History of Tamil people...
PART-7
"sumerian mathematics"
"Numbers and letters are the two discerning eyes, For all mankind to make the best of life"-Kural 392
Do you like mathematics? No matter what your answer may be, you are not alone.Mathematics is a challenging subject. Its basic concepts began to emerge when the world's very first civilization took root in Mesopotamia more than 5,000 years ago.Today we use numbers for showing prices,telling time,marking addresses,using the telephone,identifying cars and knowing the different players of a sports team.Although today most of the world uses the same decimal number system with Arabic numerals,there are several other numbers systems that have been used over the past hundreds of years.The Sumerians were one of the first to use numerals or signs to represent numbers.six signs were created.With closer inspection,it can be seen that there are really only two symbols[Wedge & circle],but they are combined and resized to make the other signs
The number system was both decimal or base 10, and sexgesimal or base 60. To create numbers other than the six shown above, the symbols would be combined.For example, to create the number 73, one large wedge[60], one small circle[10], and three small wedges [1+1+1] would be used.The largest unit always appeared on the left .
This Sumerian's number system used the main base 60 and the auxiliary base 10.It was passed down to the ancient Babylonians, and it is still used — in a modified form — for measuring time, angles, and geographic coordinates.For example, have you ever wondered why an hour has 60 minutes and a minute has 60 seconds? Have you ever thought about why a full circle has 360 degrees? As a matter of fact, sumerian number system still plays a critical role in our everyday life.
Given our own deci-centric society this choice of 60 may at first seem very strange, but it was actually extremely natural and functional. Because 60 has so many factors, the Sumerians were able to handle many division problems with ease: fractions of one unit of measure were often whole amounts of another. It's exactly why we still divide an hour into 60 minutes, a circle into 360 degrees and so on.The number 60, a superior highly composite number, has twelve factors, namely {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}, of which 2, 3, and 5 are prime numbers. With so many factors, many fractions involving sexagesimal numbers are simplified.60 is the smallest number that is divisible by every number from 1 to 6;
Later,At some point, around 2000BC, Mesopotamian people adopted this system but modified it so that it became positional (like ours).When we write numbers,in our decimal system the place of each symbol matters a lot.For example,In 278, the 2 is worth 200, the 7 is worth 70, and only the 8 is worth just that, 8.Here the place value is based on the number ten.So in 278, the 2 is worth 2 x 10 x 10 , and the 7 is worth 7x 10 & 8 is worth just 8.So, for every position a digit moves to the left, it is increased by a power of 10.This way of notation is for the Arabic numerals. But since both the Sumerians and the Babylonians used a sexagesimal system,as based on sixty,like our seconds, minutes and hours, each of their digits would be increased by a power of 60 as it moved along to the left. For example,4892 in decimal system is worth in sumerian system as:
Sixty lots of 60x60 Sixty lots of 60 Sixties Units
(60x60x60) (60x60) (60) (1)
x 60 x 60 10 x 60 23
= 3600 600 23
= 4223
This reduced the system to only two distinct symbols,the wine glass-like symbol[] is the sumerian number 1, and the horizontal A symbol []is the sumerian number 10 and the position a sign occur within a number changes its quanity, just like "1" in the number "100" is different from the "1" in the number "10,000" in our modern system.This units digits were in base ten (Y, YY, YYY, YYYY, ... YYYYYYYYY) & tens digits were in base six (<, <<, <<<, <<<<, <<<<<) meaning (10, 20, 30, 40, 50)these unit symbol () and a ten symbol () which were combined in a similar way to the familiar system of Roman< numerals (e.g. 23 would be shown as ). Thus, represents 60 plus 23, or 83.The Mesopotamian people used this new positional notation in exactly the same manner in the decimal system.So, they had a ones place, a 60s place, a 60 x 60 = 3600s place, and so on.For example,Just as, (in the decimal system), 2 can be 2 or 20 or 200, depending on the digits place, so could a Sumerian 2 mean 2, or 120 (2 x 60), and so on, depending on the place.The 360 degree circle, the foot and its 12 inches, and the "dozen" as a unit, are but a few examples of the vestiges of Sumerian Mathematics, still evident in our daily lives...
