On the linear relation between two calendars

Given a date in the Islamic calendar , there is a formula which gives a good (approximate) numerical value of the corresponding date in the Gregorian calendar.

To find this formula or relation ,  we note that the ratio of a mean (lunar) year of The Islamic calendar to a solar year is:

$\frac {354 + \frac {11} {30}} {365.242} = 0.970224$

The first year of the Islamic (Hijra or Hegira) calendar started 19th July 622 according to the Gregorian calendar. 19th July is the 200th day of the year and is (approximately ) 0.5476 in parts of the solar year ,while the number of years elapsed is equal to ( y-1) . The days are distributed regularly in both calendars , so the date of the beginning of the year y in Gregorian years is :

$0.970224 (y-1)+622.5476$

which can be written as :

$y_ {\text {ch}} = 0.970224 y_ {\text {hj}} + 621.5774$     (1)

Solving the equation above for the year of the Islamic calendar we get:

$y_ {\text {hj}} = -640.653499 + 1.03069 y_ {\text {ch}}$    (2)

If we try to take a more accurate value for the average year length of  the Gregorian calendar  and take into account additional decimal digits , we get the following slightly more precise formulas replacing the two formulas (1) and (2) above :

yhjtoch(t) = 621.5773576247731 + 0.9702237766245703 t

and:

ychtohj(t) = -640.6536023959899 + 1.0306900573793625 t

Now if we solve the two equations (1) = (2) above or yhjtoch(t) = ychtohj(t) , we should find the date when the two calendars are equal and have the same day of the same month of the same  year . Solving (1) = (2) gives the result 20875.052079 , which corresponds to 19 days  (0.052079×365 ≈ 19) of the year 20875 , or 19th January 20875 , or 19/1/20875 .
yhjtoch(t) = ychtohj(t) gives the result 20874.956161756745132 , which corresponds approximately to the 349th day of the year 20874 , 16 days before the end of the year.
Using a calender converter (within Mathematica  or an online converter ) , we find that the dates when the two calendars are equal lie between 1/5/20874 and 30/5/20874 , i.e the first day of the 5th month (May) of the year 20874 in the Gregorian calendar is also the first day of the 5th month of the year 20874 in the Islamic calendar , and this goes on until the 30th day of the 5th month of the year 20874 in both calendars. Therefore we can see that the dates obtained by making equal the two sets of equations above deviate and are further away from the real dates verified with calendar converters.

From the beginning of the Islamic calendar to the year 20874 there is a very small increase in the difference between the days of the year for the Islamic and the Gregorian calendar, and for the year 20874 the difference between the 2 calendars (if we equate ychtohj(t) and yhjtoch(t) ) is about 215 days , so I made some calculations and I subtracted a small term (0.000001758733 times t) from the formula converting from the Islamic to Gregorian calendar yhjtoch(t) and I got the following :

yhjch(t) = 621.5773576247731 + 0.9702237766245703 t – 0.000001758733 t

yhjch(t) = 621.5773576247731 + 0.9702220178915703 t

yhjch(t) is a little more accurate than the equation (1) above , which differs from the real date by approximately one day for years and dates in the current (21st) century , then the gap widens between yhjch(t) and (1) , with yhjch(t) giving dates a little before the real date , and equation (1) a little after the real (Gregorian calendar) date.For the year 20874 yhjch(t) gives more precise dates.

Setting ychtohj(t) = ychj(t) and solving ychj(t) = yhjch(t) , we get the result 20874.349006727632514 ,which is a date that lies within the interval between 1/5/20874 and 30/5/20874 for the two calendars.

Below is a graph showing ychj(t) and yhjch(t) around t = 20874 and how they intersect :

So the two conversion formulas or linear equations ychj(t) and yhjch(t) can be considered to be the two most accurate ones for converting between the Islamic and Gregorian calendars.

Additional reference work related to this post :
Time measurement and calendar construction , by Broughton Richmond.

Pi in the sky , and on the computer

Results and things  related to pi are usually published or made known on Pi Day . But you never know when you get inspired by π or find time to explore this ‘venerable’ math constant , and I have on Pi Day . By the way there is also a Pi approximation Day (July 22) , so there’s more than one date to talk or write  about π .

Let’s kick off with an image of the value of pi (in the clouds ) with 12 decimal digits , made with Photoshop.I tried to make it realistic and show π and its numerical value ( 3. 141592653589) as part of the clouds and the sky.

Click on the image above to see and enlarged version.

