Tag Archives: Mathematics

How Did the Present-day Numerals Derive Their Form?


Myself 

 

 

BT.V. Antony Raj

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Numbers (Source: express.co.uk)

Numbers (Source: express.co.uk)

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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.

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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

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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.

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Numerals and angles
Numerals and angles

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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?

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Rapidity of Exponential Growth


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Myself By T.V. Antony Raj

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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.

Whew…

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)
Comment

0

0.1 * 10^-6

1

0.2 * 10^-6

2

0.4 * 10^-6

3

0.8 * 10^-6 Thickness of finger nail.

4

1.6 * 10^-6

5

3.2 * 10^-6

6

6.4 * 10^-6

7

12.8 * 10^-6 Thickness of a notebook.

8

25.6 * 10^-6

9

51.2 * 10^-6

10

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

11

0.2 * 10^-3

12

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

13

0.8 * 10^-3

14

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

15

3.3 * 10^-3

16

6.6 * 10^-3

17

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

18

26.2 * 10^-3

19

52.4 * 10^-3

20

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

…. ….

25

3.4 * 10^0 Taller than the Matterhorn.

30

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

35

3.4 * 10^3

40

109.9 * 10^3

45

3.5 * 10^6

50

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

55

3.6 * 10^9

60

115.3 * 10^9 size of the solar system?

65

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

70

118.1 * 10^12 11 light years.

75

3.8 * 10^15 377 light years.

80

120.9 * 10^15 12,000 light years.

85

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

90

123.8 * 10^18 12 million light years.

95

4.0 * 10^21

100

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.

Fractions used by Ancient Tamils


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Myself .

By T. V. Antony Raj
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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.

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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.

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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?


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Myself By T.V. Antony Raj
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Recently I saw this picture in Facebook posted by Mr. Lenin Pugal. I think he must have downloaded it from a Tamil website (www.natpu.in) 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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