Category Archives: Mathematics

How Did the Present-day Numerals Derive Their Form?




BT.V. Antony Raj


Numbers (Source:

Numbers (Source:


A number in mathematics is an object used to count and measure. 1, 2, 3, and so forth are examples of natural numbers. In common usage, the term number may refer to a symbol, a word, or a mathematical abstraction.

The English names for the cardinal numbers were derived ultimately from Proto-Indo-European (PIE) language, the supposed proto-language that existed anywhere between 4000 and 8000 years ago. PIE was the first proposed proto-language to be widely accepted by linguists. With time, the pronunciation shifted and changed.


Numeral Modern English Old English Proto-Germanic Proto-Indo-Germanic
1 one an ainaz oi-no
(originally meaning one, unique)originally meaning one, unique)
2 two twa twai duwo
3 three þ reoreoreoreo
(þ  here is the orthography for “th” as in “thing”)
thrijiz tris-
4 four feower petwor Kwetwer
5 five fif fimfe Penkwe-
6 six siex sekhs seks
7 seven seofon Sebum septm
8 eight eahta or æhta akhto Okto(u)-
9 nine nigen (the /g/ here is pronounced lije the y in “young”. Petwor- newn
10 ten ten tekhan dekm


A numeral is a notational symbol that represents a number. We use the Hindu-Arabic numerals 0 to 9 every day. But how did these Hindu-Arabic numerals derive their form? It is a puzzle to me.

Some folk etymologies have argued that the original forms of these symbols indicated their value through the number of angles they contained, but no evidence exists of any such origin.

Recently I came across a statement that elaborated on the folk etymologies. It said:

“Numbers were named after the number of angles they represented, and each angle represented a quantity. For example, the number one has one angle, number two has two angles and so on. They have to be written with straight lines (not curved).”

I found the following image on Facebook.


Numerals and angles
Numerals and angles


Many have copied and propagated this image – the concept of angles associated with numbers. One can find them on Facebook and on many websites, explaining that this is how the numerals obtained their values.

But this claim seems to be spurious like many other urban legends. For example, 0 (zero) would have four angles if it is written with straight lines like the other numerals. So, here lies the fallacy.

So, I am still in a quandary.

Are there any authentic, rational explanation for how the present form of the Hindu-Arabic numerals we use today was derived?




Mystery of Malaysia Airlines Flight MH370 – Inmarsat’s Satellite Data


By T. V. Antony Raj

Wild ride of MH370 (Source:
Wild ride of MH370 (Source:


The search for the missing Malaysia Airlines Flight MH370 is now on in a section of the southern Indian Ocean known as the “Roaring Forties” where strong westerly winds generally blow between latitude 40° and 50°. The strong west-to-east air currents are induced by the combination of the Earth’s rotation and air being displaced from the Equator towards the South Pole, with just a few landmasses to act as windbreaks. The area is characterized by cold fronts that sweep east every four to five days, causing  13 to 30 feet (4 to 9 meters) pounding waves that churn the icy sea.

International Mobile Satellite Organization (Inmarsat) is a British satellite telecommunications company, offering global, mobile services. Inmarsat started playing an import role immediately after Malaysian Airlines Flight MH370 disappeared.

One of Inmarsat’s satellites continued to pick up a series of automated hourly ‘pings’ from the missing aircraft which would normally be used to synchronize timing information even after the Aircraft Communications Addressing and Reporting System (ACARS), which would usually transmit the plane’s position, was switched off, suggesting the plane flew to the Indian Ocean.


How Inmarsat tracked down Flight MH370 (Source:
How Inmarsat tracked down Flight MH370 (Source:


By analyzing these pings, Inmarsat established that the aircraft continued to fly for at least five hours after the aircraft left Malaysian airspace and that it had flown along one of two ‘corridors’ – one arcing north and the other south. The plane was reportedly flying at a cruising height above 30,000 feet. See my article “Mystery of Malaysia Airlines Flight MH370 – If Hijacked, Where Did It Go?

Using complex mathematical processes, Inmarsat’s engineers analyzed the tiny shifts in the frequency of the pings from the missing aircraft and came up with a detailed Doppler effect model for the northern and southern paths and inferred the aircraft’s likely final location though their method had never been used before to investigate an air disaster.

