OB surveying, number systems and Si.427 | Old Babylonian mathematics & Plimpton 322 | N J Wildberger

[@GoatzAreEpic]

I just wanted to say you are the first mathematician on youtube I look up to and I aspire to be just as knowledgable as you are one day.

[@draconyster]

This video is great. A lot of math on youtube rehashes same concepts but this is really fresh!

[@carloscalvimontesr.1601]

The Istanbul tablet is Si.427 and it is nothing more than simple surveying work with right angles and consequently the presence of right triangles can be inferred. But, for the unseen, a fuss is being made pretending it's a fabulous discovery.

[@hellenakinyi7942]

Video Content 00:00 Introduction 00:56 Old Babylonian period 01:47 OB Surveying 05:04 OB geometry (Basic shapes) 06:48 Scalling and similarity 08:25 OB sexagesimal (base 60) system 11:09 Our number systems 16:19 Practical problem (scalling a given triangle)

[@sharonjuniorchess]

I recently came across some mention of ancient Indian writings which suggested that they used a sexigesimal system in addition to their base ten system. Whether they got it from the Babylonians or the Babylonians got it from the Indus people has still yet to be determined. But they did trade and that is how ideas travel. On the matter of division in base 10 because you only have factors of 2 & 5 you only need to divide by multiples of 2 or 5. I came across a paper where they were teaching division using under and over dividing of numbers. It made things a lot easier to do once you got used to the idea of adding instead of subtracting when doing over division.

[@uncannyvalley2350]

