Theorems · Theorem · complex analysis
Complex.angle_eq_abs_arg
∀ {x y : ℂ}, x ≠ 0 → y ≠ 0 → InnerProductGeometry.angle x y = |(x / y).arg|The angle between two non-zero complex numbers is the absolute value of the argument of their
quotient.
Note that this does not hold when x or y is 0 as the LHS is π / 2 while the RHS is 0.
- Defined in
- Mathlib.Analysis.Complex.Angle
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites35
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Realstatement and proof · cited by 25,697
- Complexstatement and proof · cited by 5,565
- Norm.normproof · cited by 5,413
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
- absstatement · cited by 1,814
- Inner.innerproof · cited by 1,089
- one_ne_zeroproof · cited by 885
- Complex.reproof · cited by 882
- div_eq_mul_invproof · cited by 715
- starRingEndproof · cited by 671
Cited by4
Results whose statement or proof uses this declaration.
- Complex.angle_mul_leftproof · cited by 1
- Complex.angle_exp_expproof · cited by 1
- Complex.angle_one_leftproof · cited by 0
- Complex.angle_one_rightproof · cited by 0