Axioms for the Real Numbers

Axioms for the Real Numbers

• The real numbers form a field, denoted by $\mathbb{R}$.

  • This means that the real numbers are a set with two operations, addition (+) and multiplication (×), satisfying the following axioms:
    • Closure: For all real numbers $a$ and $b$, $a + b$ and $a \times b$ are real numbers.
    • Commutativity: For all real numbers $a$ and $b$, $a + b = b + a$ and $a \times b = b \times a$.
    • Associativity: For all real numbers $a$, $b$, and $c$, $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$.
    • Distributivity: For all real numbers $a$, $b$, and $c$, $a \times (b + c) = (a \times b) + (a \times c)$.
    • Additive identity: There is a unique real number 0 such that for all real numbers $a$, $a + 0 = a$.
    • Multiplicative identity: There is a unique real number 1 such that for all real numbers $a$, $a\times 1 = a$.
    • Additive inverse: For every real number $a$, there is a unique real number $-a$ such that $a + (-a) = 0$.
    • Multiplicative inverse: For every nonzero real number $a$, there is a unique real number $\frac{1}{a}$ such that $a \times \frac{1}{a} = 1$.

• The real numbers are ordered.

  • This means that there is a relation $<$ (less than) that satisfies the following:
    • For any real numbers $a$ and $b$, exactly one of the following holds: $a<b$, $a=b$, or $a>b$.
    • If $a<b$ and $b<c$, then $a<c$.
    • If $a<b$, then $a+c<b+c$ for any real number $c$.
    • If $a<b$ and $c>0$, then $ac<bc$.

• The real numbers are complete.

  • This means that every non-empty set of real numbers that is bounded above has a least upper bound (supremum).
  • Equivalent statement: Every non-empty set of real numbers that is bounded below has a greatest lower bound (infimum).

• The field of real numbers $\mathbb{R}$ contains the field of rational numbers $\mathbb{Q}$.

  • That is, $\mathbb{Q}$ is a subset of $\mathbb{R}$.
  • $\mathbb{Q} \subset \mathbb{R}$

• The real numbers satisfy the existence of a square root for every non-negative real number.

  • That is, for every $x \geq 0$, there is exactly one non-negative real number $y$ such that $y^2=x$. This root is denoted by $\sqrt{x}$.

• The real number line can be constructed.

  • This involves geometric and algebraic development, to visualize the order and completeness properties.
    • It allows every real number to be assigned a point on a line.
    • This construction is crucial for understanding the geometric interpretation of inequalities.

Key Differences of Real Numbers and Rational Numbers

• Real numbers include rational numbers and irrational numbers (numbers that cannot be expressed as a fraction of two integers).
• Real numbers exhibit completeness; rational numbers do not.
• Rational numbers are countable, and real numbers are uncountable.

Consequences and Importance

• The axioms provide a fundamental framework for the properties of real numbers.
• The axioms are essential for understanding and working with real number calculations, equations, and inequalities.
• These axioms form the basis of mathematical analysis, calculus, and other advanced mathematical fields. The order axiom, for instance, allows us to model and discuss concepts such as intervals and inequalities.
• The completeness axiom lets us handle concepts like limits and continuity in a rigorous way.
• Real-world problems, including those in engineering, physics, and economics, often involve real numbers, emphasizing the significance of these basic properties.

Test your understanding of the axioms governing the field of real numbers. This quiz covers key properties such as closure, commutativity, associativity, and more. Perfect for students looking to strengthen their knowledge in real analysis.