Ireland, and the Bru na Boinne* The relationship of the Megalithic second to our current form is mathematically proportional to the ratio between the Sun and Moon. A Megalithic second is 6.66 minutes (400 seconds). A Megalithic Minute is 40 minutes, or 2,400 seconds. Divide that by 60 and you get 1,440 seconds. Divide by 60 again and get 24. There are 216 Megalithic seconds in 24 hours, and after 216 years of 60 6 day weeks (plus five days [plus 1 every 4 years]) this calendar produces 1 extra day. After 630 years the Phoenix comes and lands on top of the Pyramid (Solomon= Mountain of the Sun) Alter of Ra or Holy Grail, on the first New Moon of Spring when three days are removed, bringing the calendar back to perfection. The Pyramid Capstone was even called a Benben, named for the Egyptian Phoenix, but the Phoenix appears in every major religion, even in the Americas. The Moon is 2160 miles in diameter, the Sun is 864,000 miles in diameter. There are 400 seconds to a Megalithic second, the Sun is 400 times the size of the Moon. One Megalithic hour is 240 minutes, or 14,400 seconds. There are 6 Megalithic hours to the day, each made up of 6 minutes, each of which is 6 seconds long. If the Megalithic hour was divided into 60 minutes, each would be 1,440 of our seconds, times 100 is 144,000. The height of the Great Pyramid times 43200 equals the Polar Circumference of the Earth. The Circumference of the Base of the Great Pyramid times 43200 equals the Equatorial Circumference of the Earth. Half the base is equal to one modern second or 1/86,400 The base divided by the height is Pi, the Base minus the height is Pi times 100. 100 Megalithic Yards is 1 Arc degree of the Moon The height in feet divided by Euler's number gives the square root of Pi, times 2 is 354, the number of days in 12 Lunar months of 29.5 days. Enoch buried his 36,525 scrolls under Pillars of Gold and Emerald, representing the Gold and Silver Gates of the Milky Way, found between Orion and Serpentis, at 33 degrees and 216 degrees respectively, representing Osiris and Isis in Orion and Taurus, and Horus/Hercules in Ophiuchus; the Serpent Bearer, which is also shaped like a cave, where Jesus is born and dies, just as Apollo Aulaites, Son of Typhon in Cilicia, which was the home of St Paul/ Josephus. Apollo Aulaites is also known as Zeus of the Cave. Thoth was also seen as the Son of Tphens, Horus as the Sun/ Son of the Dawn (or Dusk) The only Temple in the world that had pillars of Emerald and Gold was the Temple of Melqart in Tyre, Tyre means Rock, and sat just offshore from Urshu Shalom, City of Peace, City of the New Moon, the root of the word Jerusalem. Enoch, Metatron, Hermes Trismegistus, and Amunhotep III all appear to have the same nomeclature, Amunhotep means Happy is the Moon, the New Moon representing Peace and a Full Moon representing War. Hebrew and Iber as in Iberia also have the same root, meaning over, as in over seas. Anglo comes from Ankh, meaning Angle, or to bend, perhaps denoting around the corner, as Plato describes Atlantis being beyond the Pillars of Hercules, being the same Pillars of Solomon, or Melqart, the Gates of Gibralter, or Gabriel's Alter, due West from Tyre, the direction of the rising New Moon. The Christian Cross derives from the Cross of Baal, the Phoenician Sun God, their Almanac, and secret to navigating by the stars was preserved in the Byblos Baal, the book we now call the Bible. All mythology is based on the transit of stars through the seasons, and even today in India and New Zealand, at opposite ends of the earth, they celebrate the Summer and Winter Soltices based on the appearance of Pleaides, seen as a bird, sitting at 33 degrees of the Zodiac. The famous Hummingbird of Nasca is a representation of Pleaides, used by navigators to find land in the Pacific, also the Phoenician Princess Europa rode the Bull of Taurus just as Pleaides, Paris too is derived from Par Bull and Isii isis, being founded by Celts around 500BC. We still use the Phoenician alphabet, sound out our words phonetically. In fact the written word itself derives from their use of sounds instead of Heiroglyphs. The Quran states the Tower of Babel was built by Pharaoh to meet the God of Moses, Babel is the ancient name for Cairo. Babel is also a term for English, and Babylon derives from the first cry of a newborn baby. Britain too is a Phoenician word, the Tin they sourced there Brightened their copper, making Bronze, the material they traded through their maritime network, secretly guarded against would be competitors. This sacred Enochian calendar is encoded within a Pythagorean Triangle 216 high, 630 wide, and 666 on the slope. This comes from a sacred Jewish family secret, and is based on the 19 year Metonic cycle, it took 19 years to train a Druid, England, Scotland and Ireland being home to the same Ancient Mystery schools as found at Edfu, Harran, Cilicia, Jehdi, and Tatooine. There is also an etymological correlation between the words Mathematician and Magician. The most sacred numbers of the Celts are 3 and 11, the proportions of the Great Pyramid in relation to the Earth and Moon is also 3/11. Egyptian beads have been found in both Ireland and Norway dating to the time of Arkenaten, the only 3 Mummies they've been able to DNA test all matched the modern British population. The measurement is derived from measuring the time it takes for the full moon to pass through an aperture formed by two horizontals, as seen in Japanese Shinto Shrines, the rope being used as a pendulum to mark a short period of time. If we made a halo of full moons around the earth there would be 720 of them, divided by 2 is 360, thus we have the 24hr period broken into 2 sections of 12, which themselves divide into 360. Japan was also the last place where Dolmens and Megalithic structures were built, right up until 1000AD. The Shinto tradition is analog to Judiac tradition, and uses the same metonic calendar, and noting each year by the Reign of the presiding Emperor, much like Ancient Egypt. This is why the number 13 is sacred, as 360 divided by 13 is 12. Hence the clock actually has 13 hours on the face, just as Ophiuchus the 13th Starsign completes the circle, marking the Jewish New Year, Rosh Hashanah, Resurrection of the Dead, when Orion and Pleiades rise in the September sky. The Great Pyramid, and it's twin in Teotihuacan are 13 acres square at the base. It means our understanding of the metrics of time and distance, the foot and mile, kilometers and metres, seconds, minutes and hours all derive from the size of a full moon in the sky, especially as expressed in an eclipse, and relates to the Saros or eclipse cycle (322). The leap year is solved by the Olympic games, occurring once every 1461 days, just as the Sothic cycle is 1461 years. The games of Lugh in Ireland go back 5000 years. Even Antikythera means opposite the Great Mother, Kythera. Genesis 8:4 _and the Ark came to rest on the Mountains of Ararat on the 17th day of the 7th month_ "The NWO will begin Sept 17th 2001 during the first Crescent of the New Moon, Rosh Hashanah Feast of Trumpets, Resurrection of the Dead, the year 6000 of the Great Pyramid Calendar, in order to cleanse the Earth and Humanity in preparation for his Kingdom on Earth" British Israel Foundation memo 1922. Urshu Shalom was the name of the city next to Tyre, which means Rock. Urshu Shalom means City of Peace, or City of the New Moon, the root word of Jerusalem. Islam derives from this same root. Shiva is a Hebrew word for 7. Hebrew and Iberia also share the same root, meaning over, as in Overseas, or the ability to sail overseas, like the Phoenicians, who pioneered stellar navigation. The Pole star is called the Pole Star because of its use in navigation by the Phoenicians, it's old English name being _Ship Stare._ In ancient Egypt the Priests who watched the stars were called Horoscopoi, Watchers of the Hour, Hour being related to Horus, the Son of the Dragon, or Dawn/Dusk. A rising star is Masculine, and a setting one feminine, Atum, and Evening, being androgynous at the Zenith, and responsible for what happens on Earth at the time. So the Sun and Moon and their Metonic Calendar bring Order out of Chaos, the Ancients also saw stars as eyes, hence Ezekiel's vision is a representation of the Zodiac, or Mazzaroth in Hebrew, and the strange descriptions of Angels... Daemons being the stars of winter, or the Underworld; Invernos, the Sun and Moon were seen as the Left and Right Eyes of Horus. The Arc of the Covenant, and Garden of Eden are metaphors for the Square of Pegasus, Typhon, aka Ladon, aka Tphens or Jorgumandr are Gaia, or Hydra, the 100 headed, which takes up 100 degrees of the Zodiac. Hence the Son of the Dragon denotes both the first star at night, and Fomalhaut in Aquarius, shaped like an eye, the Sign of John the Baptist, a Setian, the root word of Saturn, or Satan. The Pheonician God of Healing was Eschmun, his Egyptian Sarcophagus found in Sidon (means fish, the Phoenician Symbol for fish is *X* hence Xmas) is inscribed with a short poem, the longest unbroken piece of Phoenician language we have, in which Eschmun, whose Temples were built over hot water springs, was known as the *_Scion of the Widow_* (Isis). His symbol is the Caduceus, the Serpent Staff of Asclepius, which can be found in the hands of Hercules, at 210 degrees. Around the staff is Cerberus, the 3 headed Serpent and guardian of the entry to the underworld. The same symbol you see on the side of medical vehicles; the Winged Serpent. Sidon is also where Jezebel daughter of Hiram and wife of Solomon came from, as well as Pythagoras. This points a lot of fingers at Rameses II being King Ahaz, King of Tyre, Regent of israel, aka Solomon, who built the Temple of Melqart to mark his alliance with Hiram Abiff, King of Assyria, creating the Neo Assyrian Empire in 911BC. /abs/2009ASPC..409..487R/abstract

