Math 2.0

Field theory and polynomials Proof

Proof that the complex numbers cannot be ordered

Builds on Ordered fieldi squared is minus oneTotal orderPartial orderThe complex numbers form a fieldComplex numbers

Proof

We argue by contradiction.

  1. Assume that ≤\le is a total order on ℂ\C that makes ℂ\C an ordered field. Its two axioms are: a≤ba \le b implies a+c≤b+ca + c \le b + c, and 0≤a0 \le a, 0≤b0 \le b imply 0≤ab0 \le a b.
  2. Claim: for each x∈ℂx \in \C, we have (−x)(−x)=xx(-x)(-x) = x x. Write x=p+qix = p + q i with p,qp, q real. Then −x=(−p)+(−q)i-x = (-p) + (-q) i. The product rule for complex numbers gives (−x)(−x)=(p2−q2)+2pq i=xx(-x)(-x) = (p^2 - q^2) + 2 p q \, i = x x.
  3. Claim: 0≤xx0 \le x x for each x∈ℂx \in \C. Since the order is total, 0≤x0 \le x or x≤0x \le 0.
  4. If 0≤x0 \le x, then the second axiom with a=b=xa = b = x gives 0≤xx0 \le x x.
  5. If x≤0x \le 0, the first axiom with c=−xc = -x gives 0≤−x0 \le -x. The second axiom gives 0≤(−x)(−x)0 \le (-x)(-x), which is xxx x by step 2.
  6. By step 3 with x=1x = 1, we get 0≤10 \le 1.
  7. By step 3 with x=ix = i, we get 0≤i20 \le i^2. By i squared is minus one, this says 0≤−10 \le -1.
  8. Adding 11 to both sides of 0≤−10 \le -1 gives 1≤01 \le 0, by the first axiom.
  9. A total order is a partial order, so it is antisymmetric. Steps 6 and 8 then give 0=10 = 1.
  10. But 0≠10 \neq 1 in ℂ\C, because ℂ\C is a field. This contradiction shows that no such order exists.

Idea

In any ordered field every square is nonnegative, so 1=12≥01 = 1^2 \ge 0. In ℂ\C the number −1-1 is also a square, namely i2i^2, so −1≥0-1 \ge 0 as well. These two facts force 0=10 = 1, which is impossible.

Reading path

To understand this page from the ground up, read these 35 pages in order. The foundations come first. The path takes the proofs Proof of the equality of complex numbers, Proof that i squared is minus one, Proof that the complex numbers form a field.

Show the path as a graph
Complex numbers, object, no evidence levelComplex numbersobjectComplex numbers as pairs of reals, concept, no evidence levelComplex numbers as pa…conceptElement, definition, no evidence levelElementdefinitionGroup, concept, no evidence levelGroupconceptImaginary unit, object, no evidence levelImaginary unitobjectOrdered field, definition, no evidence levelOrdered fielddefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionProof that i squared is minus one, proof, no evidence levelProof that i squared …proofSet, definition, no evidence levelSetdefinitionRing, definition, no evidence levelRingdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionThe reals form a field, statement, no evidence levelThe reals form a fieldstatementi squared is minus one, statement, no evidence leveli squared is minus onestatementOrdered pair, definition, no evidence levelOrdered pairdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionProof of the equality of complex numbers, proof, no evidence levelProof of the equality…proofField, definition, no evidence levelFielddefinitionSubset, definition, no evidence levelSubsetdefinitionEquality of complex numbers, statement, no evidence levelEquality of complex n…statementBinary relation, definition, no evidence levelBinary relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionProof that the complex numbers form a field, proof, no evidence levelProof that the comple…proofPreorder, definition, no evidence levelPreorderdefinitionThe complex numbers form a field, statement, no evidence levelThe complex numbers f…statementPartial order, definition, no evidence levelPartial orderdefinitionTotal order, definition, no evidence levelTotal orderdefinitionProof that the complex numbers cannot be ordered, proof, no evidence levelProof that the comple…proof
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex numbers object
  2. Complex numbers as pairs of reals concept
  3. Element definition
  4. Group concept
  5. Imaginary unit object
  6. Ordered field definition
  7. Proposition concept
  8. Real numbers object
  9. Real and imaginary parts definition
  10. Proof that i squared is minus one proof
  11. Set definition
  12. Ring definition
  13. Implication definition
  14. Predicate definition
  15. The reals form a field statement
  16. i squared is minus one statement
  17. Ordered pair definition
  18. Commutative ring definition
  19. Unit of a ring definition
  20. Universal quantifier definition
  21. Cartesian product definition
  22. Proof of the equality of complex numbers proof
  23. Field definition
  24. Subset definition
  25. Equality of complex numbers statement
  26. Binary relation definition
  27. Antisymmetric relation definition
  28. Reflexive relation definition
  29. Transitive relation definition
  30. Proof that the complex numbers form a field proof
  31. Preorder definition
  32. The complex numbers form a field statement
  33. Partial order definition
  34. Total order definition
  35. Proof that the complex numbers cannot be ordered proof

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ordered_field, Orderability of fields
About this page
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URIm2://9DM7JGR0K4
Kindproof
WritingW0 stub
MSC12J15
ProvesThe complex numbers cannot be ordered
UsesOrdered field, i squared is minus one, Total order, Partial order, The complex numbers form a field, Complex numbers
Changed4 November 2026

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