# Mathematics: A Refined Explanation of Basic Additions

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## Chapter 1: Understanding Basic Arithmetic

When we think of simple additions like 1 + 1 = 2 or 2 + 2 = 4, they might seem trivial—calculations a child could easily perform. However, there is a rigorous mathematical framework established by past scholars that underpins the truth of these equations. Dive into this article to wow your friends and family with your newfound knowledge!

Mathematics is characterized by its precision and grace, and this article will showcase those qualities.

### Chapter 1.1: The Peano Axioms

The Peano axioms, introduced by the 19th-century mathematician Giuseppe Peano, are fundamental principles in mathematical logic concerning natural numbers. Axioms are foundational statements assumed to be true, serving as the groundwork for further logical reasoning and arguments.

To grasp why 1 + 1 equals 2, we must first clarify the meaning of the equality sign. This is crucial to avoid any potential misunderstandings.

The equality sign '=' is simply a symbol until we define its properties.

Below are the four axioms that will guide our understanding:

- The equality sign '=' signifies that any natural number is equal to itself.
- For any two natural numbers x and y, if x equals y, then y also equals x. This is known as the symmetry axiom.
- For any three natural numbers x, y, and z, if x equals y and y equals z, then x equals z. This is referred to as the transitive axiom.
- For any natural numbers x and y, if x equals y, then both belong to the set of natural numbers.

These initial axioms are akin to the ‘baby’ Peano axioms, as they lay the groundwork for understanding what constitutes a ‘number’.

### Chapter 1.2: Defining Natural Numbers

- The number 0 is considered a natural number. Some mathematicians may exclude zero, but we will adhere to the convention that it is included.
- For every natural number x, its successor S(x) is also a natural number. Here, S(x) can be thought of as x + 1, although we must avoid using addition until we clarify its meaning.
- The successor of any natural number x cannot be zero. Thus, S(0) is defined as 1.
- If S(x) equals S(y), then x must equal y, which is essential for maintaining the structure of natural numbers.

Through this reasoning, we can conclude that there are at least two natural numbers: 0 and 1. To confirm the existence of additional natural numbers, we introduce a fifth axiom:

## Axiom 5: Induction

This axiom allows us to build upon the previous four, showing that:

- S(0) = 1
- S(S(0)) = S(1) = 2, and so forth.

This method is known as induction, enabling us to define the entire set of natural numbers.

## Chapter 2: What Does Addition Represent?

Now that we have established the foundation of natural numbers, we can define what addition means:

- Adding any number to 0 yields that number.
- When adding a number a to the successor of b, the result is the successor of (a + b).

To illustrate, let’s calculate 1 + 1:

1 + 1 can be rewritten using the successor function. Here’s how we can demonstrate this step-by-step:

- We know that 1 is the successor of 0.
- According to our rules, the successor of 1 (the result of 1 + 1) is defined as 2.

This confirms that 1 + 1 equals 2.

To extend this concept, we can also verify that 2 + 2 equals 4 by applying the same principles. Start with the addition of 2 and 2, using our defined rules and the successor function.

In the end, we see that this lengthy exploration has revealed a simple yet profound truth about arithmetic. Now that we’ve laid down the basics, take a moment to step outside and refresh your mind before diving deeper into the world of mathematics.

Thank you for accompanying me on this journey!

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