[@danielfmansfield]

Nice video, Norman. I like that you chose to use a decimal version of regular and the 3-4-5 triple. That gets the ideas across nicely without being bogged down in the details of the base 60 system.

[@njwildberger]

Thanks Daniel! While making this video, the following question occurred to me: the 3-4-5 triangle is pleasantly completely regular in the OB system, as 3,4 and 5 are all regular (wrt base 60). My question: apart from obvious re-scalings of the 3-4-5, are there any other completely regular right triangles? If so, do we have a list of them?

[@danielfmansfield]

@@njwildberger The 3-4-5 is the only one that is completely regular. But triples with two regular sides are also useful, such as 5-12-13 and 8-15-17 which appear in Si.427, and 7-24-25 which appears in MS-3971 (you can think of parts 3 and 4 of MS-3971 as a mini-Plimpton 322).

[@njwildberger]

@@danielfmansfield I can see that if we have a completely regular OB triangle, then no two of the sides can have a common factor. But how do we eliminate the possibility that we have a power of 2, a power of 3 and a power of 5 --- other than the trivial 3-4-5- case?

[@ronancarroll8554]

@@njwildberger This is a rough explanation. 2*3*5=30. using the list 1,7,11,13,17,19,23,29. These are the numbers that are coprime to 2,3,5 less than 30. Yes they are prime but not the issue here and 1 must be included. Add an integer multiple of 30 to this list i.e. 2,3,4,5..... and you will generate a list of all numbers (in as far as you want to go) that are coprime to 2,3,5. Search this extensive list for other Pythagorean triples.

[@michielkarskens2284]

Dear Daniel and Norman, loved the article in Foundations of Science and this video. Would it be possible to do a sequel, showing how easy it is to rescale in the sexagesimal system by multiplying the regular side with its reciprocal? Both the article and the video show hardly anything on the actual tablet Si 427. I am really interested to learn Daniel's reinterptetation of this tablet and how it is connected to Plimpton 322. Another 20 minutes on this subject would be wonderfull and enlightening!!

