Ancient India pioneered the concept of Negative Numbers, in this documentary it is explained in detail – that will change our understanding about the history of Negative Numbers
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Происхождение отрицательных чисел и вклад Бхараты
Текст рассказывает об истории развития отрицательных чисел и вкладе индийской цивилизации Бхарата в их становление. Вот основные моменты:
Позиционная система счисления
- Позиционная система счисления была усовершенствована в Бхарате, хотя ее истоки восходят к более ранним цивилизациям1.
- Число имеет три атрибута: абсолютное значение (неизменное), позиционное значение (меняется в зависимости от позиции) и направленное значение (положительное или отрицательное)1.
Вклад Брахмагупты
- Основным источником по происхождению отрицательных чисел является трактат “Брахмасидхантика”, написанный Брахмагуптой в 7 веке н.э.1
- Брахмагупта кодифицировал правила арифметических операций с положительными, отрицательными числами и нулем: сложение, вычитание, умножение, деление1.
- Согласно Брахмагупте, положительные и отрицательные числа – это переходные состояния относительно нуля, который является абсолютно стабильным1.
- Брахмагупта первым записал правила деления на ноль, хотя их смысл неясен1.
Более ранние упоминания
- Рукопись Бакшали, датируемая 200-250 гг. н.э., является древнейшим сохранившимся манускриптом по индийской математике1.
- В рукописи Бакшали встречаются знаки сложения, записанные в шарадском письме, происходящем от брахми1.
- Знак плюс впервые появляется в эдиктах Ашоки 3 века до н.э., написанных письмом брахми1.
Таким образом, хотя первые упоминания отрицательных чисел встречаются в более ранних источниках, именно Брахмагупта систематизировал и кодифицировал правила работы с ними, что стало важным вкладом в развитие математики.
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[Music]
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[Music] in the history of human beings once we
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started learning how to count probably that is the first step when we also started to understand the nature in a
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very scientific manner almost every civilization on the face of Earth and the history of time had their own ways
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and means of keeping count of things things having their own number systems many of them we understand and some of
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them we are yet to all the number systems have one template in common that is to keep count of things in the real
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world but that is just one side of the coin let’s flip to the other side of the coin when we simply call numbers whole
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or natural the purpose is pretty clear and straight and all of us have a very good understanding about it but what
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about the negative numbers why do we need them many civilizations could would not comprehend the purpose or even the
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meaning of a negative number as far as the recorded history goes only two civilizations Barat and China have their
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contributions towards negative numbers well documented and in this documentary let us try and understand the origins of
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negative numbers and the contributions that came in from bhat I’m sure that you
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have a good understanding about the positional value system or the Hindu number system that we are using across
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the world and for starters here is a quick metaphor imagine that we have five men of different economic classes sorted
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from poorest to the richest and then we have a 2,000 rupee note now this 2,000
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rupee note has a value in the market that it can buy certain amount of things no matter who gives the 2,000 rupe note
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the worth of it remains exactly the same and that is the absolute value now
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imagine that this 2,000 rupe note is jumping across the pockets right from poorest to the rich
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then the value of this 2,000 rupee note will reduce what it means is a poor man
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having a 2,000 rupee note it is very valuable to him but as it jumps the pockets towards the richest person the
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value of this 2,000 rupee note when it is in the pocket of a rich person is very less for him because he has a lot
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of money this scenario metaphorically explains how the Hindu number system works or the positional number system a
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number has an absolute value which never changes and and also a positional value changes depending on which position it
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is occupying in a complete number and if we try to transpose the
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2,000 rupe note scenario to the Hindu number system this is how it works seven as a number has an absolute value that
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will never change but s also has a positional value which will change as a
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function of its position that it occupies in a larger number before we try to understand
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negative numbers let us try to understand what is a number in first place imagine we have seven discrete
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objects to deal with those seven discrete objects put together the collection of it is called as a number
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now when we try to represent this number on paper by writing it in one or the other language that is called a numeral
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no matter which language you write your numerals in the number will always
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remain constant the absolute value that’s what it is called as the absolute
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value of the number will never change the absolute value is a fixed attribute
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of a number and then comes a variable attribute of the same number called as the positional value and you know what
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it means there are many civilizations which developed this positional value system but it is bhat who perfected it
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so it is fair to say that bhat only perfected the positional value system but was not necessarily the first one to
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invent it and there’s a good reason why I’m saying this which you’ll understand eventually down the line in addition to
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the absolute value and the positional value there is a third attribute of the number which is called as directional
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value and this attribute defines whether the magnitude of the number is a
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positive or A negative you could have 7,235 as a number but the directional
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value which attributes a positive sign or a negative sign to that number significantly varies the value of that
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number so if we go from bottom to top absolute value is less significant positional value is more significant
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than absolute value and directional value is even more significant than the positional value now what sets bhut
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apart in its contributions in the field of mathematics is number one it perfected the positional value system
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and number two bhat very well comprehended this directional value system how to deal with positive and
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negative quantities and including zero all the arithmetic operations with your integer
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is very well perfected in bhat and that story of directional value especially