Now for some computer-based mathematical explorations related to π.
There is a known relation between e ( the base of the natural logarithm ) and π :

$e^{\pi }-\pi =19.999099979$ and is approximately equal to 20 or almost 20 , which is known as an almost integer.
If we try to be more accurate and find the first 1000 decimal digits for the expression above we get (with the help of Mathematica) the following number:

19.99909997918947576726644298466904449606893684322510617247010181721652594440424378488893717172543215169380461828780546649733419980514325361299208647148136824787768176096730370916343136911881572947102843075505750157713461345968680161070464780150721176248631484786057786790083331108325695374657291368002032330492961850463283115054452239990730318010838062172626769958035434209665854687644987964315998803435936569779503997342833135008957566815879735578133492779192490846222394896357465468950148911891909347185826596341254678588264050033689529697396648300564585855142666534919457239163444586998081050100236576797224041127139639108211122123659510905094871070706680635934325684092946890616346767578519812785089761055789304041857980123101280905543416254404987679233496308302396952371198509012175432057419088516489412743155057902167919927734272964964116423666794634333328342687902907792168390827162859622042360176355034576875485783678406122447755263475337650755251536818489395213976127148481818560841182505647  ,
which shows that $e^{\pi }-\pi$ does not approach 20 completely or uniformly.

Here are some more expressions and calculations  involving π (calculated with Mathematica):

I think it remains to be seen if the numbers above are transcendental. The Mathematica (version 10)  command Element[z , Algebraics ] cannot determine whether these numbers belong to the domain of algebraic numbers or not.

Now let’s consider an expression containing π , e and i (the imaginary unit complex number). The following expressions are equivalent:

A 2D complex plot of the last function above (with z between -4 and 4) gives the following graph (made with Maple):

A 3 D complex plot of the same function ( with z between -4-4i and 4+4i)   gives the following graph (with the help of Maple):

A general solution of the function f(z) above for f(z) = 0 is (calculated with Mathematica):

Here is an interesting result:

Using the Mathematica commands Element[z , Algebraics ] and Not[Element[z , Algebraics ]] , it seems that the solutions z of f(z)=0 above ( for different values of the constants ) do not belong to the domain of algebraic numbers , and are therefore  transcendental numbers.

And with this I bid π farewell for now.

Calendars ,days of the week , and dates

There are different calendar systems that are still used or were used in the past . Examples include the Julian calendar , the Gregorian calendar , the Hebrew calendar , the Islamic and the Iranian calendars , the French Republican calendar , etc. Most calendars use astronomical events and cycles as their basis: the day is based on the rotation of planet Earth on its axis , the month is based on the Moon revolving around the Earth , and the year is based on the period of revolution of the Earth around the Sun.The influence of the gravitational force from other planets may cause the length of a particular year to vary by several minutes. The Gregorian calendar was introduced in 1582 .Dates before October 1582 are usually given in the Julian calendar .

Here is a list of leap years in the Gregorian calendar from 2016 to 2416 (done with the help of Mathematica) :

2016 , 2020 , 2024 , 2028 , 2032 , 2036 , 2040 , 2044 , 2048 , 2052 , 2056 , 2060 , 2064 , 2068 , 2072 , 2076 , 2080 , 2084 , 2088 , 2092 , 2096 , 2104 , 2108 , 2112 , 2116 , 2120 , 2124 , 2128 , 2132 , 2136 , 2140 , 2144 , 2148 , 2152 , 2156 , 2160 , 2164 , 2168 , 2172 , 2176 , 2180 , 2184 , 2188 , 2192 , 2196 , 2204 , 2208 , 2212 , 2216 , 2220 , 2224 , 2228 , 2232 , 2236 , 2240 , 2244 , 2248 , 2252 , 2256 , 2260 , 2264 , 2268 , 2272 , 2276 , 2280 , 2284 , 2288 , 2292 , 2296 , 2304 , 2308 , 2312 , 2316 , 2320 , 2324 , 2328 , 2332 , 2336 , 2340 , 2344 , 2348 , 2352 , 2356 , 2360 , 2364 , 2368 , 2372 , 2376 , 2380 , 2384 , 2388 , 2392 , 2396 , 2400 , 2404 , 2408 , 2412 , 2416 .

In the Gregorian calendar , most years that are multiples of 4 are leap years . The years 2100, 2200, 2300 are not leap years .

Below is a list of the days of the week each one corresponding to the first day of the year (Gregorian calendar) from the year  2016 to 2036 :

• 1/1/2016 : Friday.
• 1/1/2017 : Sunday.
• 1/1/2018 :Monday.
• 1/1/2019: Tuesday.
• 1/1/2020: Wednesday.
• 1/1/2021: Friday.
• 1/1/2022: Saturday.
• 1/1/2023: Sunday.
• 1/1/2024: Monday.
• 1/1/2025: Wednesday.
• 1/1/2026: Thursday.
• 1/1/2027: Friday.
• 1/1/2028: Saturday.
• 1/1/2029: Monday.
• 1/1/2030: Tuesday.
• 1/1/2031: Wednesday.
• 1/1/2032: Thursday.
• 1/1/2033: Saturday.
• 1/1/2034: Sunday.
• 1/1/2035: Monday.
• 1/1/2036: Tuesday.

