Numbers perhaps were the earliest mathematical abstraction by man. How man first came to use numbers is still shrouded in mystery. But by all evidence it seems it was the earliest attempt by man to keep count of his livestock. Often he placed a stone against every animal that left his cave in the morning, when the cattle returned to his cave after a day’s grazing he casts away a stone, that he collected in the morning against his cattles, for every animal that entered the cave. If some stones remained in the heap without being cast for an animal then he understood that some of his animals are still at large. Lines drawn on a bone of a young wolf dated 30,000 years back discovered in Czechoslovakia in groups of five suggest man perhaps had started using his figures for counting. It is most likely that number signs were discovered before the number words as it is easier to draw a line on stone, cut a notch on a piece of wood; rather than creat a well modulated
All these suggest that man discovered numbers to keep count of things. In a way it sound to identify numbers. was the “cardinal” aspect of numbers. But anthropological studies suggest an alternative ordinal approach. It is suggested that order of precedence during ancient religious rituals led to invention of numbers. This is an interesting theory but yet to be established. Till now we do not know for certain how it all began.
Once it began, the mankind must have experienced enormous difficulty in writing and pronouncing the symbols. Egyptians, Babylonian’s, Mayans, Romans, Chinese, Indians developed these method of symbols of numeration. The concept of zero originated in India along with decimal system of enumeration and powerful place value system. The bulk of Indian mathematics were generated and perfected largely by the following Indian mathematicians :
Aryabhata (476-550 CE): Credited with introducing the decimal place-value system and the concept of zero as a placeholder. In his work Aryabhatiya, he calculated an accurate approximation of \(\pi\) (3.1416) and developed methods for trigonometric functions, square roots, and cube roots.
Brahmagupta (598-668 CE): A mathematician and astronomer who wrote the Brahmasphutasiddhanta, in which he treated zero as a number in its own right, establishing rules for its use in calculations (addition, subtraction, multiplication, and division) and introducing negative numbers.
Baskara I (c. 600-680 CE): A prominent commentator on Aryabhata’s work who created a close rational approximation for the sine function.
Bhaskara II (1114-1185 CE): Also known as Bhaskaracharya, he wrote Siddhanta Shiromani, which covered algebra, arithmetic, and astronomy, and he further explored the concept of zero and division, suggesting that division by zero yields infinity.
Madhava of Sangamagrama (c. 1350-1425 CE): Founder of the Kerala School of Mathematics, who made significant contributions to infinite series, trigonometry, and the development of power series expansions for \(\pi\).
Varahamihira (6th Century CE): A noted astronomer and mathematician who wrote the Pancasiddhantika, a significant text on astronomical, trigonometric, and mathematical calculations.
Pingala (around 200 BCE): Known for his Chhandas-Sutra, where he described a binary system and, according to some interpretations, early concepts related to the decimal place-value system.
These scholars, working largely in Sanskrit, built the foundation for the decimal, place value, and zero-based system that defines modern mathematics.
The Evolution of Zero
It is worth noting that while many civilizations (like the Mayans or Babylonians) used a placeholder to signify “nothing,” the Indian mathematicians were the first to treat Zero (Shunya) as a number in its own right-a value that could be added, subtracted, and multiplied.
This shift from a mere “gap” to a mathematical “object” is what allowed for the development of the place value system we use globally today. Without it, the complex calculations required for modern physics and computer science would be nearly impossible to write down.
Natural Numbers and Integers:
Based on the text provided, the set of natural numbers, denoted by \(\mathbb{N}\) or \(\mathbb{N}^*\), is defined using the Peano Axioms to establish a rigorous, non-circular foundation for counting and arithmetic.
The set of natural numbers (denoted by \(\mathbb{N}\)) consists of the positive integers used for counting, starting from 1 and continuing infinitely: \(\{1,2,3,4,5, \ldots\}\). While some mathematicians include 0 (making them whole numbers or non-negative integers), the traditional definition, and the one most commonly used for “counting numbers,” begins with 1.
Here is an analysis of the rules stipulated in the text:
Successor Function \(\left(n^{+}\right)\): Every natural number \(n\) has a unique successor, denoted by \(n^{+}\), which acts as the “next” number. Consequently, \(n\) is considered the predecessor of \(n^{+}\).
