Documentation

Mathlib.Data.Real.Sign

Real sign function #

This file introduces and contains some results about Real.sign which maps negative real numbers to -1, positive real numbers to 1, and 0 to 0.

Main definitions #

Tags #

sign function

noncomputable def Real.sign (r : ) :

The sign function that maps negative real numbers to -1, positive numbers to 1, and 0 otherwise.

Equations
Instances For
    theorem Real.sign_of_neg {r : } (hr : r < 0) :
    theorem Real.sign_of_pos {r : } (hr : 0 < r) :
    @[simp]
    @[simp]
    theorem Real.sign_apply_eq_of_ne_zero (r : ) (h : r 0) :

    This lemma is useful for working with ℝˣ

    @[simp]
    theorem Real.sign_eq_zero_iff {r : } :
    Real.sign r = 0 r = 0
    theorem Real.sign_int_cast (z : ) :
    Real.sign z = (Int.sign z)
    theorem Real.sign_mul_pos_of_ne_zero (r : ) (hr : r 0) :
    0 < Real.sign r * r
    @[simp]
    @[simp]