Chris McLaughlin, senior vice-president of external affairs at Inmarsat said:

“We looked at the Doppler effect, which is the change in frequency due to the movement of a satellite in its orbit. What that then gave us was a predicted path for the northerly route and a predicted path the southerly route…

That’s never been done before; our engineers came up with it as a unique contribution… By yesterday they were able to definitively say that the plane had undoubtedly taken the southern route…

We worked out where the last ping was, and we knew that the plane must have run out of fuel before the next automated ping, but we didn’t know what speed the aircraft was flying at – we assumed about 450 knots. We can’t know when the fuel actually ran out, we can’t know whether the plane plunged or glided, and we can’t know whether the plane at the end of the time in the air was flying more slowly because it was on fumes.”


Pings to Inmarsat (video grab from Wall Street Journal)
Pings to Inmarsat (video grab from Wall Street Journal)


According to the Wall Street Journal, Inmarsat relayed their findings to the Malaysian officials and the British security and air-safety officials on March 12, 2014. But the Malaysian government concerned about corroborating the data and dealing with internal disagreements about how much information to release did not publicly acknowledge Inmarsat’s information until four days later. On Saturday, March 15, 2014, during a news conference, Malaysian Prime Minister Najib Razak accepted for the first time that deliberate actions were involved in the disappearance of the aircraft. He said:

“Based on new satellite information, we can say with a high degree of certainty that the Aircraft Communications Addressing and Reporting System (ACARS) was disabled just before the aircraft reached the east coast of Peninsular Malaysia. Shortly afterwards, near the border between Malaysian and Vietnamese air traffic control, the aircraft’s transponder was switched off.”

 He added that the search effort was redirected from that day to focus on the areas the Inmarsat information described:

“From this point onwards, the Royal Malaysian Air Force primary radar showed that an aircraft which was believed – but not confirmed – to be MH370 did indeed turn back. It then flew in a westerly direction back over Peninsular Malaysia before turning north-west. Up until the point at which it left military primary radar coverage, these movements are consistent with deliberate action by someone on the plane.

Today, based on raw satellite data that was obtained from the satellite data service provider, we can confirm that the aircraft shown in the primary radar data was flight MH370. After much forensic work and deliberation, the F.A.A., N.T.S.B., A.A.I.B. and the Malaysian authorities, working separately on the same data, concur.

According to the new data, the last confirmed communication between the plane and the satellite was at 8:11 a.m. Malaysian time on Saturday 8th March. The investigations team is making further calculations which will indicate how far the aircraft may have flown after this last point of contact. This will help us to refine the search.

Due to the type of satellite data, we are unable to confirm the precise location of the plane when it last made contact with the satellite.

However, based on this new data, the aviation authorities of Malaysia and their international counterparts have determined that the plane’s last communication with the satellite was in one of two possible corridors: a northern corridor stretching approximately from the border of Kazakhstan and Turkmenistan to northern Thailand, or a southern corridor stretching approximately from Indonesia to the southern Indian Ocean. The investigation team is working to further refine the information.

In view of this latest development, the Malaysian authorities have refocused their investigation into the crew and passengers on board. Despite media reports that the plane was hijacked, I wish to be very clear: we are still investigating all possibilities as to what caused MH370 to deviate from its original flight path.

This new satellite information has a significant impact on the nature and scope of the search operation. We are ending our operations in the South China Sea and reassessing the redeployment of our assets. We are working with the relevant countries to request all information relevant to the search, including radar data.

As the two new corridors involve many countries, the relevant foreign embassies have been invited to a briefing on the new information today by the Malaysian Foreign Ministry and the technical experts. I have also instructed the Foreign Ministry to provide a full briefing to foreign governments which had passengers on the plane. This morning, Malaysia Airlines has been informing the families of the passengers and crew of these new developments.”

On March 18, 2014, Australia and the US National Transportation Safety Board narrowed down the search area to just three per cent of the southern corridor by taking into consideration Inmarsat’s inference from the satellite pings, along with assumptions about the plane’s speed.

