A review I wrote 20 years ago.
Georges Ifrah is a Frenchman of Moroccan origin who was an ordinary schoolteacher of mathematics before his students sparked one of the great intellectual quests of our time (or, indeed, of any time). His students asked him where numbers came from? Who invented them and why? How did they take their modern form? When he tried to answer these simple questions, he found that the information found in standard textbooks was highly unsatisfactory and frequently contradictory. Not content with passing on half-truths and conjectures, Mr. Ifrah abandoned his job and embarked on a ten-year quest to uncover the history of numbers. He traveled to the four corners of the world, read thousands of books, visited hundreds of libraries and museums and asked questions of countless scholars. All this research was supported by odd jobs as delivery boy, chauffer, waiter, night watchman and so on. The result was a book called FROM ONE TO ZERO A Universal History of Numbers, (published in English translation in 1985 ). The book was a hit and brought fame and fortune and the chance to do more research. This led to a much larger book, The Universal History of Numbers: From Prehistory to the Invention of the Computer, which was translated into English in 1998 (after initial publication in French in 1994) and is now available in either one or two volumes.
These books have earned Mr. Ifrah the title of “Indiana Jones of numbers” and worldwide celebrity. After reading the book, I can only add that he deserves every superlative that has been used, and more. To quote a reviewer from “The Guardian”: “Georges Ifrah is the man, and this book, quite simply, rules.” This is not just a history of numbers, it is universal history disguised as the history of numbers. Mr. Ifrah starts with the most basic questions; what kind of “counting sense” do animals possess? What do we know about the number sense of our pre-human ancestors? When we evolved into Homo sapiens sapiens, what kind of numerical ability was “hard-wired” into our brains? He presents fascinating information about the most primitive counting systems, using tally marks, fingers, body parts etc. from these simple beginnings, we move to the abstract concepts of number and its notations. The detail provided is astounding. We learn about the earliest systems of numbers used in the Middle East, India, china, and the ancient Maya etc.etc. And not only do we learn about the numbers, Mr. Ifrah slips in his humanistic, sensitive and very very detailed knowledge of history so smoothly that we hardly notice that we are learning, not just the history of numbers, but the history of mankind; told by a very fair, very balanced and deeply sympathetic observer.
The book is designed to be a reference work and thus contains more detail than the casual reader may require, but unlike most reference works, it is written in an accessible style and every concept is beautifully explained from the bottom up. You can read it from beginning to end (and enjoy every minute) or just jump to the matter that interests you and learn about that. If you have ever wondered how “primitive” people added and multiplied on their fingers, look no further, if you want to know how the abacus is used and how the Roman numerals can be (or cannot be!) manipulated, step right in. Mr. Ifrah has the answers. He will also tell you all about the use of letters to represent numbers and the number values of every letter in Hebrew, Greek, Arabic etc. I learned for the first time why the huroof-e-abjad are in an order different from the order of letters in the modern Arabic alphabet (aleph, bey, geem; rather than aleph, bey, tey) and how and when these changes came about. The section on magic and mystery tells us about the occult significances of numbers and arcane topics like chronograms (words that express a particular date). E.g. the chronogram “zaatish murd” literally means “died of burns” but when the abjud values are added together, gives us the date 952…. the hijrah year in which king Sher of Bihar died in a fire!
Mr. Ifrah tackles the question of where our modern number system came from and gives an unambiguous answer: from India. Somewhere between the second and fifth century CE, Indian mathematicians worked out the revolutionary system of using 9 numbers and a zero that we still use in the same form today. This system traveled from India to the Middle East, as did the nine Indian numerals. In the course of their travels, the numerals were gradually modified into their current forms, and Mr. Ifrah provides detailed (and graphic) evidence of how this happened. This chapter also puts to rest the theory that the number forms have anything to do with counting the lines or angles to equal their value. The forms are modifications of the original Indian “Brahmi” forms and nothing more. The Arabs who took up these numerals made no attempt to hide their origin and writers like Khwarizmi gave full credit to the Indians for these discoveries, but by the time the news reached Europe in another two hundred years, the Indian origin had become obscured and thus we still call our numerals “Arabic” numerals. It’s interesting that the first attempt to introduce these to Europe was made by the progressive pope Sylvester the second in 1000 CE but failed due to resistance from conservative elements. It was only after the crusades and the work of the famous Italian mathematician Fibonacci that the new system started to take hold, even then, it was several hundred years before the church fully accepted the new invention. In fact, resistance from the church led to the zero based system being used in secret. A bit of history from which we get the word “cipher” – which originally meant zero, but took on connotations of code and secrecy because the zero based system was used in secret!