"He is also numbers and letters So coveted by those of the sea-girt earth,He is The nature (of all things)"-Thirunavukkarasar;
In above sumerian system, to represent "59" we still requires five tens and nine ones.But other civilizations, for example the Maya of Central America and the Romans, were more adapt at combining a main base and auxiliary base in ways that avoided the repetition of symbols beyond five.Roman numbers are much easier to read than Sumerian numbers because they use the decimal system familiar to us, and because they are not written in pictorial form but use letters of the alphabet. The well-known Roman numerals are
A Roman number never requires the repeat of more than 4 identical symbols, which makes reading Roman numbers relatively easy.
[ஆந்திரா மாநில, தொண்டுர் பகுதியில் கி பி 300 ஆண்டை சேர்ந்த "குகை தமிழ் பிராமி கல்வெட்டு" ஒன்று இது.இங்கு இலக்கம் 3,இரண்டு வரி கல்வெட்டின் இறுதியில் மூன்று கிடைமட்டமாக ஒன்றன் மேல் ஒன்று அடுக்கிய இணை கோடுகளால் குறிக்கப்பட்டு உள்ளது./Tamil-Brahmi cave inscription from Tondur(3rd century C.E.) Here The numeral 3 engraved at the end of a short two-line inscription in the cave is represented by three horizontal parallel lines.The inscription records that the village of Agalur gifted three stone beds in the cave chiselled by Mosi]
[இராமேஸ்வரத்திற்கு அருகில் உள்ள அழகன்குளம் பகுதியில் கண்டு எடுக்கப்பட்ட கி பி 100/200 ஆண்டை சேர்ந்த மட்பாண்ட துண்டு இது.ஐராவதம் மகாதேவன் அங்கு குறிக்கப்பட்ட எண்ணை 408 என அடையாளம் காண்கிறார். என்றாலும் என்.கணேசன் இதை 804 என்கிறார்./A well-preserved pottery inscription from Alagankulam near Rameswaram.The inscription is dated to the 1st or 2nd century C.E. IRAVATHAM MAHADEVAN said the number is read 408,The first digit at right looking like the cross is the symbol for 4.It is followed by the symbol for 100 (resembling the Brahmi letter sa) and the last symbol at left is 8, incised in reversed direction.However N.Ganesan stressed it is 804,]
It is easy to see the origins of a decimal (base 10) number system. Our hands have 10 digits [fingers] to count on, so a decimal system follows naturally.With the addition of the toes on our feet a vigesimal (base 20) number system, like that of the Maya, also makes sense. But understanding a sexagesimal (base 60) number system, as used by the Sumerians, takes a little more thought .By using the thumb as a pointer, and marking off the each finger bone[the distal phalanx, middle phalanx and proximal phalanx] on the four fingers in turn, we can count up to 12 on one hand,For example,Each of your fingers except thumb has three distinct segments. and, touching now the middle segment of your right index finger with your right thumb,means: “Two.”Now, similarly count all segments in your right hand fingers with right hand thumb and having successfully counted to 12, make a thumbs-up sign with your left hand.As in . . . “that’s one set of 12.” Count another set of twelve with your right hand and you earn an unfolded left index finger . “That’s two sets of 12.”Keep doing this until you have unfolded all five fingers of your left hand, and you’ve got 60!
When the Sumerians first came up with their numerals, they did not have a specific symbol for zero and no concept of 0 as a number, which created ambiguities in positional notation.Neither the Sumerians nor other people in Mesopotamia (most notably, the Babylonians) were able to come up with a solution at the time. This issue would remain unsolved until around 500 A.D. when the Indians developed the Arabic numerals that we are still using today.As such the sumerian number 60 was represented by the same symbol as the number 1 & the actual place value of a symbol often had to be inferred from the context!