On Monday, March 24, 2014, Prime Minister Najib Razak said that according to Inmarsat the aircraft flew along the southern corridor and ended its journey in the middle of the southern Indian Ocean. He said:

“Based on new analysis… MH370 flew along the southern corridor and that its last position was in the middle of the Indian Ocean west of Perth… It is therefore, with deep sadness and regret, that I must inform you that according to this new data that flight MH370 ended in the southern Indian Ocean.”

On the same day, Australian and Chinese search planes separately spotted a few objects in the southern Indian Ocean and alleged they were possible debris from the missing aircraft and reported the coordinates to the Australian Maritime Safety Authority (AMSA), which is coordinating the multinational search, and also to the Chinese icebreaker Snow Dragon, which is en route to the area. Half a dozen other Chinese ships along with 20 fishing vessels have been ordered to move toward the search zone.

Australian Prime Minister Tony Abbott said the crew of an Australian P3 Orion plane had located and two objects in the search zone, but it was unclear if they were part of an aircraft. He said the first object was grey or green and circular, the second orange and rectangular. The crew was able to photograph the objects.


Search suspended ... this satellite image shows severe tropical cyclone Gillian off the Western Australian coast. Credit: Bureau of Meteorology
Search suspended this satellite image shows severe tropical cyclone Gillian off the Western Australian coast. Credit: Bureau of Meteorology


An Australian Navy supply ship, the HMAS Success, was on the scene on Monday trying to locate and retrieve the objects. However, according to AMSA, due to rough seas, the vessel left the search area early Tuesday morning since conducting the search in such conditions would be hazardous and pose a risk to crews.AMSA said the vessel is now in transit south of the search area until the sea calms down and if weather conditions permit the search would be resumed tomorrow, otherwise, if weather conditions continue to deteriorate it could be several days before the search is resumed.

Meanwhile, the United States prepared to move into the region a special device that can locate black boxes.



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Unicode Technical Note #21: Tamil Numbers


By T. V. Antony Raj

Recently, I found the Unicode Technical Notes that provide information on a variety of topics related to Unicode and Internationalization technologies. The website stresses that these technical notes are independent publications, not approved by any of the Unicode Technical Committees, nor are they part of the Unicode Standard or any other Unicode specification and publication and does not imply endorsement by the Unicode Consortium in any way. These documents are not subject to the Unicode Patent Policy nor updated regularly.

Being a Tamil, Unicode Technical Note (UTN) #21: Tamil Numbers by Michael Kaplan, fascinated and impressed me.

Originally, Tamils did not use zero, nor did they use positional digits. They have separate symbols for the numbers 10, 100 and 1000. Symbols similar to other Tamil letters, with some minor changes. For example, the number 3782 not written as ௩௭௮௨ as in modern usage but as ௩ ௲ ௭ ௱ ௮ ௰ ௨.

This would be read as they are written as Three Thousands, Seven Hundreds, Eight Tens, Two; and in Tamil as மூன்று-ஆயிரத்து-எழு-நூற்று-எண்-பத்து-இரண்டு.

௧ = 1
௨ = 2
௩ = 3
௪ = 4
௫ = 5
௬ = 6
௭ = 7
௮ = 8
௯ = 9
௰ = 10
௰௧ = 11
௰௨ = 12
௰௩ = 13
௰௪ = 14
௰௫ = 15
௰௬ = 16
௰௭ = 17
௰௮ = 18
௰௯ = 19
௨௰ = 20
௱ = 100
௨௱ = 200
௩௱ = 300
௱௫௰௬ = 156
௲ = 1000
௲௧ = 1001
௲௪௰ = 1040
௮௲ = 8000
௰௲ = 10,000
௭௰௲ = 70,000
௯௰௲ = 90,000
௱௲ = 100,000 (lakh)
௮௱௲ = 800,000
௰௱௲ = 1,000,000 (10 lakhs)
௯௰௱௲ = 9,000,000
௱௱௲ = 10,000,000 (crore)
௰௱௱௲ = 100,000,000 (10 crore)
௱௱௱௲ = 1,000,000,000 (100 crore)
௲௱௱௲ = 10,000,000,000 (thousand crore)
௰௲௱௱௲ = 100,000,000,000 (10 thousand crore)
௱௲௱௱௲ = 1,000,000,000,000 (lakh crore)
௱௱௲௱௱௲ = 100,000,000,000,000 (crore crore)


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Rapidity of Exponential Growth

Myself By T.V. Antony Raj


A few days ago, during a discussion, a friend of mine told us that the damage caused by a nuclear catastrophe would be exponential. What does the term ‘exponential’ mean? How could we show the rapidity of exponential growth?  Usual growth is just a few percentage points but here is a demonstration of how rapidly exponential growth would be.