At the end to the book, Mr. Ifrah provides a brilliant after word, where he incidentally demolishes all the theories of prehistoric alien invaders with the simple question: if ancient invaders came to teach us the secrets of how to build the pyramids, why didn’t they teach us about zero and place value notation? Is it conceivable that any advanced civilization that flew around in spaceships did not have the zero? The evidence is very clear that it took mankind a very long time and many failed experiments before our Indian ancestors finally solved this riddle. If arithmetic was invented slowly and painfully, with no outside help; then so was everything else, because, before any other science there is the science of numbers!
46 thoughts on “Book Review: The Universal History of Numbers”
Thanks for the review. Looks like an interesting book to read. Does the author spend time in locating the place in India where the decimal system was first used? As I understand the Brahmi script was used predominantly in the Gangetic plains. Does he give a more specific location for this? Nalanda was created in the fifth century CE and there might have been other centers of higher learning. Perhaps a brilliant young student came up the decimal system under pressure to secure tenure!
The Arabs who took up these numerals made no attempt to hide their origin and writers like Khwarizmi gave full credit to the Indians for these discoveries, but by the time the news reached Europe in another two hundred years, the Indian origin had become obscured and thus we still call our numerals “Arabic” numerals.
I was taught in primary school that these were Arabic numerals as well and I remember my dad telling me they were Hindu numerals. Subsequently the first BJP government of Vajpayee brought in academic changes and later it was changed to Hindu-Arabic numerals.
When I was taught how people figured out irrational numbers to fill in the ‘holes’ in the number line which in-turn was necessary to calculate Riemann/Lebesgue integrals I was mind blown at how awesome mathematicians are. It is my firm belief that with decimal system and series convergence results, conjecturing value of pi and other work on irrational numbers … Indians would have eventually figured out measure theory on our own if we had not been over-run by invaders.
My mind was blown when we had to prove that probability of picking a rational number on a real number line is zero. Similarly the probability that the random number that you pick will be an irrational number is 1. We did this in my probability theory class.
Ever realized that in Hindi 1 to 100 numbers are virtually different making it a 100 based system? In contrast, in Telugu and English numbers are combined after 20 making them a bi-decimal systems at least verbally. Read more about it in this wonderful blog by shrikant talageri https://talageri.blogspot.com/2018/
I am sure that Indian mathematicians could have achieved a lot without invaders, but measure theory seems to be a step too far to be achieved in isolation to rest of the mathematics.
Indian mathematicians had decimal system and algebra far longer than others but haven’t gotten to calculus or probability theory. (In “Against the gods, the remarkable story of risk” Bernstein notes that Roman numerals were a hinderance for development of algebra, set theory, and consequently probability calculus. Hence, the relatively young history of probability theory/risk. Makes me wonder why Indians didn’t have a leg up on everyone else on this).
We were still stuck on how to take different “pramanaas” for induction without any quantification of empirical observations.
Makes me wonder why Indians didn’t have a leg up on everyone else on this).
I think South Asia on a whole, not just Math, Medicine and Engineering.
My opinion has been that the publishing, copyright and patent system has allowed the west to make huge leaps in theory and technology. I would even argue the single most concept (coupled with printing) that propelled the development of the west.
Copyright and Patent, allows the inventors and authors to protect their brain child and have a monopoly on profits for a limited time. At end of patent or copyright. Publishing science, math or whatever allows exchange of ideas developing. i.e. The ability to stand on the shoulder of giant and not reinvent the wheel ever couple of decades.
In contrast, in South Asia knowledge was kept within the family (or disciples). Rarely shared. Often knowledge was lost with death.
Patent and copyright are the typical capitalist tools which put monetary value on knowledge, apart from rescuiing knowledge from anonynity and giving credit where it is due.