"By My eyes have lost their brightness, sight is dimmed; my fingers worn,
With nothing on the wall the days since I was left forlorn."-Thirukkural 1261
Which says:My finger has worn away by marking (on the wall) the days he has been absent while my eyes have lost their lustre and begin to fail.So there must be a number system/numerals during sangam period.Further,The Ancient Poetry (anthology) ” Paripadal” of the Sangam period reveals to us the truth where numbers should start.Paripaadal (பரிபாடல்) is mentioned as the 5th group of verses in the Sangam compilation of "ettu-th-thogai" (எட்டுத் -தொகை ) and is possibly the oldest religious composition in Tamil.That is,Tamils learnt the art of counting (numerals) first. This they called mathematics (Kanakku). They gave figures to indicate it. This mathematics had only a few figures. But alphabetic has more letters, so, they was called it lengthy mathematics. (Nedung Kanakku)
Paazh (Zero/ go to ruin ,land waste by drought . dead-loss/ஒன்றுமில்லா ஊழி/பாழ் என்றால் எதுவுமற்ற வெறுமை என அர்த்தம்.) and Kaal (Quarter) and Paagu (half) and oneru (one) and Erandu (Two) and Moonru (Three) And Nangu (Four) and inthu (Five) and aaru (Six) and Ezhu (seven)and Ettu (Eight) and Thonndu (Nine) ……(PARIPADAL 3 ; 77-80).
Also we do have ancient literary references specifically mentioning eN (‘numeral') as distinguished from ezhuttu [‘letter of the alphabet,' for example, Tolkappiyam 655.4, Tirukkural 392.]...
"எண்ணாகி யெண்ணுக்கோ ரெழுத்து மாகி.."/"சொல்லாகிச் சொல்லுக்கோர் பொருளு மாகிச்"-திருநாவுக்கரசர
"Maggie ennaki yennukko reluttu .." / "collakic collukkor text makic" tirunavukkaracar
We have "300" refers to the fact that we apply the two markers. "3" for a pair of "0" is.
But we do not have a code of Sumerian model puccciyatti what happens? Their "60" [Babylonian 1.svg] i, "1" [Babylonian 1.svg] elutuvarkalate model like the "10" [Babylonian 10.svg], "10" or "10 x 60" ... this perumappati its value if it is a big problem, why did not they? தெரிகிறது.சந்தர்ப்பம் situations and circumstances may have the opportunity to help them because we assume that the numbers on the signs will be located. The presentation will give an example below.
"As a poor fisherman" Y "only had boats. Avanirku If everything is correct, he one day" << "fish pitippanoru month he almost" "minkalai pitittaninta his family during the" eat "the fish he needed patukiratumikutiyai He sold it to others. "We can now offer the potential below for details.
Boats: "Y" or a 1 to 1 x 60 or 1 x 60 x 60 up to ... something like this kurikkuman minavanakave he was a poor man can not keep the 60 boats or more. The possible answer is that he had only one boat
The amount of the catch of the day: "<<" is 20 or 20 x 60 or 20 x 60 x 60, ....... kurikkumavan something like this is only a small boat, or fish in the 20 x 60 = 1200 And you can not be stored around for almost 20 fish per day, only to catch him.
In a month, the amount of the catch for a 30-day month is typically கொண்டிருக்கும்.ஆகவே he nearly 600 [= 20 x 30] catch fish. "" "Somewhat of a 9 x 60 + 50 = 590 kurikkumatu the cariyeallatu is 9 x 60 x 60 + 50 x 60 =, .... one of the biggest tokaiakave he is devoid of practicality kurikkumitu 590 a month is the favorite fish.
His family is the amount of fish eaten in a month, "" is a 5, which is the lowest amount or 5 x 60 = 300, which is acceptable to the possible amount, or 5 x 60 x 60 = 18,000, is much greater amounts. So he is in the family 300 a month cappattatu.
The amount of fish sold in one month: the fisherman's boat has 1 a day for a total of 590 fish in his nearly 20 fish pitittanoru pitittanat 300 fish for their dinner for his family, he is using his vittatuakave 590-300 = 290 fish a day selling ullanata was 9 or 10 fish.
We have "the numbers related to the" [positional notation], not just the big numbers, small numbers signify. Ie 1/4 [quarter], 0.25 to 2 to 5, in response to the hundredths place, we can write anywhere. In an analogous manner to the people of Mesopotamia 1/60, 1/3600, introduced ..... the place. The number 60 has many factors in smaller numbers than us, simply stated. For example, the fraction is 1/4 = 15/60 to 1/60 at 15, not only as the people of Mesopotamia காட்டினார்கள்.இந்த take advantage of the system can identify most of our modern times, for example, to show the ailing 1/3 etukkalamnam the while, they about it Nothing to be worried very easily 1/60 where 20 is particularly striking ass.