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dealing with negative numbers that is one of the greatest historical contributions that came in from bat but
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unfortunately many of us are not aware of it and this doc film is all about just that so before we move forward a
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quick side note the absolute value positional value and directional value all three attributes Define what a
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number is but do we understand a number completely or could there be another thing like dimensional value that means
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as you transpose up into different dimensions will there be any other attribute which we can call just like a
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dimensional value that would further deepen our understanding about numbers let me know your thoughts in the
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comments below I would like to read them owing to the subject of this doc
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film the origins of negative numbers the most important source that we have is
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brahmaa sidhant written by brahmagupta in the 7th Century CE this work on
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mathematics and astronomy is one of the crown jewels in the history of science that was developed in bhat brahmagupta
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who is a great astronomer and mathematician in this book called brahmas sidhant he elaborated a diverse
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set of subjects related to mathematics definition of positive numbers and negative numbers including zero and how
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to perform arithmetic operations with all three principles of algebra dealing with indeterminate equations principles
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of J omry trigonometry and the list is very long in this doc film our main interest is purely about the integers
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the positive numbers zero and the negative numbers as defined by
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brahmagupta cutting across straight to the Crux of the topic in the 18th chapter of brahmas santam brahmagupta
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codified the rules for the arithmetic operations with positive numbers zero
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and negative numbers how to add them how to subtract them how to multiply and divide them and what you’re seeing on
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your screen are just that a well codified set of samskritam slokas in
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which brahmagupta codified the arithmetic operations of positive numbers zero and negative numbers now
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let us try to break it down and understand across the set of slokas
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brahmagupta repeatedly uses three shabas or three words dham which essentially
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represents the positive quantity shum which represents zero and runam which
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represents the negative quantity I colorcoded all these three words dham in
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red shunum in green and runam in blue across the slokas just a simple glance
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will give us a feeling like how tongue twisting these slokas are but all these slokas when translated into a language
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that we can understand this is how it looks like a long list of the principles
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of arithmetic operations on how to deal with positive quantities zero and
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negative quantities now let us Deep dive into this if we want to understand the rules
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on the arithmetic operations codified by brahmagupta we need to have a clearcut
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understanding rather an in-depth understanding about three words
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dhamam and runam because fundamentally samsam is not a simple language to deal
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with the depth of the language is just uncomprehended so these three words the
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real meaning of it has an incredibly important role to play in these rules of
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arithmetic operations so starting with the first word dham dham in srum what it
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means is wealth accumulation or credit and in that connotation this word dham
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would be used for example we have this word called pramana which means
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progression and development and the word pramana is built on this root word dham
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moving next to sunam this is pretty clear and straight for all of us sunum means something that is void emptiness
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and also a stable state in those connotations this word shum in samsum would be used and moving to the third
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one renim this is very interesting the word renim means to return or obligatory
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or debt in such connotations this word renim would be used in fact in samsam
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this root word Ru means to give back that you owe something to someone or to
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return back and exactly the Latin word for same connotation is also r that is
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the reason we have refunded written redeem repent retrospect all these
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English words originate from the Latin root word re and all these words
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interestingly represent which has something to deal with the past starts with the word r so in this context the
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Latin word R and samsum R has exactly the same meaning so that is renim so I
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will leave this here but if you really want to understand in detail about brahmagupta’s rules on arithmetic
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operations it is very important that we go much more deeper in understanding The samrm Words dham sham and
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rum now here comes the most thought-provoking part on understanding
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positive and negative numbers according to the codification done by brahmagupta
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sunum or zero is a stable State and both dham and runam are transient states that
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will go up and down but never below shum now this understanding is in stark
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contrast with our understanding of negative numbers where today we perceive that -2 -3 -4 or so on are the numbers
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which are less than zero this is where brahmagupta’s explanation about a
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positive and negative quantities opens up totally a new different direction of thinking on understanding numbers I
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repeat according to brahmagupta’s codification the positive numbers which is called as dham and the negative
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numbers which is rum are transient states with regards to zero and zero is
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the ultim stable state so for you to think much deeper
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into this topic I just left couple of notes Here the first thing is brahmagupta’s illustration of positives
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and negatives is different and challenges modern day understanding of negatives and positives so towards the
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right what you see here the number line today we understand that minus 1 -2 – 3
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lies towards the left hand side of zero which essentially means that the value is is less than Z and + 1 + 2 + 3 is
