Show me content of introduction of algebra and number theory

Dear student,

The introduction of algebra and number theory is a vast topic, which, I'm afraid cannot be contained here alone. 

But, let me tell you about the fundamental basis, core concepts, or underlying principles that define each field, the document outlines them:

For Algebra:

• Basis: The generalisation of arithmetic. It moves from specific numbers to abstract symbols (variables) to represent quantities and relationships.
• Core Concepts/Formulas:
  • Variables: Using letters (e.g., x,y) to represent unknown or changing values.
  • Expressions: Combinations of variables, constants, and operations (e.g., 2x+5).
  • Equations: Statements of equality between expressions (e.g., A=l×w for area, or solving 2x+5=11). The "calculation" in algebra often involves manipulating these equations to solve for variables.
  • Operations: The fundamental arithmetic operations (addition, subtraction, multiplication, division) extended to variables.

For Number Theory:

• Basis: The study of the properties and relationships of integers (whole numbers and their negatives).
• Core Concepts/Formulas:
  • Divisibility: The concept of one integer dividing another without a remainder.
  • Prime Numbers: Integers greater than 1 divisible only by 1 and themselves. This is a foundational concept.
  • Fundamental Theorem of Arithmetic: The unique prime factorization of every integer greater than 1. This is a cornerstone "formula" or principle.
  • Modular Arithmetic: A system of "clock arithmetic" based on remainders, expressed as congruences (e.g., a≡b(modn)).

In essence, the "basis" of algebra lies in its abstract notation and equation-solving methods, while the "basis" of number theory is the deep and often surprising properties discovered within the set of integers.