Next I give the day of the week corresponding to the 4th of July from the year 2015 to 2020:

• 2015: 4th of July is a Saturday.
• 2016: 4th of July is a Monday.
• 2017: Tuesday.
• 2018: Wednesday.
• 2019: Thursday.
• 2020: Saturday.

The day of the week corresponding to the 14th of July (Bastille day in France) from the year 2015 to 2020:

• 2015: 14th of July is a Tuesday.
• 2016: Thursday.
• 2017: Friday.
• 2018: Saturday.
• 2019: Sunday.
• 2020: Tuesday.

Easter Sunday date (Western churches and Greek Orthodox churches) for 4 years:

• 27 March 2016 (Western), 1 May 2016 (Greek Orthodox).
• 16 April 2017 , same date for Greek Orthodox churches.
• 1 April 2018 , 8 April 2018.
• 21 April 2019 , 28 April 2019.

Here are days of the week corresponding to some important historical dates and people (after 1582 the dates are in the Gregorian calendar) :

• The most accepted date for the Death of Genghis Khan  ( Julian calendar) is 18 August 1227 , which was a Wednesday.The date in the Gregorian calendar would be  Wednesday 25 August 1227.
• Columbus reached America : Friday 10 October 1492 (Julian calendar). The date in the Gregorian calendar would be Friday 21 October 1492.
• First publication of Philosophiæ Naturalis Principia Mathematica by Isaac Newton: Saturday 5 July 1687 ( Gregorian).
• Date of birth of Carl Friedrich Gauss : Wednesday 30 April 1777.
• Date of death of Carl Friedrich Gauss: Friday 23 February 1855.
• Death of Napoleon Bonaparte : Saturday 5 May 1821.
• Date of birth of Jules Verne : Friday 8 February 1828.
• Date of birth of Pierre Simon Laplace : Wednesday 23 April 1749.
• Death of Pierre Simon Laplace : Thursday 5 April 1827.

Another day of the week and date: Sunday , 31 July 2072 (this date is somewhat important for me).

The Islamic calendar is a lunar calendar ( based on the motion and phases of the Moon) of 12 months  with alternating months of 29 and 30 days , and  a year made of 354 days , which is about 11 days shorter than the length of the year in the Gregorian calendar. In the table below I give the day of the week corresponding to the first day of the year (Islamic calendar) and the corresponding Gregorian date for a number of consecutive years:

Year(Islamic calendar) First day of the year Gregorian date
1/1/1437  Thursday  15 October 2015
1/1/1438  Monday  3 October 2016
1439  Friday  22 September 2017
1440  Wednesday  12 September 2018
1441  Sunday  1 September 2019
1442  Thursday  20 August 2020
1443  Tuesday  10 August 2021
1444  Saturday  30 July 2022
1445  Wednesday  19 July 2023
1446  Monday  8 July 2024
1447  Friday  27 June 2025
1448  Wednesday  17 June 2026
1449  Sunday  6 June 2027
1450  Thursday  25 May 2028
1451  Tuesday  15 May 2029
1452  Saturday  4 May 2030
1453  Wednesday  23 April 2031
1454  Monday  12 April 2032
1455  Friday  1 April 2033
1456  Tuesday  21 March 2034
1457 Sunday 11 March 2035
1458 Thursday 28 February 2036
1459 Tuesday 17 February 2037
1460 Saturday 6 February 2038
1461 Wednesday 26 January 2039
1462 Monday 16 January 2040
1463 Friday 4 January 2041
1464 Tuesday 24 December 2041
1465 Sunday 14 December 2042
1466 Thursday 3 December 2043
1467 Tuesday 22 November 2044

A good online calendar converter can be found in here.

A number of scholars , thinkers and scientists in the past  have written books and studies about calendars , the chronology of world history and historical events (such as Isaac Newton’s The Chronology of Ancient Kingdoms and al Biruni‘s The Chronology of Ancient Nations , also known as The Remaining Signs of Past Centuries).

However , historical chronology and dates before the beginning of the Christian Era are mostly approximate , uncertain or imprecise. Therefore I think a better and more accurate chronology of ancient History (and of World History in general) ought to be constructed , using rigorous historiographical and historical methods , and unbiased scientific reasoning , analysis , and methods.