[@Tonjevic]

Though it does not really undermine your broader point here, it might be worth pointing out that the "floating point" numbers you describe here are actually fixed point numbers. The floating point numbers used in everywhere in computing do not hold a fixed number of digits after the decimal point, but rather they hold a fixed number of significant digits and an exponent, allowing the decimal point to float around in order to scale the number as necessary. The major difference here is the distribution of numbers within the representable interval. Due to the difference in storage, for a given number of bits floating point numbers tend to be able to represent a greater range than fixed point numbers do. On the other hand, floating point numbers cluster more densely around 0, with the gaps between consecutive numbers widening further from the origin; by contrast, fixed point numbers are distributed evenly within their range. Fixed point arithmetic is sometimes used for financial and some scientific applications, but is not typically built into hardware like floating point operations tend to be as it ends up just boiling down to a slight elaboration on integer math anyway.

[@Juanbaez_]

Love your videos. I created an number system, but when I researched about it, I found out that the maya and Babylonian are fairly similar to my numbers. However, all that mathematics you show about the Babylonian’s math and Greeks fits perfectly with my numbers. Note: I like the way how you are taking the time to give some practical problems to understand better. I would like to see an course from your where you show how Babylonian trig works, in detail. Adding solving modern practical problems we face in the modern trig college. Thank you so much for the effort.

[@kallebundsgaard6885]

thank you for this video I have a big project in school, and it helped me understand the complicated text

[@MarkMasters...]

Excellent. Thank you.

[@pickeyberry4060]

Video Content 00:00 Introduction 1:48 Old Babylonian surveying 5:05 Old Babylonian geometry 6:47 Scaling and similarity 8:24 OB sexagesimal (base 60) system 11:08 Our number systems 16:18 Scaling the (3,4,5) triangle

[@geraldillo]

Amazing discovery! I love the fact that the Babylonians used precise math!

[@carloscalvimontesr.1601]

The Istanbul tablet is Si.427 and it is nothing more than simple surveying work with right angles and consequently the presence of right triangles can be inferred. But, for the unseen, a fuss is being made pretending it's a fabulous discovery.

[@carloscalvimontesr.1601]

@Gennady Arshad Notowidigdo ¡¡Excelente! Totalmente correcto.

[@brendawilliams8062]

I like that too.

[@RizzyG007]

I appreciate the effort sir. Btw is your foundations of mathematics playlist good for people who went through highschool math and just want to brush up on the basics all over again to build a more stronger foundation?

[@njwildberger]

@Grand fire expert: Absolutely -- the Math Foundations Playlist at kzbin.info/www/bejne/e5rNqJqfbdd2iMk is chalk full of valuable information to get ordinary people up to speed on a wide range of basic mathematical thinking!

[@robharwood3538]

I would say yes, definitely! I've gone through it more than once, and I'm going through it again recently. Every time I do I get something new out of it. It's really quite interesting and amazing how it all can be made to fit together so well.

[@AhmadAhmad-qx6fp]

It's very important to correlate all this sexigenary numerical systems into meaningful contexts of practical everyday useful entities. Sure! Professeur Norman talks about area calculation and its corresponding geometrical constructs. But remember! Not all Babylonians were on being assets appraisal or in that particular manner, tax collection officers. Lets talk about more elementary notion of say, calendar and time. Just because they used base-60, they still had 12 months within their lunisolar calendar system. Moreover, there's standing problem in deciphering the meaning of 24,29 in Si.427 artefact, to which I think was quite simple. Just as any document, with legal and financial implication, it's quintessential to mark the date to which it's designated. A. We do know that Babylonians used lunisolar calendar, in which, as consequence, was subjected to inevitable intercalation. B. The number 24 and 29 inscribed within the tablet, might be indicating of the calendar they used. Meaning, month and year. But of no casual interpretation from our era. C. The intercalation was done on each 19-year cycle, through adding one "additional month", in year of 3, 6, 8, 11, 14, 17, and 19. D. Babylonians counted their years according to each reigning king. And it was commonly taught tht Hammurabi was the longest reigning king of Babylonian dynasties. E. 24 is the counted month: month of 24; of year 29 from whoever king at the time reigned at Babylonian Empire. If we look upon palû babili, there were several kings whose reigning years were more than 29 years. E. As for the intercalation, month-24 meant as the second month of year 29 of king... Why? Because, there had been 1 month added consecutively on year 3, 6, 8, 11, 14, 17 and 19 (first cycle of 19 year) and then the other 3, 6, and 8. Thus 7+3 additional months. 24 - 10 = 14 => 2nd month of year 29 of King.... The name for second month in Babylonian was, Ayaru.