Under the head of the table shown above, which is 60 feet to take advantage of the system, which explains to me that you have a complete lack of medicine, or 1 / 7th the number of head to head முடியாது.இப்படியான fully describe the numbers, for example, 1/7, 1/11, 1/13, etc. without considering them, they abandoned the agenda were published within a few vanai தயாரித்தார்கள்.இதன் their card.
Cumeriyar fraction upside-number multiples of கையாண்டார்கள்.உதாரணம் 19 to 12, by dividing the replacement of 19 to 12 under the head of the பெருக்கினார்கள்.அதா the a / b c instead of a / b = ax (1 / b) as karutinarkalitan above them and the efficient, upside-down table தயாரித்தார்கள்.அது not alone in their classes, kanankal the class source, kanamulat the tables produced with the kettira mathematically trained, splitting puzzles, with the tablets, BC for the 2600 year வைத்திருந்ததார்கள்.என்றாலும் their kettira math practice for some time, error and இருந்தன.இயூபிரட்டீசு River area CE was found in 1854 BC and 2000 year-classes of the two lenses and the number 32 to number 59 to number kotukkiratuata to honor கொடுக்கின்றன.இந்த Table 82 = 1,4 = (1 x 60) + 4 = 64, and so .59² = 58, 1 = (58 x 60) +1 = 3481 kotukkiratuinta mathematics to boost the numbers of the two different ab = [(a + b) 2 - a2 - b2] / 2 bus used in the equalizer. Their classes through the table because we know the time for them to be easily and iruntatuatan the CE to 1800 from 1600 papiloniyan lenses and many mathematical issues கொண்டிருந்தன.உதாரணம் the algebra, one step [linear equation], the two-step, some cubic equations solving methods was familiar. Iruntanarinnum know the size of the circle is a papiloniyan [YBC 7289] √2 served for the match to 5 decimal places, giving the approximate value of π ullatuinnum has a 3 1/8 (3.125] is rated at. Π is the true value of 3.1416.
The cycle of the ancient Tamil book and some of the properties of the light does காக்கைப்பாடினியம் tolkappiyar காக்கைப்பாடினியார் earlier period. (.711 BC is heavily Tolkappiyar time.) It's a wonderful math book is written in verse form பட்டுள்ளது.ஏரம்பம் disappeared and now it is the oldest accounting nulenrum pavanar tevaneyap said. Karinayanarum accounting books are written beside it patiniyarum crows.
"Seven to vittamor
The four, along with tikaivara
Quickly replicate Chain
Tikaippana curruttane "kakkaip patiniyam
Accordingly, when "V" as a given,
Tikaivara = "V" is the
Seven vittamor done = V / 7
Four inclusion = V +, 4 * (V / 7) is
Quickly replicate Chain = (2 (V +, (4 V / 7) is.
Accordingly, if (2 * ((11 V) / 7) = 22/7 * v. The circumference of the circle is now given πD).
There is, therefore, π is 22/7. We apply this estimate is less precise perumanameinrum குறிப்பிடத்தக்கது.அது It is only today that we use the formula for the circumference = 2 π r several hundred years ago, our ancestors were aware of the fact that we know that can boast!
Kanakkatikaram the periphery of the periphery of the ancient text விளக்குகின்றது.இதிலும் vattati cycle in the form of verse is saying.
Kanakkatikarap Song: 50
"Having matanai accelerate yirattittu
In mattunan mariye Madhavan - in ettatan
Erriye ceppiyati lerum vattattalavum
Torrumep punkoti Tell You "
Accordingly,
Vittamtanai accelerate yirattittu = twice the diameter = 2r + 2r = 4r (diameter = 2r); Madhavan modular chord sweep by the mariye = 4; Multiply by the ettatan erriye = 8; Ceppiyati = divide by 20.
Circle's circumference = (4r x 4 x 8) / 20 = 32/5 r = 2 (16/5) r = 2 π r
Where π = 16/5 = 32ituvum perumaname a rather accurate!....
