Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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The complex numbers form a field, and every nonzero has inverse
Statement
with the preceding operations is a field. For , The conventions and prerequisite facts used below are recorded in The complex numbers as , with their arithmetic, real embedding, and imaginary unit, The reals form a totally ordered field, Field.
Facts & Assumptions
Given: Complex numbers and .
Proof
Coordinate expansion verifies associativity, commutativity, distributivity, and the identities and .
If , then and direct multiplication gives .
Dividing by this nonzero real proves the inverse formula and the field axioms.
Depends on
Used by
- Complex de Moivre formula for every integer exponent Corollary
- For n≥2, the sum of all n-th roots of unity is zero Corollary
- Complex logarithms, the principal logarithm, and principal and multivalued complex powers Definition
- Complex series, absolute convergence, complex power series, and radius of convergence Definition
- Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential Definition
- Formal complex polynomials, evaluation, degree, leading coefficient, and monic polynomials Definition
- Integer powers in the complex field Definition
- The complex exponential by its power series Definition
- The complex geometric power series has radius 1 and sums to 1/(1-z) for |z|<1 Example
- Conjugation laws, z overline z=|z|², multiplicativity of modulus, and the triangle inequality Lemma
- The binomial theorem over the complex field Lemma
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 19 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)