Key Components
Examples:
Basic Examples:
Successor of \(0: 0^{+}=0+1=1\)
Successor of \(1 : 1^{+}=1+1=2\)
Successor of \(5: 5^{+}=5+1=6\)
Predecessor Examples:
If \(n^{+}=10\), then \(n\) (the predecessor) is 9.
If \(n^{+}=51\), then \(n\) (the predecessor) is 50.
The Unique Zero \((0)\) : There exists a unique element 0 in \(\mathbb{N}^*\) that acts as the starting point. It is the only element that has no predecessor, meaning no natural number \(n\) exists such that \(n+1=0\).
Uniqueness of Predecessors (Injectivity): If the successors of two numbers are the same ( \(n^{+}=m^{+}\)), then the numbers themselves are the same ( \(n=m\) ). The Uniqueness of Predecessors, often referred to as the injectivity of the successor function (Axiom 4 in many formulations of Peano’s Axioms), states that if the successors of two natural numbers (\(n\) and \(m\)) are the same, then the numbers themselves are the same:
If \(S(n)=S(m)\), then \(n=m\)
Example: \((\mathbf{S}(n)=n+1)\) :
If the “next number” (successor) of \(n\) is 5 , and the “next number” of \(m\) is also 5 , then \(n\) and \(m\) must be 4.
\(S(4)=5\)
\(S(m)=5\)
Therefore, \(m=4\).
Principle of Mathematical Induction: If a subset \(A\) of \(\mathbb{N}^*\) contains 0 (i.e., \(0 \in A\)) and the inclusion of a number \(n\) in \(A\) implies that its successor \(n^{+}\)is also in \(A\) (\(n \in A \Longrightarrow n^{+} \in A\)), then \(A\) must be the entire set of natural numbers (\(A=\mathbb{N}^*\)).
It proves statements true for all natural numbers by establishing a base case (0) and showing that if the statement holds for \(n\), it holds for \(n+1\) (inductive step), acting like falling dominoes.
Examples of Induction:
Sum of Natural Numbers: To prove \(0+1+2+\ldots+n=\frac{n(n+1)}{2}\) for all \(n \in \mathbb{N}^*\).
Base Case: Check for \(n=0: 0=\frac{0(0+1)}{2}=0\). The equation holds.
These axioms formally define the set \(\mathbb{N}^*=\{0,1,2,3, \ldots\}\), allowing for the definition of operations like addition and multiplication, which forms the basis of familiar arithmetic.
Addition
(i) \(m+0=m \quad \forall m \in \mathbb{~N}^*\)
(ii) \(m+n^{+}=(m+n)^{+}\)
The rule \(m+n^{+}=(m+n)^{+}\)tells us that adding the successor of \(n\) is the same as taking the successor of the sum \((m+n)\).
Example: To find \(2+1\) :
1. We know 1 is the successor of 0 , so \(1=0^{+}\).
2. Therefore, \(2+1=2+0^{+}\).
3. By your rule (ii), \(2+0^{+}=(2+0)^{+}\).
4. By your rule (i), \(2+0=2\).
5. So, \(2+1=2^{+}\), which we call 3.
So by principle of mathematical induction stated above one sees that the sum \(m+n\) is defined for every \(m, n \in \mathbb{~N}^*\)
Using the above definition it is not hard to show that
(i) \(m+n=n+m~ \forall m, n \in \mathbb{~N}^*\)
(ii) \((m+n)+p=m+(n+p)~ \forall m, n, p \in \mathbb{~N}^*\)
(iii) \(m+0=m~ \forall m \in \mathbb{~N}^*\)
(iv) \(m+p=n+p \Rightarrow m=n~ \forall m, n, p \in \mathbb{~N}^*\)
Order in \(\mathbb{N}^*\). There is an ordering in \(\mathbb{N}^*\) which is easily discernible :
(i) \(0<n\) if \(n \in \mathbb{~N}^*-\{0\}\)
(ii) \(n<n^*~ \forall n \in \mathbb{~N}^*\)
(iii) \(m<n \Rightarrow m^{+}<n\) or \(m^{+}=n~, \forall m, n \in \mathbb{N}^*\)
Examples of Ordering
1. Comparing 3 and 5: We know that \(3+2=5\). Since 2 is a natural number (and \(2 \neq 0\)), we conclude that \(3<5\).
2. The Role of Zero: Since any \(n \in \mathbb{N}^*\) (where \(n \neq 0\)) can be written as \(0+n=n\), it follows that \(0<n\) for all non-zero natural numbers. Zero is the “least” element in this set.
