Concept Of Numbers
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भास्कराचार्य (Bhaskaracharya) in his work, बिजगणितं (Bijaganitam), mentioned that how the numbers exist
रूपत्रयं means “an appearance of three quantities (3)”. The main thing to notice here is “रूप” meaning “appearance”.
स्वं means “something that belongs to you; something that you are to acquire, or you acquired”
क्षयगं means “something that you are to lose or lost; something that is to be removed”
खं means “a sense of nothing”.
Here, Bhaskaracharya clearly states that a quantity is neither positive nor negative, but it can be represented by an appearance only after certain action is performed on them or bounded to them, which we call as numbers, and these numbers reflects the actual measurement that we take.
One important thing to note which Brahmagupta did not explicitly address, yet his insights imply—is this: if there is only negative number, then what does it represent? To remove something, we must have something.
The above insight is crucial for understanding ऋण संख्या (negative number), a concept that many पश्चिम गणितज्ञाः (Western mathematicians) have struggled to fully grasp.
When we talk about behaviors, this concept makes more sense. If we assign a tendency to go out of the system, then the first thing we must do is to observe a quantity to remove. As soon as you observe that quantity, it came under your measurement space and should be removed. So, a quantity can be removed out of the system only after it comes inside the system. What negative numbers tell us is that whenever a quantity enters our system, it must be removed.
“To remove something, you must have something!”
So, what happens if it does not leave the measurement space? Yes, it will become the part of the measurement. This is why multiplying a negative with a negative result in a positive and it behaves like a positive. This is the most foundational explanation you can think of.
Positive Numbers: The observed quantity must come inside the measurement space
Negative Numbers: The observed quantity must go outside the measurement space
In other words, a quantity can be assigned with two types of nature represented by numbers. Basically, two distinct natures is discussed by Bhaskaracharya, Brahmagupta and many others. These are स्वं (positive) and क्षयगं (negative).
A number (संख्या) is just an appearance, representing the magnitude of a quantity with certain action involved, enlightening us with the most fundamental measurement that an observer can take.
What do we mean by संख्या (Samkhya) is quite interesting:
“संख्या” is a Saṃskṛt word which can be understood as “number” in English. However, the word ‘संख्या’ is not the same as the word ‘number’. The English word ‘number’ comes from the Middle English ‘nombre’. This, in turn, comes from the Old French word ‘nombre’. The Old French term is derived from the Latin word ‘numerus’, which means "quantity or amount".
When we split the word संख्या in its root forms, we get:
सं → "Sam" (prefix indicating completeness or totality)
ख्या → "Khya" (root word meaning to know, declare, or enumerate)
Hence, संख्या is itself representing a measurement, the most fundamental measurement, indicating the totality till now and then declaring it by a sound or a letter.
What a beautiful word! It conveys the exact meaning of the concept. This small word has a deep meaning confined to it.
Quantity can be represented by a visual appearance or a symbol. Measurement of a physical quantity involves adding to or removing from a system.
What do I mean by appearance?
Six apples are simply represented as 6 apples, and a hundred lemons as 100 lemons.
Here, “6” is the appearance representing the quantity of Apples and “100” is the appearance representing the quantity of Lemons.
Since when you started counting those apples, you consider them as inside your system represented by positive numbers.
Negative numbers, in themselves, lack a tangible existence in the physical realm in modern mathematics and serve primarily as conceptual tools rather than representations of reality. But this is not true.
When a quantity is removed, what removed is the quantity itself. To represent this removal, a special notation—such as a dot placed above the symbol used to represent a positive number—can be used to signify the removal. This approach underscores that negative numbers can have an independent existence just like positive numbers and can serve as symbolic representations of a removal in the measured quantity.
Hence, Negative numbers can attain their own significance, or their own appearance just as positive numbers do. Positive and negative are just the behaviours bind to a quantity. Negative numbers too reflect measurable entities in the same direct manner. They are those quantities themselves but with opposite actions performed on them (a different nature opposite to positive). This perspective reinforces the idea that positive numbers are not the only one bearing the foundation of all physical measurements and representations. (Will be discussed later in this paper).
This removal effect of quantities can be given a separate identity, representing quantities with opposite behaviour. It is an approach to represent the same physical quantity with different perspective. So, negative numbers represents the physical quantities like positive numbers do but assigned with opposite nature.
Hence, the initial state is but after observing a quantity (+1), the measurement state transforms to 0.
But what does signifies? Well, it talks about the removal of . Remember the statement?
tells that whenever a quantity is observed it should be removed from the measurement space but tells that whenever a quantity is observed, it should not be removed from the measurement space. Basically, the negative sign (-) indicates the reversal of behaviour.
For example, if you have seven apples (7) and you consume three, this removal of apples can be represented as , where the dot notation signifies the removal or reduction of three apples. In this context, does denote a standalone negative number which fundamentally represent the act of removing three from the existing seven.