Patent and copyright are the typical capitalist tools which put monetary value on knowledge
Monetizing knowledge is easier in a capitalist system. Even in other political systems, those with knowledge*, gain respect and possibly power. Those two in turn can be monetized.
Blackmail is knowledge too.
Thanks for the link on Indian math. Downloaded and will peruse.
The probability of randomly chosen number from a real line being an rational number is ZERO, similarly probability of randomly chosen number from a real line being an irrational number is ONE, this just came as a shock to me when we had to prove this in my Probability Theory class in my PhD.
A good regarding the history of numbers is in the link
Have you ever noticed that in Hindi numbers from 1-100 have different names and you combine names for 1-100 that come after 100 thus making it a 100 based system. On the other hand English and Telugu have unique names till 20 and then combinations of names from 1-20 for the numbers that come after 20, thus a “bi-decimal” system. On the other hand in sanskrit 1-10 have unique names and combinations of words of 1-10 for numbers that come after 10 like. For example: 12 in sanskrit is dva-dasa (dva is 2 and dasa is 10). In Telugu, 1 is okati, 2 is rendu and 10 is padi however 11 is padakondu, 12 is pannendu….20 is Iravi. But, 21 is Iravi Okati (20+1) similar to english.
These ancient Indian mathematicians never get the credit they honestly deserve. Its all sort of reduced to some joke about zero that’s never fully explained. But the introduction of the decimal number system changed humanity in such profound ways. If Brahmagupta, Bhaskara, etc. were Europeans, they’d be household names worldwide like Euclid and Pythagoras.
And quite frankly some credit also goes to the Persian mathematicians who recognized the value and then gave due respect and credit to source while popularizing the system and spreading it throughout the middle east.
This is also why I hate the “Vimanas and Nuclear bombs in the Mahabharata” crowd. If you look, there is great amazing actual stuff in Indian past… no need to invent wild theories that ruin credibility.
If Brahmagupta, Bhaskara, etc. were Europeans, they’d be household names worldwide like Euclid and Pythagoras.
idk. euclid is special in a way cuz *elements* is still taught as a text at some modern universities! that’s incredible.
Euclid must have been a god-tier mathematician. Greeks in general owned Geometry.
But whoever figured out zero (Brahmagupta?) was the greatest of them all (until current reigning champion Carl Friedrich Gauss came along and wagged his anaconda whereas most people are endowed with an earthworm at best, that guy was ‘The God’ incarnate) figuring out zero was the original ‘big breakthrough’ of humanity. Someone would have eventually figured out geometry but abstractions are really difficult. I will almost certainly never manage even one in my whole life. But 2000 years ago give me (or anyone reasonably smart) some idea about proofs and some push with ideas like locus, straight line etc and I would give you substantial amounts of Euclidean Geometry on my own. Invention of number zero ‘0’ is a singular achievement of mankind.
Also, there is nothing called Hindu-mathematics. Like there is no Hindu-water just water, there is just mathematics or **trolling** like there is no ‘South Asia’ only Indian-subcontinent.
Euclid and Pythagoras were names known (hated by some) by those who did O/L Science and Math (not Commercial Arithmetic).
We started learning Geometry (and Algebra) in Grade 7.
Durrels Geometry was used till O/L (grade 10). Had to know how the prove some 100 odd theorems, including Pythagoras Theorem.
Halls Algebra, the same Grade 7 to O/L’s
Durrels Elementary Geometery was anything but elementary
Halls Algebra For those who like to relive school days.
Grade 7 to O/L’s (Grade 10)
Excellent review but this is another book which missed to notice the existence of the oldest civilisation in the world with the oldest alphabet. The origin of the numbers in India is from Vinchan alphabet.
Thanks Omar; shared on FB (and email to classmates and Uni friends)
Because was looking at Asoka this connection was fresh in my mind.
The first examples of the Hindu-Arabic numeral system appeared in the Brahmi numerals used in the Edicts of Ashoka, in which a few numerals are found, although the system is not yet positional (the zero, together with a mature positional system, was invented much later around the 6th century CE) and involves different symbols for units, dozens or hundreds. This system is later further documented with more numerals in the Nanaghat inscriptions (1st century BCE), and later in the Nasik Caves inscriptions (2nd century CE), to acquire designs which are largely similar to the Hindu-Arabic numerals used today.