For this experiment, take a sheet of ordinary  letter size paper (A4). A sheet of paper weighing 80 gm/sq.m is about 0.1 mm thick.

Fold this paper into half. Fold it a second time, and a third time. The paper will now be  as thick as our finger nail.

At the 7th fold it will be as thick as a notebook and it would be barely possible to fold further.

If we were to fold the paper 10 times, it would be as thick as the width of our hand.

Unfortunately, it would not be possible to fold more than 12 times.

Hypothetically, if it was possible to fold further …

At seventeen folds it would be taller than an average house.

Three more folds and that sheet of paper is a quarter height of the Sears Tower (a skyscraper renamed as Willis Tower in 2009 is 1,729 feet from Franklin Street Entrance,  in Chicago, Illinois).

Ten more folds will make it cross the outer atmosphere.

Add another twenty folds to reach the sun.

At the sixtieth fold it would have the diameter of our solar system.

At 100 folds it would have reached the radius of our universe.


If you do not have a good scientific calculator, then here is a table that shows the rapidity of growth on an exponential scale. In this table that I have used the caret symbol to represent the exponentiation operator. This table

n In kilometres
(0.1*10^-6 * 2^n)


0.1 * 10^-6


0.2 * 10^-6


0.4 * 10^-6


0.8 * 10^-6 Thickness of finger nail.


1.6 * 10^-6


3.2 * 10^-6


6.4 * 10^-6


12.8 * 10^-6 Thickness of a notebook.


25.6 * 10^-6


51.2 * 10^-6


0.1 * 10^-3 Width of a hand including the thumb.


0.2 * 10^-3


0.4 * 10^-3 Height of a stool – 0.4 m.


0.8 * 10^-3


1.6 * 10^-3 An average person’s height – 1.6 m.


3.3 * 10^-3


6.6 * 10^-3


13.1 * 10^-3 Height of a two story house – 13 m.


26.2 * 10^-3


52.4 * 10^-3


104.9 * 10^-3 Quarter height of the Sears Tower.

…. ….


3.4 * 10^0 Taller than the Matterhorn.


107.4 * 10^0 Reach the outer limits of the atmosphere.


3.4 * 10^3


109.9 * 10^3


3.5 * 10^6


112.5 * 10^6 ~ distance to the sun (95 million miles).


3.6 * 10^9


115.3 * 10^9 size of the solar system?


3.7 * 10^12 one-third of a light year.


118.1 * 10^12 11 light years.


3.8 * 10^15 377 light years.


120.9 * 10^15 12,000 light years.


3.9 * 10^18 4x the diameter of our galaxy.


123.8 * 10^18 12 million light years.


4.0 * 10^21


126.8 * 10^21 (12 billion light years) approx. radius of the known universe?

I came across the following video clip while surfing the internet. Click on this link: Paper folding to the Moon.The exponential growth also works inversely for the width of the paper. Each time the paper is folded, its width is halved. If we begin folding with a large piece of newspaper let’s say 50 cm wide, after 10 folds, the paper would be 0.05cm wide. After 20 folds, it would be 0.000048 cm wide. After 30 folds, 0.000000047 cm wide. And suppose we could fold it 33 times (which we can never accomplish), the width would be less than an atom.

Pythagoras vs Bothaināyaṉār


Myself By T.V. Antony Raj


Pythagoras of Samos


In Euclidean geometry, the Pythagoras’ theorem (or Pythagorean theorem) is a relation among the three sides of a right triangle (or right-angled triangle). In terms of areas, it states:

In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Animated geometric proof of the Pythagoras the...
Click image to view an animated geometric proof of the Pythagoras theorem. (Photo credit: Wikipedia)

Pythagoras’ theorem can be written as an equation relating the lengths of the sides a, b and c:

where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

If the length of both a and b is known, then c can be calculated as follows:

If the length of hypotenuse c and any one side (a or b) are known, then the length of the other side can be calculated with the following equations:




The Pythagorean theorem is named after the Greek mathematician Pythagoras of Samos, an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism whose central tenet was that numbers constitute the true nature of things.