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towards the right which means it is greater than zero but brahmagupta’s codification states that a zero could
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start anywhere but a positive essentially represents forward movement
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and a negative represents a backward mement all with regards to the position
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of reference that you choose and second thing is according to brahmagupta’s illustration positives and negatives are
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not just magnitudes greater or lesser than zero but they are transient
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magnitudes with reference to zero this is very crucial to understand and in
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this concept of brahmagupta there is no value absolutely there is no value which
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is lesser than sunum sunum is the ultimate stable state it really feels
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good when a 1500 years old mathematical text challenges the status quo that you
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and me have been learning since our school day it is not about who is right it’s all
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about what is right and a quick touch and go the rules
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on additions of positives negatives and zero as the first one
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reads which means positive plus positive gives a positive negative plus negative
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gives a negative I literally have to spell what brahmagupta wrote dham plus dham gives out dham Ram plus rum gives
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out rum also to highlight the fourth principle 0 + 0 is equal to 0 or shum
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plus shum is equal to shum now this is very important in the history of mathematics brahmas sidhanta is the
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oldest known mathematical scripture which codified the rules on arithmetic
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operations with zero up next we have the rules on
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subtracting positives negatives and zero for instance the first one
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reads it means a larger positive minus smaller positive results in positive
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same way another example for instance the third one dham dham it means when you subtract
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Zero from a positive or A negative quantity you’ll be just left with the same positive or negative
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quantity up next we have the rules on multipli ation for instance if you see the second
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one which means negative multiplied by negative results in positive the
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immediate next one positive multiplied by positive
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results in positive and so on and lastly we have the rules on
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division of positives negatives and zero as you can read the first one D when you divide a positive by positive
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the result is positive take a look at the last one k kak which means 0 / 0 is equal to 0 we
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know that anything divided by zero is undefined so let’s spark that argument for a second and let’s try and
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understand the literary sense of this why did brahmagupta mentioned it as K
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kakam and not sh till now we understood that means
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zero and that’s how he artic calulated in rest all Locas in many places but why
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K is also representing zero this is very important to understand hold this and
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we’ll come back now owing to the subject of this doc film the origins of negative numbers
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so a simple question that we should ask our is is brahmagupta the first one who
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constituted this idea of negative numbers or how to deal with this element
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of subtraction subtraction of number numbers is where we encounter the
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negative numbers it’s pretty obvious right so let us do a bit of time traveling 450 years before brahmagupta
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so that’s around 200 to 250 CE that is when this manuscript called the bakali
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manuscript was written as far as bhattia ganum or the ancient Indian mathematics
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is concerned this is the oldest manuscript which is almost 18800 years
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old is the oldest one that’s available to us as of today and this is where
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Another Story related to the negative numbers would start I already made a very detailed documentary about bakali
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manuscript it’s in the channel or I’ll leave the link in the description below should you wish to watch it so getting
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back to the current context negative numbers and subtraction what does bakar
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manuscript tell us about these two aspects first of all the writing style
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or the script in bhat well over thousands of years we have this culture and tradition of naming the script of
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samskritam after the name of Saraswati Davi so the oldest known script about
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samsam is brahi and from brahmi came sharada and from sharada came sidria and
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from there we have DEA nagari which we are currently using today and this manuscript the bakali manuscript is
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written in sharada script what you’re seeing here on the screen is one of the
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oldest written forms of samsam written in sharada script and here towards the
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left in the white box I gave how the numbers 1 2 3 4 5 6 7 8 9 and Zer all
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the 10 digits are written in Shara script throughout this
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manuscript take a look at this part of the manuscript that I highlighted it is two different arithmetic operations 3 +
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3 and 8 + 8 it’s quite interesting to observe these arithmetic operations from
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a different standpoint let us ask a question why and how the arithmetic symbols that we are using today came
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into existence with this question and this piece of the manuscript that we have on the screen 3 + 3 one written on
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top of the other one and 8 + 8 what does it mean and how did this plus symbol
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came into existence let us try to find it out like I already mentioned couple of
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minutes ago the bakali manuscript is written in sharada writing style or
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sharada script which originated from its predecessor the brahmi script of samrm
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so to understand this 3 + 3 or 8 + 8 from the bakali manuscript the best
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available source for us to rely on is the pillars of Ashoka the edicts of
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Ashoka which date back to third Century BC and is written in brahi and if you
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see on these addicts you have something like a plus symbol where I pointed with
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this white triangles and there’s a bit of Zoom as well towards the right so there is something like this plus symbol