There are few public texts about Vincan Number System (about 5500BC) (even in Wiki) which are usually pretty conservative, probably trying to be politically correct and avoid to directly oppose the official falsifications in the world history.
One of the public papers I found interesting is – The Number System of the Old European Script, written by – Eric Lewin Altschuler, M.D., Ph.D. Mt. Sinai School of Medicine, NY, and Brain and Perception Laboratory, University of California, San Diego AND Nicholas Christenfeld, Ph.D. Department of Psychology, University of California, San Diego.
The following is their conclusion:
“In conclusion we find (1) that many signs in the OES (MT: =Vincan Script) seem to represent a number system (2) with 10 apparently an important base or unit. (3) Scratched score marks on the bottom of a pot, in particular, and other OES signs convey no religious meaning, and (4) possibly could have had some economic purpose. (5) The delineation of the number signs of the Old European Script should facilitate further understanding of the rest of the script and of the Old European culture, especially as new archaeologic findings emerge. (6) The beauty and power of numbers wrought by our ancestors’ hand so long ago speaks to us today with great clarity”.
The book can be downloaded freely from the link:
The author mentioned that the oldest numerical system is Sumerian, 3200BC. It is more than 2000 years after Vinca. It is interesting presentation of various numerical systems. The author did not know that Phoenician, Etruscan, Lydian, Lycian and some other are actually versions of the Vincan numerical system. Also, he did not know that Greeks got their alphabet from indigenous people. They had to adopt also letters they did not have in their language (e.g. ‘Ч’ = CH = 90) to preserve underlining numerical system. That is the proof that Etruscan was not influenced by Greek because their system is older and that Latin alphabet was not based on Greek than on Etruscan, i.e. Vincan alphabet.
Still, very good book (American Scientist’s list of “100 or so Books that shaped a Century of Science”, referring to the 20th century) and lot of efforts invested in its writing. It is unfortunate that he did not go to the roots because the book about Vincan alphabet was written in mid-80ies.
The author unfortunately died several months ago. I hope, one day, Omar can write updated review based on new information obtained in meantime.
Comprehensiove book on Hindu mathematics
Mathematics in India
by Kim Plofker
PRINCETON UNIVERSITY PRESS
Also few books are available free
History Of Hindu Mathematics A Source Book
History Of Hindu Mathematics A Source Book Part 1
Avadesh Narain Singh 1935
https://twitter.com/michael_nielsen/status/1174420006907473920 Great thread by a physicist, he considers whether it was possible for romans to come up with same number system. Archimedes, comes close.
Aryabhata comes with own notation of playing around with representing numbers with letters. I believe this was the key that unlocked decimal representation. A lot of maths is India has to do with sanskrit and music.
Severus Sebokht 575-667 CE
Notes d’Astronome Syrienne, Journal Asiatique, series 10, vol. 16 (1910), pp. 225
Bulletin of the American Mathematical Society, May 1917, pp 368
Ekadyekothara pada sankalitham
Samam padavargathinte pakuthi
The integral of a variable equals half of its square.
Modern mathematical journals are coming around to acknowledge the first origins of calculus in the Kerala school by Jyeshthadeva in Yukthibasa (16th Century AD) much before Newton or Leibnitz. This was absorbed by Jesuits and transmitted to Europe – https://twitter.com/JoeAgneya
How many of you knew this already?
I am somewhat amazed, that the hellenics, much like their philosophy, have a mathematical system that is philosophical with their method of proofs. It seems their style of doing math or philosophy is fundamentally focused on not making errors.
Has anyone wondered how our ancient daddies figured negative numbers? Do you know most of the world didn’t really use negative numbers till recently?
My advisor challenged me to prove (-2) + (-2) = -4. There is no intuitive explanation for this, you have to follow step by step procedures using basic axioms of arithmetic.
It was -2x-2=4 not (-2) + (-2) = -4
Multiplication (by a unit) is vector rotation.
Multiplication by -1 rotates the vector by 180deg. Multiplication by -1 again gives 360deg, ie same sign.This also explains why imaginary axis is orthogonal to the real line. Because if multiplying by some unit i rotates by 90deg, then i*i must be equivalent to 180deg rotation (which is equivalent to -1 multiplication), ergo i^2 = -1, i.e. i must be defined as sqrt(-1).