Pythagoras is credited with the discovery and proof of the theorem. But it is often argued that the knowledge of the theorem predates him. Some claim that Babylonian mathematicians understood the equation, but there is not much of evidence for this claim.


Today, while surfing the internet I read a post in Facebook in Tamil and I was impressed by the following Tamil quatrain:

ஓடும் நீளம் தனை ஒரே எட்டுக்
கூறு ஆக்கி கூறிலே ஒன்றைத்
தள்ளி குன்றத்தில் பாதியாய்ச் சேர்த்தால்
வருவது கர்ணம் தானே.

oadum neelam thanai orae ettuk
kuru aakki koorilae ondraith
thalli kundraththil paathiyaaych cherthaal
varuvathu karnam thaanae

Divide the running length into eight equal parts. Discard one of the divided parts and add half the height. Isn’t the result the hypotenuse?

And here is another example:

a = 4
b = 3
So, c = (4 – 4/8) + (3/2) = 5

The article says that the author of this quatrain was a sage, mathematician, and poet named Bothaināyaṉār, and that the advantage of this Bothaināyaṉār‘s theorem over Pythagorean theorem is that the calculations can be easily done without calculating the square root.

By the way, this quatrain failed to produce the answer if a is less than b, for example if a = 3 and b = 4.

Next I tried the following:

Try #1: a = 12, b = 6

Modern mathematics:
sqr((12 x 12) + ( 6 x 6)) = 13.416407864998738178455042012388

Bothaināyaṉār’s method:
(12 – (12 / 8))  + (6 / 2) = 13.5

Try #2: a = 13, b = 9

Modern mathematics:
sqr((13 x 13) + (9 x 9)) = 15.811388300841896659994467722164

Bothaināyaṉār’s method:
(13 – (13 / 8)) + (9 / 2) = 15.875

Try #3: a = 15, b = 12

Modern mathematics:
sqr((15 x 15) + (12 x 12)) = 19.209372712298546059464653023865

Bothaināyaṉār’s method:
(15 – (15 / 8)) + (12 / 2) = 19.125

In most cases, the results obtained using Bothaināyaṉār‘s method was not accurate even to the first decimal place. So, I think I’ll better stick to the Pythagorean theorem.

Today, I spent a good amount of my valuable time on the net to learn about this person named Bothaināyaṉār, but was not able to gather any information about him. I doubt whether this person ever existed.

It’s funny that the Tamil word “Bothai” means inebriation and the word “nāyaṉār” translates to lord, master, or devotee. So, is someone playing a prank using the name Bothaināyaṉār (Devotee of Inebriation)?

The Tamil community and I would be glad if anyone out there could give any relevant and useful information on this subject. Your comments are welcome.


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Fractions used by Ancient Tamils


Myself .

By T. V. Antony Raj

In our conversations, we, Tamils, use words denoting fractions very frequently without batting an eyelid.

A Tamil goldsmith will assure his client, “இம்மி அளவேனும் குறையாது (immi alavaenum kurayathu) meaning “not a fraction less”. Here, the word இம்மி (immi) is the Tamil word for the fraction 1/2150400 ≈ 4.6502976190476190476190476190476e-07.



The word “aNu“, meaning “Atom”, is used very frequently by the Tamils to denote very minute portions or particles. In ayurvedic and sidda medicines the naattu vaithiyar (the country doctor) might give instructions to his assistant to add “அணு அளவு பாதரசம்” (aNu alavu padharasam) meaning “an atom sized mercury”. Here the word அணு (aNu) is the Tamil word for the fraction 1/165580800 ≈ 6.0393475572047000618429189857761e-09.

Now, I wonder why the ancient Tamils had such names for these particular fractions. The smallest fraction to be named being தேர்த்துகள் (thaertthugal) which is

1/2323824530227200000000  ≈ 4.3032508995084501477534881372607e-22

I am not able to fathom the underlying reason for the ancient Tamils to use such minute fractions and name them too. I read somewhere that the only place where such minute fractions are used nowadays is in NASA but I am not sure.