[@robharwood3538]

Fascinating! I hope Prof. Daniel Mansfield hears of this interpretation and is able to confirm it!

[@3Iraq]

Thank you for mentioning the history of my country, Iraq ⚘

[@RedShiftedDollar]

Do one on highly composite/anti-prime numbers and their influence on the reducibility of fractions. I was always confused by why the Babylonians chose base 60, clocks have 12 hrs, 60 mins, etc. until I understood the significance of highly composite numbers. Intervals are so much easier to manage in base 60.

[@carloscalvimontesr.1601]

The Istanbul tablet is Si.427 and it is nothing more than simple surveying work with right angles and consequently the presence of right triangles can be inferred. But, for the unseen, a fuss is being made pretending it's a fabulous discovery.

[@BoomBoom-ym5oy]

shut up

[@dsm5d723]

God bless you, modeling the 3D mind of an Ancient is the hardest thing for binary insect brains to do. This IS that Hammurabi math, period specific. The inequity of the first Amorite Dynasty began with 26th century BCE inscriptions from Sumer/Mesopotamia, which called them Levantine nomads with no homes or agriculture. Abram met two Kings in the Valley, one was the occulted good governance God-King, the other Pharaoh, to a Near East Ancient who knew all of the stories. Only after the return from Babylonian Exile and the Book of Daniel do we get the scrolls and manuscripts of the OT. And the loop of the Amorites' first Dynasty being the city Abram was to model completes to my mind. The highest Priesthood, that of life, with Melchezidek praising God with bread and wine (NO guy in the sky, but ambient yeast, life in the air). Sodom's king wants power over persons, and gives their possessions as a second thought. Reputation is why Abram says keep all the things, not even a shoe buckle, lest you say behind me you made me a great nation. As great as Hammurabi's?

[@uncannyvalley2350]

Divide the world's Circumference by six and get 6,666. Divide the number of seconds in a day by 400 and get 216, 2160 is the length of an astrological age. The Moon is 400 times smaller than the sun, which is 864,000 miles wide, just like there are 86,400 seconds in a day. The planet orbits the Sun at 108,000 km/hr or 66,600 miles an hour. If we divide the angled side of the Pyramid (186.6 x 100 = 18,660) by 21.6 we get 863.8, plus .2 is 864. The base of the Pyramid minus the height is also Pi times 100, and Al Nitak follows Sirius past the King's Chamber in 100 minutes. If we divide the height by Euler's number we get the square root of Pi, times 2 is 354, the number of days in 12 lunar months. If we divide the diameter of the Sun by 6 we get 144,000 The hands of our 24 hr clock go around 60 times 60 times 10, which is 36,000, the number of Arc degrees in one second times 10, which means each second is one 360th of a circle, times 100. This shows that the Star of David was used as a kind of calculator to devise time and do complex equations using a hexadecimal system. The Egyptian number of perfection is 100, we divide 400 by 100 to get 4, we divide 600 by 100 we get 6. 4/6 is equal to 2/3 and 3/9, all of which have a ratio of 66.666666666, by which they can divide the Horizon down to seconds, and thus navigate the globe knowing both its dimensions and be able to make accurate maps. 86,400 ÷ 400 is 216 216 x 2 is 432 432 + 2160 is 6840 432 x 2 is 864 So rather than divide 864,000 by 2160, they divide 86,400 by 216, which is 400, rather than 40,000. This means a Megalithic Clock would go around 40 times, with each second broken down into tenths. 6 times 6 times 10; 3600. 400 ÷ 6 is 66.66666 These numbers all divide into each other. Half of 216 is 108, just as the Earth orbits the Sun at 108,000 km/hr. The interior angles of a regular Pentagon are also 108, and the interior angles of a Star of David add up to 1,440, times 100 is 144,000. Half of 108 is 54. It takes 360 Full Moons to span the night sky Horizon to Horizon, 720 total, 72 times 3 is 216. 6 x 6 x 6 x 4 = 864 Which means a full moon is equivalent to 300 seconds, or 5 minutes, meaning 2 Full moons per 10 minutes. This means seconds represent tenths of the Moon, a Minute (6 times 10 times 10) being 2 Full Moons or 1 degree of arc. Multiply the Moon's diameter by 18.6, the number of years in a Metonic Cyle, and get 40,175, the diameter of the Earth plus 100. 40,000 times 100 is 4,000,000, the Earth's circumference in meters. Multiply 18.6 by 2150 (actual diameter of the Earth) by 18.6 and you get 39,990, just 10 km short. This means they measured the Earth with the Orbit of the Moon, and based their metrics off of the Full Moon, cubing and squaring it to fund the relationships between the heavenly bodies. Half of the Pyramid's base equals one 86,400th of the Earth's Circumference. Dived the Base by the height and get Pi. The height of the Great Pyramid times 43200 equals the Polar Circumference of the Earth. Also the Circumference of the Base of the Great Pyramid times 43200 equals the Equatorial Circumference of the Earth. An equilateral Triangle formed within the face of the Great Pyramid is 6666 inches along each side, it represents one half of the Star of David, 720 degrees, as above so below, so we double it, 1440 ÷ 6 is 240, the number of hours on a clock times ten. 24 being 6 x 4, combining both ratios of of Sun and Moon, hence Solomon. The Pyramid itself is Squaring the Circle, by reducing the proportions of the cosmos to squares and roots based on Phi and Pi and Euler's number as a ratio to feet, and the Star of David is what allows then to do it, like a proto Antikythera mechanism. I can't say if they went to hundredths of a second, because I'm not even that much of a mathematician (majician) but they definitely did tenths, and it equates to the same nautical metrics we use today. Enoch also buries 36,525 scrolls, the number of days in a year, times 100. Oh by the way, this shows that our current measure of time is based on the principle of 1/6, the basis of an Egyptian Royal Cubit, but first they built the first ring at Stonehenge, which is 100 metres (330 ft) wide, with an area of 2160 square feet 😳

