The real numbers form an ordered field, satisfying both algebraic axioms (closure, commutativity, associativity, distributivity, identity and inverse elements) and order axioms (total order ≤ compatible with addition and multiplication). These axioms provide the foundational rules for all manipulations in real analysis.
Every algebraic manipulation you have ever performed — moving terms across an equals sign, multiplying both sides by a constant, factoring — was justified by some combination of a small list of rules. The field axioms name these rules explicitly. A field is a set with two operations (addition and multiplication) where both operations are commutative and associative, multiplication distributes over addition, and every nonzero element has a multiplicative inverse. The rationals ℚ and the reals ℝ both satisfy these. What this buys you in analysis is certainty: proofs can cite specific axioms rather than relying on informal arithmetic intuition.
The order axioms add a compatible comparison structure. The relation ≤ must be a total order (any two elements are comparable), and it must interact sensibly with the algebraic operations: if a ≤ b then a + c ≤ b + c for any c, and if a ≤ b and 0 ≤ c then ac ≤ bc. These two compatibility conditions encode why the intuitive sign rules work. For example, multiplying both sides of an inequality by a negative number reverses the direction — a consequence of the order axioms, not a separate memorized rule.
The rationals also form an ordered field, so why do we need the reals? The answer is gaps. In ℚ, the set {x : x² < 2} is bounded above (by 2, say) but has no least upper bound in ℚ — there is no rational number that is the smallest rational ≥ all elements of the set. The equation x² = 2 has no rational solution; there is a "hole" where √2 should be. The ordered field axioms alone do not close these gaps. That is precisely the job of the next axiom: completeness (the least-upper-bound property), which asserts that every nonempty set bounded above has a supremum in ℝ. The ordered field structure you are studying now is the foundation; completeness is what distinguishes ℝ from ℚ and makes calculus coherent.
This is a foundational topic with no prerequisites.
No prerequisites — this is a starting point.