Here is the list that I gathered of fractions used by the ancient Tamils:

1 – ஒன்று – onRu
3/4 = 0.75 – முக்கால் – mukkaal
1/2= 0.5 – அரை – arai
1/4 = 0.25 – கால் – kaal
1/5 = 0.2 – நாலுமா – naalumaa
3/16 = 0.1875 – மும்மாகாணி – mummaakaani
3/20 = 0.15 – மும்மா – mummaa
1/8 = 0.125 – அரைக்கால் – araikkaal
1/10 = 0.1 – இருமா – irumaa
1/16 = 0.0625 – மாகாணி (வீசம்) – maakaaNi (veesam)
1/20 = 0.05 – ஒருமா – orumaa
3/64 = 0.046875 – முக்கால்வீசம் – mukkaal veesam
3/80 = 0.0375 – முக்காணி – mukkaaNi
1/32 = 0.03125 – அரைவீசம் – araiveesam
1/40 = 0.025 – அரைமா – araimaa
1/64 = 0.015625 – கால் வீசம் – kaal veesam
1/80 = 0.0125 – காணி – kaaNi
3/320 = 0.009375 – அரைக்காணி முந்திரி – araikkaaNi munthiri
1/160 = 0.00625 – அரைக்காணி – araikkaaNi
1/320 = 0.003125 – முந்திரி – munthiri
3/1280 = 0.00234375 – கீழ் முக்கால் – keel mukkal
1/640 = 0.0015625 – கீழரை – keelArai
1/1280 = 7.8125e-04 – கீழ் கால் – keel kaal
1/1600 = 0.000625 – கீழ் நாலுமா – keel nalumaa
3/5120 ≈ 5.85938e-04 – கீழ் மூன்று வீசம் – keel moondru veesam
3/6400 = 4.6875e-04 – கீழ் மும்மா – keel mummaa
1/2500 = 0.0004 – கீழ் அரைக்கால் – keel araikkaal
1/3200 = 3.12500e-04 – கீழ் இருமா – keel irumaa
1/5120 ≈ 1.95313e-04 – கீழ் வீசம் – keel veesam
1/6400 = 1.56250e-04 – கீழொருமா – keelorumaa
1/102400 ≈ 9.76563e-06 – கீழ்முந்திரி – keezh munthiri
1/2150400 ≈ 4.65030e-07 – இம்மி – immi
1/23654400 ≈ 4.22754e-08 – மும்மி – mummi
1/165580800 ≈ 6.03935e-09 – அணு – aNu
1/1490227200 ≈ 6.71039e-10 – குணம் – kuNam
1/7451136000 ≈ 1.34208e-10 – பந்தம் – pantham
1/44706816000 ≈ 2.23680e-11 – பாகம் – paagam
1/312947712000 ≈ 3.19542e-12 – விந்தம் – vintham
1/5320111104000 ≈ 1.87966e-13 – நாகவிந்தம் – naagavintham
1/74481555456000 ≈ 1.34261e-14 – சிந்தை – sinthai
1/1489631109120000 ≈ 6.71307e-16 – கதிர்முனை – kathirmunai
1/59585244364800000 ≈ 1.67827e-17 – குரல்வளைப்படி – kuralvaLaippidi
1/3575114661888000000 ≈ 2.79711e-19 -வெள்ளம் – veLLam
1/357511466188800000000 ≈ 2.79711e-21 – நுண்மணல் – nuNNmaNal
1/2323824530227200000000 ≈ 4.30325e-22 – தேர்த்துகள் – thaertthugal



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Did Decimal Numerals originate from Tamil Numerals?


Myself By T.V. Antony Raj

Recently I saw this picture in Facebook posted by Mr. Lenin Pugal. I think he must have downloaded it from a Tamil website ( because the watermark says so.

However, being a Tamil, this picture impressed me very much.

According to this post, the decimal numeral system that we use had been derived from the Tamil numerals. The heading says,

Eyes of arithmetic! Tamil numerals!

  • In the first column we see the Tamil numerals.
  • In the second column certain portions of the Tamil numerals are erased, indicated in red.
  • The third column is the resulting decimal numbers.

Isn’t this an eye opener?



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