[@robharwood3538]

Prof. W., I just realized: 3 x 4 x 5 = 60! Could it be that the whole reason for base 60 was because of the convenience and simplicity (and scaling) of 345 triangles???

[@robharwood3538]

Probably not, but it's an intriguing thought anyway. 😅

[@WildEggmathematicscourses]

@Rob Harwood, Yes that is an intriguing idea, but likely to not fit with the historical record. We have a lot of evidence of metrological (measurement) systems long before the OB times in which 10's and 6's feature prominently, typically alternating. So I think it has maybe more to do with their measurements, of grain, area etc. But it is a really interesting question for sure!

[@brendawilliams8062]

427/432 96/16

[@SenorMorgenStern]

regardless of the base for a number system, wouldn't we eventually run into "periodic" expansion past the period point? So eventually, regardless of base form number system, we'll have to deal with repeating or non-repeating "past base point" expansion?

[@robharwood3538]

Yes, that's correct. I haven't watched the video yet, but from my past viewings of Prof. Wildberger's videos, and his objections to 'infinite decimals', my understanding is that he doesn't have any problem whatsoever with 'decimal' (or any other base) numbers which are used *as approximations.* His main objection is in regards to people using infinite decimals to 'pretend' that they are representing *exact* numbers, especially in the field of Pure Maths, as opposed to Applied Maths. For example, pretending that infinite decimals (or infinite base-60 digits) can represent, say, pi, or e, or some other transcendental number. For 'repeating decimals', you can augment a decimal system with notation for repeating parts, but there are two issues. The first, and most difficult is that the repeating part can be very long indeed, e.g. 1/23 needs 22 digits repeated, and in the worst case in general, the reciprocal of some prime p will need p-1 digits in its 'repeating decimal' form. So, for even medium sized primes, the number of digits grows really fast! The second issue, which really drives the point home, and shows why Wildberger's emphasis on Rational numbers makes good sense, is that the whole issue with repeating 'decimals' (or whatever base) can be more or less completely bypassed by simply using a rational number representation. Instead of needing 22 digits, you can just express 1/23 as "1/23" which is much much shorter. And even shorter still with bigger primes.

[@carloscalvimontesr.1601]

The Istanbul tablet is Si.427 and it is nothing more than simple surveying work with right angles and consequently the presence of right triangles can be inferred. But, for the unseen, a fuss is being made pretending it's a fabulous discovery.

[@whig01]

If we'd use base 420 we could divide by 7 too.

[@JoeKnapp614]

The multiplication table would be a bear.