The equation \(1 + 1 = 2\) is a basic arithmetic fact that forms the foundation of our number system. Here’s a detailed explanation:

1. **Definition of Numbers**: In mathematics, numbers are defined and used according to certain rules and axioms. The number \(1\) is the first positive integer, and \(2\) is the next integer after \(1\).

2. **Peano Axioms**: The Peano axioms provide a foundation for the natural numbers. One of these axioms states that every natural number has a unique successor. According to these axioms:
   - \(1\) is a natural number.
   - The successor of \(1\) is \(2\).

3. **Addition**: Addition is a basic operation in arithmetic. According to the definition of addition, adding \(1\) to a number means finding the successor of that number. 
   - If we start with \(1\), adding \(1\) means finding the successor of \(1\), which is \(2\).

4. **Intuitive Understanding**: From an intuitive perspective, if you have one object and you add another object to it, you end up with two objects. This is visually and conceptually straightforward.

5. **Symbolic Representation**: The symbols we use (\(1\), \(+\), \(=\), \(2\)) are part of a formal system to represent these ideas. When we write \(1 + 1 = 2\), we're using this symbolic language to express a basic truth about numbers and addition.

In summary, \(1 + 1 = 2\) because, by the rules and definitions of our number system and arithmetic operations, adding the number \(1\) to itself yields the number \(2\).

Would you like more details or have any questions?

Here are some related questions:

1. What are the Peano axioms?
2. How is addition defined for natural numbers?
3. What are natural numbers and their properties?
4. How is the concept of number defined in mathematics?
5. Can you explain the successor function in the context of natural numbers?
6. How are arithmetic operations like addition formalized in mathematics?
7. What is the historical development of the number system?
8. How does the concept of addition differ in various number systems?

**Tip:** Understanding the foundational axioms of mathematics, such as the Peano axioms, helps to grasp why basic arithmetic operations work the way they do.

Explore why 1 + 1 equals 2 in arithmetic through foundational concepts like number systems, Peano axioms, and basic arithmetic operations. Gain a deeper understanding of how numbers are defined and the rules governing addition. Suitable for students in Grades 7-9.

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