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we don’t know what exactly it means yet but definitely in bakali manuscript there is a plus sign and in the adicts
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of Ashoka there is also a plus sign and both essentially are very related writing styles the pillars of Ashoka are
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in brahmi and the bakali manuscript is in sharada brahmi gave birth to shardas
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script so let’s try and understand what is the meaning of this plus
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symbol luckily the brahi script is well decoded and we can figure out very
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easily that this plus symbol which is on the edicts of Ashoka is actually a
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samsam shda which represents K so the 3 + 3 it’s not a plus sign but it is a
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samsam shabda which is written in brahi and it should be pronounced as 3 c three
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the plus is nothing but C in fact you can try it out the unic code for brahi
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it is + 111013 try out this Unicode in your laptop and you will be able to type the
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brahi letter car which exactly looks like plus but what is the meaning of car
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what is the meaning of three three as written in bhali manuscript that is most
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important for us to understand let’s move forward so finally the cat is out of the
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bag and here is the meaning of three k three the word Ka or the shab KA in
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samsam means diminish or reduce so going top down step by step superficially what
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appears to be like 3 + 3 should be pronounced as three k three and the word
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Ka or the meaning of the shabda KA means diminished so three diminished by 3
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essentially 3 minus 3 so basically it is not 3 + 3 in modern paralin but is
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actually 3 minus 3 in shardas script and here is a snippet of the meaning of this
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shabda from a samsum dictionary so it is not a symbol but a samrm letter what is
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even more important to note here is generally in any language a group of letters form a word and the word has a
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meaning but in samsam single letter individual letters can also hold meaning
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for instance here Ka means to reduce and until and unless one understands what is
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the in-depth meaning of samsum letters one will not be able to decode what’s
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actually written in the scriptures so anyways coming back to this topic so from bakali manuscript it is very clear
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and evident that subtraction as an operation existed in bakali manuscript
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but did negative numbers also exist in bakali manuscript in my research I could not find that yet because we don’t have
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access to the complete baksh manuscript but we cannot rule out the possibility of existence of negative numbers or the
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usage of negative numbers because clearly subtraction as an operation existed in bakali manuscript let me play
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a bit of Devil’s Advocate am I cooking up all this or what is the truth if anyone gets this doubt the first thing
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that he or she should know is or at least try to do is learn about bakali manuscript for yourself and then you’ll
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be able to understand the truth for yourself now if we take a quick look
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back into the rules of division codified by brahmagupta k kakam k which means 0 /
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0 is equal to 0 so we asked that question why did he use C instead of
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shun because in samsum like I said discrete letters single letters also
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hold meanings very specific meanings c means diminish same way C represents Sky
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hope you can find the difference between the sounds the first one is C which means diminished and c means sky so here
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brahmagupta is saying that Sky divided by Sky is equal to Sky so basically it
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means that avoid divided by void is equal to void so this is just to give
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you a picture of how cryptic the language of samsum could be now if we just keep looking only for the
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shabda then one will not be able to understand what is this K kabak it means
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void ided by void is equal to void and here in this context sky is being
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represented as void that’s how cryptic it is so again owing to to the objective of
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this doc film the origins of negative numbers we started with brahmagupta
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where he clearly codified the rules of how to deal with negatives and positives and zero but was brahmagupta the first
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one to do that not necessarily because bakali manuscript also quote the
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operation of subtraction and as a benefit of doubt maybe that negative numbers also existed in bakali
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manuscript which belongs to around 230 CE but is bakali manuscript the oldest
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source of subtraction and by extension negative numbers not necessarily let us
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go further back into time and then we have one of the world’s oldest scripture
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on astronomy Surya santum till today there is no proper dating attributed to
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sua sidhant and there are varied opinions of 1,000 BC 1500 BC 3,000 BC
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some dated in ad so it’s whatever be the case it is the oldest trise on astronomy
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that is available to the humankind at this point in time and this what you’re seeing on your screen is an example of
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how sua santam conceives the foundation for the concept of negative numbers here
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we go the fifth slokum from the second chapter of su santam says that when a
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planet is pulled strongly in their UT stanum epicus the planet’s orbital
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velocity will be increased and this this is called as dham and when they move
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farther their orbital velocity will be decreased and this is called as runam
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dham and runam remember these words these are the exact words used by
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brahmagupta in his codification of positives and negatives so anyways coming back to this one this particular
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slokum is the beginning of orbital velocity calculations of different planets in their orbits and dham is
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referred to as increase in the orbital velocity when it comes closer to the
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object around which any given planetary body is revolving so the highest orbital
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velocity will be achieved when it is closest to the object that it is pulling like I showed here in the diagram