Properties of the Order
The ordering in \(\mathbb{N}^*\) follows three very important logical rules:
1. Trichotomy Law: For any two numbers \(m\) and \(n\), exactly one of these must be true:
\(m<n\)
\(m=n\)
\(m>n\)
2. Transitivity: If \(m<n\) and \(n<p\), then \(m<p\).
Example: Since \(2<5\) and \(5<10\), then \(2<10\).
3. Monotonicity: If \(m<n\), then \(m+p<n+p\) for any \(p\).
Example: Since \(3<7\), then \(3+10<7+10(13<17)\).
Subtraction
Given \(m, n \in \mathbb{~N}^*\) and \(m \geq n\) there is a unique element \(d \in \mathbb{~N}^*\) such that \(n+d=m\)
We write \(d=m-n\). With this new notation we can now answer the restricted inverse problem : which number added to \(m\) gives \(n\) ? This has an answer if \(m \leq n\).
Examples in \(\mathbb{N}^*\)
1. A Valid Subtraction (When \(m \geq n\) ): Suppose we want to find \(5-3\).
Here \(m=5\) and \(n=3\).
Since \(5 \geq 3\), we look for a \(d \in \mathbb{N}^*\) such that \(3+d=5\).
We find \(d=2\), because \(3+2=5\).
Thus, \(5-3=2\).
2. The Identity Case: Suppose we want to find \(m-m\).
We look for a \(d\) such that \(m+d=m\).
From your addition rule (i), we know \(m+0=m\).
Therefore, \(d=0\). So, \(m-m=0\) for any natural number.
The “Restricted” Inverse Problem
As you noted, the question “Which number \(x\) added to \(m\) gives \(n\) ?” (mathematically \(m+x= n\)) only has an answer in \(\mathbb{N}^*\) if \(m \leq n\).
Example of the Restriction:
If we ask: “What number added to 10 gives 7?”
Equation: \(10+x=7\).
If we search within \(\mathbb{N}^*=\{0,1,2,3, \ldots\}\), we find no such number.
Even the smallest natural number, 0 , results in \(10+0=10\), which is already greater than 7.
This “failure” of the system to provide an answer for all cases is exactly why mathematicians had to invent Integers (\(\mathbb{Z}\)).
Transition to Integers
To solve \(m+x=n\) for any \(m\) and \(n\), we introduce negative numbers. If we define the integer -3, we can finally solve the problem above:
\(
10+(-3)=7
\)
Multiplication
For \(m \in \mathbb{~N}^*\) define
\(
\begin{aligned}
& m \cdot 0=0 \\
& m \cdot n^{+}-m \cdot n+m
\end{aligned}
\)
Additive inverse problem can be solved if we extend the natural numbers to the set of integers \(\mathbb{Z}=\{0, \pm 1, \pm 2, \ldots \ldots\}\).
Example: Proving \(2 \cdot 2=4\)
Using your axioms, let’s calculate \(2 \cdot 2\) (where 2 is \(1^{+}\)and 1 is \(0^{+}\)):
1. Base Rule: \(2 \cdot 0=0\)
2. Next Step: \(2 \cdot 1=2 \cdot 0^{+}\). According to your rule, this is \((2 \cdot 0)+2\).
\(0+2=2\). So, \(2 \cdot 1=2\).
3. Final Step: \(2 \cdot 2=2 \cdot 1^{+}\). According to your rule, this is \((2 \cdot 1)+2\).
\(2+2=4\).
Solving the Additive Inverse Problem
As you noted, the “inverse problem” for addition asks: Given \(a\) and \(b\), find \(x\) such that \(a+x=b\).
Examples:
The Problem \((a+x=b)\) Solution in \(\mathbb{N}^*\) Solution in \(\mathbb{Z}\) Explanation
\(
5+x=8 \quad x=3 \quad x=3, \text { Since } 8>5, \text { a natural number exists. }
\)
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