Tell your advisor you have received enlightenment on brown pundits from an English grad 😉
PS: In case some of you are wondering multiplication by sqrt(i) is 45deg rotation, and so it goes like clockwork… it is amazing how deep the beauty and structure of mathematics is.
This would be a pretty good explanation in most cases. I guess it’s challenging when one is supposed to start from the bottom up and state all the assumptions. In this case, you’re starting with the observation that the set of real numbers, or just the integers form a vector space. Of course this needs a prior definition of a vector space. Once you have that structure, in which you demand that the associative property hold for multiplication, you can then get the intuitive understanding that multiplying by -1 is rotating a vector by 180 degrees.
The theory of vector spaces (or linear algebra etc) is not assumptional to this view. You are mixing cause and effect.
The Cartesian view (16c) of representing numbers on a real line is centuries older than vector spaces (19c). So people knew that +1 can be represented on a line right of zero and -1 in the opposite direction (right-left is just conventional of course)
The key insight is that -1 does *not* represent a directional flip or mirror transformation (as it is intuitively understood by many people) but a rotation. Once we “see” -1 as rotation, imaginary numbers / complex numbers are a mere corollary. And all of this requires nothing more than basic Euclidean geometry and Cartesian representation of numbers on an axis. No funky vector spaces – which really are a formalisation of this insight and so follow from it.
Now, you’re using vectors to prove. I am not sure if the ancients had any access to that.
Wait, I taught that to my 6 year old ages ago. Negative numbers are just direction change and multiplication is just repeated addition.
So, we did magnitude and direction separately. You will get four as magnitude and original direction as final if you make the child “walk” the number line.
The thing that is difficult is fraction division and how dividing 1/2 with 1/2 is one and not 1/4 since we are halving the half’s. I am still stumped at how to make this intuitive other than just say flip it and multiply.
Wait till he asks you to prove Complex numbers theorems. 😛
Lol, I handed over all higher math teaching to 3blue1brown on YouTube.
As I understand, the spirit of the question would be to prove that multiplying by minus 1 just changes the direction while maintaining the magnitude. That seems trivial to us because we imagine the number line, however, the number line already assumes this property in the first place. So this reasoning is tautological.
I am only an engineer who uses math for expression (like a poet uses language to convey a feeling) rather than create it (like a grammarian or linguist). Pardon me that I still don’t get the rigour of answer needed.
I haven’t taught inherent properties of number line at all. We started with counting, went to addition (extended counting by more than singles), and by learning subtraction (again knowing only counting), the direction is already established. (How counting moved objects between two sets – in or out). Subtraction is just addition with changed direction (I.e. negative numbers)
If multiplication is repeated addition, then direction is already established from learning subtraction. (I.e. you don’t remove two things twice).
Is the question about negative numbers themselves and their physical meaning? Or about some first principles that I am missing?
What I am more interested in is the development of Physics in India. vaisheshika sutra was a good enough start. They came to the ideas of distance being equal to time *speed. What then, application to celestial models.
I wonder that once they got to calculus.Would it have been possible to develop physics?
There is so much amount of work to be done to collect all the works on Indian sciences together that have been worked out.
The theory of vector spaces (or linear algebra etc) is not assumptional to this view. You are mixing cause and effect.
I agree it doesn’t have to be assumptional. I mentioned it because you used the transformation properties of vectors (rotations) to argue your point. Even more fundamentally, the idea that we can understand a number as an arrow originating from zero and extending on the number line till the magnitude of a number, is a non-trivial observation which doesn’t just require a Cartesian space but the notion of a vector space.
All of this is irrelevant for almost everyone. I’m just trying to make OP’s point about why it was difficult for the ancients to understand multiplication by a negative number and would be challenging to prove in a formal setting.
I am only an engineer who uses math for expression (like a poet uses language to convey a feeling) rather than create it (like a grammarian or linguist).
Neither am I a mathematician. I’m a physics student who’s taken too many math courses for his own good!
