Concept Of Numbers

Numbers

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 bound 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.

Hence, the initial state is $\dot{1}$ but after observing a quantity (+1), the measurement state transforms to 0.

But what does $-\dot{1}$ signifies? Well, it talks about the removal of $\dot{1}$. Remember the statement?

“To remove something, you must have something!”

$\dot{1}$ tells that whenever a quantity is observed it should be removed from the measurement space but $-\dot{1}$ 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.

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.

Hence, (+) means to retain the behavior and (-) means to inverse the behavior.

What exactly are positive numbers?

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.

What exactly are negative 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).

For example, if you have seven apples (7) and you consume three, this removal of apples can be represented as $\dot{3}$, where the dot notation signifies the removal or reduction of three apples. In this context, $\dot{3}$ does denote a standalone negative number which fundamentally represent the act of removing three from the existing seven.

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.

A Great Misconception

Take any one of your hands and make a fist. Now let us count, how many fingers do you have.

1, 2, 3, 4 and 5. You have 5 fingers in total.

Hey! What was the last finger you called?

Was it 5?

Is it actually 5?

No! That was your 5th finger and here we must acknowledge the basic difference between cardinals and ordinals. The below diagram explains this well.

Ordinals

Cardinals

Types of calculative Mathematics in ancient Bharat

Two types of calculative mathematics were developed one after another in ancient Bharat:

• तत्कालगणितम् (Tatkālagaṇitam) → Mathematics of present moment (In modern context, you can relate this to the framework of monoids)

• समग्रगणितम् (Samagragaṇitam) → All types of Mathematical frameworks

Tatkālagaṇitam represents the most basic form or fundamental framework of mathematics, while Samagragaṇitam is considered its advanced or upgraded version.

Negative numbers are not required for Tatkālagaṇitam because it focuses on calculations relevant to the present moment, emphasizing actions rather than behaviors. Behavior analysis is unnecessary in this context, as predictive mathematics, which deals with future and past events, necessitates behavior consideration. Although all calculations performed by Tatkālagaṇitam can be executed using Samagragaṇitam, with the advancement of mathematics, this framework was replaced to enable more sophisticated calculations.

Let us understand this with some examples:

Suppose you are a member of a tribe and hunted some animals along with your 3 friends. Now, you want to know the total quantity you all hunted together. You will just have to add them up and for this you must do some actions, not behaviors. Initial state must be 0 and then progressively you perform the incoming action (action of adding quantities into the measurement space) to calculate the sum.

Now, suppose your tribe ate a bear, a snake and a reindeer. Then you must remove them from your measurement.

Now, assume your tribe has hunted and stored enough food for today. If your tribe plans to hunt three animals tomorrow and wants an organized approach to track progress, it is important to record measurements to compare the planned activities with the actual achievements.

Now comes the role of behaviors.

What if 3 animals are added to the current measurement? It will give the estimate for tomorrow. So, those 3 animals tend to come inside our measurement system (an incoming behavior).

Now, imagine a situation where the tribe has no food for the next day and has utilized all hunted animals today. If they require two deer and a bear to avoid hunger, how would you note this requirement? The solution involves using a new type of number that represents an inverse quantity. Previously, the animals entered your measurement system (incoming behavior), but now they tend to leave your measurement system (outgoing behavior), necessitating a different representation. This means that whatever animals are hunted tomorrow, three of them must be consumed to prevent hunger. The tribe currently lacks three animals.

Now assume that the hungry members of your tribe did not survive the hunger. Since everyone is now well-fed, any animals hunted will not be consumed immediately and will remain stored. Therefore, the three animals that are to be hunted by your tribe will stay in your stock, reflecting an addition to your measurement system.

Hence, the removal of outgoing behavior results in incoming behavior.

It is evident from the above examples that all the calculations of Tatkālagaṇitam can be done or achieved by Samagragaṇitam only. So, from now on we will only discuss Samagragaṇitam.

It is worth noting that Tatkālagaṇitam is way more advance than you think. You can extend this concept to matrices and other domains as well because this is the very foundation of the entire mathematical framework. We will surely discuss it later.

Conclusion on Numbers

• 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 measurement that an observer takes.

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.