-metre swimming to 48_{10} = 60_{8} m. Instead of 50_{10} m, 100_{10} m, 200_{10} m, 400_{10} m, 800_{10} m and 1500_{10} m would be to swim 48_{10} = 60_{8} m, 96_{10} = 140_{8} m, 192_{10} = 300_{8} m, 384_{10} = 600_{8} m, 768_{10} = 1400_{8} m and 1536_{10} = 3000_{8} m. These deviations are justifiable. The long hauls should be multiples of 1000_{8}.

In athletics, there are also the following distances: 3000_{10} m, 5000_{10} m, 10000_{10} m, 20000_{10} m and 50000_{10} m, which could be replaced by the following distances 3072_{10} = 6000_{8} m, 4096_{10} = 10000_{8} m, 8192_{10} = 20000_{8} m, 20480_{10} = 50000_{8} m and 49152_{10} = 140000_{8} m. The marathon will remain unchanged (42.195 km = 122323_{8} octal kilometres).

For the construction usually apply to the centimetre exactly determined measures. As in the octal system can be accurately calculated at 1/64_{10} and 1/512_{10} m, there should be no problems. The same is true for furniture, although certainly a turning away from measures of multiples of 0.1 m to multiples of 0.125 m can be accepted.

© 2001-2008 by Boris Haase

• disclaimer • mail@boris-haase.de • pdf-version • bibliography • subjects • definitions • statistics • php-code • rss-feed • top