Important Number Sets to Know for Intro to Abstract Math

Understanding important number sets lays the foundation for abstract math. These sets, from natural numbers to complex numbers, help us grasp how numbers interact, their properties, and their roles in various mathematical concepts. Each set builds on the previous one.

1. Natural Numbers (ℕ)

   • Defined as the set of positive integers starting from 1: {1, 2, 3, ...}.
   • Used for counting and ordering.
   • Closed under addition and multiplication, but not under subtraction or division.

2. Whole Numbers (W)

   • Includes all natural numbers plus zero: {0, 1, 2, 3, ...}.
   • Useful in contexts where zero is a valid count (e.g., empty sets).
   • Closed under addition, multiplication, and also includes zero in subtraction.

3. Integers (ℤ)

   • Comprises all whole numbers and their negative counterparts: {..., -3, -2, -1, 0, 1, 2, 3, ...}.
   • Allows for the representation of both positive and negative quantities.
   • Closed under addition, subtraction, and multiplication.

4. Rational Numbers (ℚ)

   • Defined as numbers that can be expressed as the quotient of two integers (a/b, where b ≠ 0).
   • Includes integers, fractions, and finite or repeating decimals.
   • Closed under addition, subtraction, multiplication, and division (except by zero).

5. Irrational Numbers

   • Numbers that cannot be expressed as a simple fraction; their decimal expansions are non-repeating and non-terminating.
   • Examples include √2, π, and e.
   • Important for understanding limits and continuity in calculus.

6. Real Numbers (ℝ)

   • The set that includes all rational and irrational numbers.
   • Represents all points on the number line, encompassing both positive and negative values.
   • Closed under addition, subtraction, multiplication, and division (except by zero).

7. Complex Numbers (ℂ)

   • Comprises numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1).
   • Extends the concept of numbers to include solutions to equations that have no real solutions (e.g., x² + 1 = 0).
   • Closed under addition, subtraction, multiplication, and division (except by zero).

8. Algebraic Numbers

   • Numbers that are roots of non-zero polynomial equations with rational coefficients.
   • Includes all rational numbers and some irrational numbers (e.g., √2).
   • Important in fields such as number theory and algebra.

9. Transcendental Numbers

   • Numbers that are not algebraic; they cannot be roots of any polynomial equation with rational coefficients.
   • Examples include π and e, which have significant implications in mathematics and science.
   • Highlight the distinction between algebraic and non-algebraic numbers in advanced mathematical studies.