The number line is itself a huge leap in our understanding of mathematics because it gives us a way to geometrically understand arithmetic operations. When one argues that the number line is a self-consistent framework to express the set of real numbers or the integers, we are really making use of this incredible achievement in math to understand something which is more elementary, i.e., multiplication. The difficulty of this question arises from the amount of mathematical structure we can assume to begin with. If we use only that math which was known in India around the fifth century CE and then try to answer it, it will be a lot more challenging than to use 21st century math to answer this. Of course, for us it is trivial but that triviality hides behind it how difficult it must have been to understand negative numbers in the first place.
// incredible achievement in math to understand something which is more elementary //
Standard error of reductionism. Expressing something commonplace or well known in terms of harder to achieve general abstraction (which often post-dates the commonplace ‘fact’) is the very point of knowing anything.
PS: my last comment on this thread.
Expressing something commonplace or well known in terms of harder to achieve general abstraction (which often post-dates the commonplace ‘fact’) is the very point of knowing anything.
The point is to not just know something but to also understand how do we know it. In this case, the harder to achieve abstraction, i.e., vector spaces or even Cartesian spaces already use the result that you are being asked to prove, i.e., multiplying by a negative number is the same as rotating by 180 degrees. They just make it easier to visualize it. Hence, you haven’t explained anything because your argument is tautological.
His argument is correct by your very own assertion that ” Of course, for us it is trivial but that triviality hides behind it how difficult it must have been to understand negative numbers in the first place.”
Cartesian coordinate system has different coordinates of y . the idea of -1 as rotation by 180 degrees is applied to a single number line of one dimension (x) with imaginary part ‘i’ in place of y-coordinates . Slapstick is correct. Time to update. And his argument is not tautological. My view.
I am not sure if Indian Mathematicians had any idea of vectors but one thing they could’ve done is:
-2(-2 + 2) = 0 #since anything multiplied by zero is a zero and IMPORTANTLY the ancient indians already knew that
(-2x-2) + (-2×2) = 0
Now -2×2, one can simply say that you doubled your debts so the answer is -4. Or 2(2 – 2) = 0, 4 + 2x-2 = 0, 2x-2 = -4
(-2x-2) + (-4) = 0
(-2x-2) + (-4+4) = 0+4
(-2x-2) + (0) = 0+4
-2x-2 = 4
Yes, associativity in multiplication is the key property.
The number system is India’s greatest contribution to humanity imo.
-Do negative numbers come from India too?
-What about imaginary numbers?
-If Europe wasn’t resistant to Hindu numerals for that long would humanity have progress further today? What if there was greater sharing of knowledge between Europe, West Asia and South Asia in general?
negative numbers were there in china. Though i am not sure in the abstract form. In algebra and in numerical problems mathematicians solved the problems knowing that negative solutions meant the direction was different. They asked question about square root of -1. Didnt take it though. And the other question is that, had christianity and islam not taken root, how much faster would there have been an advance. Also, had alexander did conquer India and there was a near total transfer of knowledge and philosophy between east and west, what then?. Because while the hellenic civilization was finished in west, certain of their ideas would have continued in India and other places. And India was in contact with china. I think the world would have been very different had Alexander conquered India and total knowledge transfer would have happened. Also, Europe got ahead because of patent rights and copyright laws and printing press. Any civilization that got those implemented irrespective of century would have kickstarted many things.
Yes they were there in China but I read they were independently invented in India. There isn’t anything that tells me humanity would have progressed more without Christianity or Islam. The same question could be asked for Hinduism btw. No math was coming out of the Arabian peninsula even if they stayed pagan.
Progress is product of diversity of ideas, in europe it came from patent and copyright laws. Yes, same question could be asked for Hinduism too. My point is there is a difference in between polytheism and monotheism. No new great mathematical result came in europe from 400 to 1200 yrs, an astounding 8 centuries of dark ages. As for Islam, after initial burst of creativity, it caused downfall in India. cut off the trade routes exchange of ideas between china and India, europeans had to find another way to trade with India rather than through land. Arabia is a desert land. Consider instead across the silk route, from greece to China and India was a rich source of creativity and exchange of ideas. Yes people could independently have discovered things. Ideas of calculating debts are common.With monotheism, one becomes suspicious of ideas. It took over 3 centuries to accept Indian numeral system in west. evolution is doubted more in lands of monotheism than other places.
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