[Babylonian Yale tablet YBC 7289 showing the sexagesimal number1;24,51,10 approximating √2]
Thus,the square root of 2, the length of the diagonal of a unit square,was approximated by the Babylonians of the Old Babylonian Period (1900 BC – 1650 BC) as
It has on it a diagram of a square with 30 on one side, the diagonals are drawn in and near the centre is written 1,24,51,10 and 42,25,35. These numbers are written in sumerian/Babylonian numerals to base 60.Now the sumerian/Babylonian numbers are always ambiguous and no indication occurs as to where the integer part ends and the fractional part begins. Assuming that the first number is 1; 24,51,10 then converting this to a decimal gives 1.414212963 while √2 = 1.414213562. Calculating 30 × [ 1;24,51,10 ] gives 42;25,35 which is the second number. The diagonal of a square of side 30 is found by multiplying 30 by the approximation to √2.
[Babylonian tablet numbered 322[Plimpton 322]]
The tablet numbered 322 has four columns with 15 rows. The last column is the simplest to understand for it gives the row number and so contains 1, 2, 3, ... , 15. In every row the square of the number c in column 3 minus the square of the number b in column 2 is a perfect square, say h. c2 - b2 = h2, So the table is a list of Pythagorean integer triples.
Have you ever studied Pythogaras theorem? It was discovered by Pythogoras who lived during 569–475 BC. Thats what our books and we say...Then, what about this below poem by a Tamil Poet called Pothayanar ?The highlight of this formula is that it gives a method to calculate hypotenuse using linear equation instead of the non-linear one given by Pythagoras! ie This method employs finding hypotneuse without using square root!!!
Hypotenuse = 7/8 * Longer Side + 1/2 * Shorter side.Say,Hypotenuse = C,Length = L,wide = W, Then C = 7/8 x L + 1/2 x W
ஓடும் நீளம் தனை ஒரேஎட்டுக்
கூறு ஆக்கி கூறிலே ஒன்றைத்
தள்ளி குன்றத்தில் பாதியாய்ச் சேர்த்தால்
வருவது கர்ணம் தானே.
This poem is said to be written by Tamil poet bothayanar.
Explanation:seven eigth of largest side[a] added with half of smallest side[b] will give Hypotenuse[c]
ie The formula is hyp = (adj - (adj/8)) + (opp/2),where the adjacent side is always longer than the opposite side [You may get approximate answer with out application of squre root) An easy method for pythagorous theorem by bothaiyanaar:
c=(a-a/8)+(b/ 2),where a>b. b>a is not possible. If possible that object may not stand very longer period.(it is against the nature.]or we may say:largest value be deducted by its 1/8 and half of the smallest value added to it.
Please note it is not always correct.It works for 3:4:5 (a=4,b=3),[6:8:10] and 5:12:13 (a=12,b=5)... but, not for 9:40:41... So, it can not be generalised as a theorem .There are so many restrictions,but may give approximate answer.The issue is not about the correctness of this formula... The issue is that he(Bothaiyanar) was atleast tried to establish a formula!!.Though this applies for restricted usage,It does not take away any credit from this great man!!!!
As we shall see above,Mesopotomian mathematics is quite impressive.However,like the ancient Egyptians,the mesopotomians never gave what we would call "PROOFS" for their results.The first people to do so were the Greeks.
Any way,With their advanced knowledge in numerals, people in Mesopotamia were excellent mathematicians. When applied to their daily life, they developed formulas to calculate weights, areas, volumes, and wages. Students from that time needed to study mathematics at school, too. They had to learn how to do addition, subtraction, multiplication, division, and fractions. During the reign of Hammurabi (1792 B.C. - 1750 B.C.) of the 1st dynasty of Babylon, there were even specific laws addressing issues such as interests and loans. Because of those codified rules, we know that people in Mesopotamia were the ones who established the world's first banking system. Without mastering mathematics, that would be entirely impossible!....
Our oldest written records come from the civilization of Sumer, which arose in around the Tigris and Euphrates Rivers in what is now southern Iraq. Sumerian cuneiform documents dating as far back as 3100 BC have been found and a flourishing literature developed, which reached its peak in the centuries around 2000 BC. This literature contains a large collection of love songs, most of them incorporated to annual fertility festivals. What were intense expressions of love between an individual man and a woman became wrapped into the larger context of the ceremonial union of male[Dumuz] and female[Inanna] fertility gods, a union considered essential to gaining flourishing crops and bountiful livestock.