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towards the right and the lowest orbital velocity will be there when it is the
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farthest from the object that it is making it to revolve in the orbit so whenever the Planet increases its
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orbital velocity that increase is called as dham and whenever the planet loses
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its orbital velocity it is called as runam now this definition exactly
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matches with the codification done by brahmagupta like I said dham and runam
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positive and negative are shown as an increase and decrease with reference to
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a stable state which is shunum so brahmagupta and Su santam are perfectly
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in line time and what’s also important to note is Su santam came much much
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before than brahmagupta again it’s not about who is first it’s just a quest to
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know what are the origins of this negative
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numbers so if we sum up all the sources the negative numbers the origins of it
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we can find in brahmagupta’s Brahma santum and then parts of it in bakali
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manuscript and most importantly the foundations of brahmagupta’s work could be found in a text which is much before
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than brahmagupta the sua sidhant which talks about the dham and runam Concepts
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and also gives formulas to calculate the orbital velocities of planets and there
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is a very widespread usage of sign function in those calculations from Suri
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santum which eventually leads to handling negative quantities so coming
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back to the subject what are the origins of of negative numbers I don’t know
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because it is much much deeper than what we could even possibly imagine saying
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that brahmagupta was the first one to constitute the idea of positives and negatives would be a factually incorrect
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statement rather what’s correct is brahmas sputa santam written by brahmagupta gives us a clearcut
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guidelines on how to deal with these arithmetic operations of positives negatives and zeros also ancient China
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contribut to the idea of negative numbers and their arithmetic operations are also well codified but when we
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really want to understand the origins of negative numbers if Chinese mathematics
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on dealing with negative numbers predates sua santum then it is a
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factually right statement to say that the concept of negative numbers was first pioneered by China but if Surya
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santam predates Chinese mathematics on negative numbers then sua santum stays
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at as the origins or the very Genesis of this concept of negative numbers another
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very important thing is it’s very interesting to see how Europe dealt with negative numbers it is quite surprising
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astonishing and amazing that negative numbers are perceived as evil taboo and
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satanic and whatnot until last 200 years ago that was the perception in Europe
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about negative numbers while in bat and China both the civilizations pioneered
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dealing with negative numbers the history of mathematics or the history of science what we read today is absolute
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crap absolute crap the Asian civilizations are totally ignored their
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contributions are totally ignored in many fields and negative number stands as an undisputable scenario where the
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contributions which came in from Barat and China were not known to the world at least the majority of them don’t know it
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and if you start speaking about negative numbers that exposes how hypocritic
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Europe has been let me quote a simple example and you will understand better remember the laws of planetary motions
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defined by John Kepler which we might have studied during our school days here is the first law of John Kepler on laws
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of planetary motion the orbit of a planet is an ellipse with the Sun at one
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of it FY so basically here the first law of planetary motion defines that all the
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orbits of planet are elliptical and this is revered as the first discovery that
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the orbits of planets are elliptical and not circular by the Western historians
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before I move any further let me clarify one point absolutely no disregard or
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disrespect towards John skler and his findings that’s not my intention in fact
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I want to expose the disregard and disrespect shown by Western historians
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towards the real history so why would I disrespect John skler in first place I want to expose such kind of hypocrisy
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and here is it you check any ancient BH scriptures on astronomy like Su sidhant
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aryaa brahmas santam everywhere the planet’s orbit is defined as an ellipse
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elliptical orbits are very common in fact the very sloka that we just saw for
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dham and runam clearly defines that the orbit of a planet is elliptical
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thousands of years before John Kepler did but nowhere that we would learn
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about these facts in any of our schools all we still learn is that John skler
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first discovered that planets revolve in elliptical orbits again I repeat no
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disrespect towards John skler all I want to expose is the hypocrisy that’s staring into our face so use your common
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sense and go beyond your textbooks and read the real history in an unbiased
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format and then you’ll get to know what is the truth so that is a very brief
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history about the origins of negative numbers and this is the third episode of
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the bhartia ganam web series from Project shiv I get a lot of emails and
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DMS where people are saying that bhara ganum they are really liking this web series thank you so much for your
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compliments but i’ would be more happy if you not just watch this videos but get to the actual scriptures read for
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yourself and spread the truth so that every student in bhat and across the world gets to know the real
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contributions that came in from bat in the field of mathematics we’ll be coming up with more episodes of BH ganit but
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for now and as always thanks for watching
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[Music]