The male god Dumuz appears to have originally been Dumuzi, a mythical Sumerian king of Erech[ Erech, Sumerian Uruk] who reigned sometime in the third millenium after Lugulbanda. In later times, the ceremony for ensuring the return of spring and successful farming required the local king to assume the role of Dumuz and a cultic priestess to take the role of Inanna. The sexual union of the two was the climax of climaxes in a city-wide celebration of several days at the New Year.The king’s performance in this Sumerian festival was essential for the well-being of an ancient community. The king should be demonstrably strong and virile, because he was the ceremonial link with the gods and the good harvests they alone could ensure.
Sexual love has been primarily responsible for the perpetuation of the human race,but it has also been responsible for the creation of literature-more so than any category of love except ,perhaps,the love of god.Hence the annual rite on new year day at sumeria developed two sets of most popular literature/poems ,namely -A long work celebrates the courtship & wedding of the sex goddess Inanna & her shepherd-lover Dumuzi,another,the mourning that attended his death.Elsewhere,lyrics lavishly praise the handsomeness of King Shu-Sin of Ur & the dutifulness of his queen.These two set of poems may indeed be related for an annual fertility festivals.
Such Love songs or poems are probably found in every culture.Also Sangam poems shows a regular alternation of male and female speakers as in ’courtship of Inanna and Dumuzi’.Similarly, the convention of lovers addressing each other as brother and sister appears here as normal terms of endearment. The similarities between these poems and those from southeast Asia and India suggests that songs of this type may have been part of the oral culture shared via trade routes between the regions in the second millenium BC.
We are giving below few such songs.Already one such songs,believed by archaeologists to be the oldest love poem found to date, with Tamil translation was given in part:15-"Man of my heart, my beloved man, your allure is a sweet thing, as sweet as honey."
It was understand that the premarital courting and wooing of Inanna by Dumuzi which became a favorite subject of the Sumerian poets and bards. One of the most charming of these consists of a two-column tablet now in the Hilprecht collection of the University of Jena in East Germany, which may not inaptly be entitled “Love Finds A Way” or “Fooling Mother.” Its two main characters are the goddess Inanna “Queen of Heaven,” the Sumerian Venus, and Dumuzi (known also by names Kulianna, Amaushumgalanna, and Kulienlil), her mortal sweetheart and husband-to-be. The first stanza begins with a soliloquy by Inanna who says:
"Last night as I, the Queen, was shining bright,
Last night as I, the Queen of Heaven, was shining bright,
As I was shining bright, was dancing about,
As I was singing away while the bright light conquered the night,
He met me, he met me,
The lord Kulianna met me,
The lord put his hand into my hand,
Amaushumgalanna embraced me."
Then follows an amorous tete-a-tete between the two lovers with Inanna pleading:
"Come now, set me free, I must go home,
Kulienlil, set me free, I must go home,
What can I say to deceive my mother,
What can I say to deceive my mother Ningal?"
But this does not stop Dumuzi who has a ready answer:
"I will tell you, I will tell you,
Inanna, most beautiful of women, I will tell you.
(Say) ‘My girl friend took me with her to the public square,
There a player entertained us with dances,
His chant, the sweet, he sang for us.’
Thus deceitfully stand up to your mother,
While we by the moonlight take our fill of love;
I will prepare for you a bed pure, sweet, and noble,
The sweet day will bring you joyful fulfillment."
We also find a sangam poem,Kaliththokai 51, where the title “Love Finds A Way” or “Fooling Mother.” fit as well & its also in the form of conversation between two.
"Listen my bright bangled friend! That wild brat
who used to kick our little sand houses
that we built, with his leg,
pull our flower strands from our hair,
and yank the striped ball from us,
and caused us agony,
came one day when mother and I were at home.
“O, people of this house,
please give me some water to drink” he said.
Mother said to me,
“Pour the water in the thick gold vessel,
and give it to him to drink,
my daughter with bright jewels”.
And so I went unknowingly,
that he had come there.
He seized my bangled arm, pressed it, and scared me.
“Mother, look at what he has done”, I shouted.
Mother came with a distress cry,
I said to her,
“he had hiccups while drinking water”.
Mother stroked his back gently,
and asked him to drink slowly.
He looked at me through the corners of his eyes,
smiled, and gave me killer looks,
And what a pleasurable union it was – that thief."....
WILL FOLLOW... PART-8...
யாழறிவன்... Yalarivan Jackson Jackie
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