mathlib documentation

data.real.irrational

Irrational real numbers

In this file we define a predicate irrational on ℝ, prove that the n-th root of an integer number is irrational if it is not integer, and that sqrt q is irrational if and only if rat.sqrt q * rat.sqrt q ≠ q ∧ 0 ≤ q.

We also provide dot-style constructors like irrational.add_rat, irrational.rat_sub etc.

def irrational (x : ℝ) :

Prop

A real number is irrational if it is not equal to any rational number.

Equations

irrational x = (x ∉ set.range coe)

theorem irrational_iff_ne_rational (x : ℝ) :

irrational x ↔ ∀ (a b : ℤ), x ≠ ↑a / ↑b

Irrationality of roots of integer and rational numbers

theorem irrational_nrt_of_notint_nrt {x : ℝ} (n : ℕ) (m : ℤ) (hxr : x ^ n = ↑m) (hv : ¬∃ (y : ℤ), x = ↑y) (hnpos : 0 < n) :

If x^n, n > 0, is integer and is not the n-th power of an integer, then x is irrational.

theorem irrational_nrt_of_n_not_dvd_multiplicity {x : ℝ} (n : ℕ) {m : ℤ} (hm : m ≠ 0) (p : ℕ) [hp : fact (nat.prime p)] (hxr : x ^ n = ↑m) (hv : (multiplicity ↑p m).get _ % n ≠ 0) :

If x^n = m is an integer and n does not divide the multiplicity p m, then x is irrational.

theorem irrational_sqrt_of_multiplicity_odd (m : ℤ) (hm : 0 < m) (p : ℕ) [hp : fact (nat.prime p)] (Hpv : (multiplicity ↑p m).get _ % 2 = 1) :

irrational (real.sqrt ↑m)

theorem nat.prime.irrational_sqrt {p : ℕ} (hp : nat.prime p) :

irrational (real.sqrt ↑p)

theorem irrational_sqrt_two :

irrational (real.sqrt 2)

theorem irrational_sqrt_rat_iff (q : ℚ) :

irrational (real.sqrt ↑q) ↔ (rat.sqrt q) * rat.sqrt q ≠ q ∧ 0 ≤ q

@[instance]

def irrational.decidable (q : ℚ) :

decidable (irrational (real.sqrt ↑q))

Equations

irrational.decidable q = decidable_of_iff' ((rat.sqrt q) * rat.sqrt q ≠ q ∧ 0 ≤ q) _

Adding/subtracting/multiplying by rational numbers

theorem rat.not_irrational (q : ℚ) :

¬irrational ↑q

theorem irrational.add_cases {x y : ℝ} (a : irrational (x + y)) :

irrational x ∨ irrational y

theorem irrational.of_rat_add (q : ℚ) {x : ℝ} (h : irrational (↑q + x)) :

theorem irrational.rat_add (q : ℚ) {x : ℝ} (h : irrational x) :

irrational (↑q + x)

theorem irrational.of_add_rat (q : ℚ) {x : ℝ} (a : irrational (x + ↑q)) :

theorem irrational.add_rat (q : ℚ) {x : ℝ} (h : irrational x) :

irrational (x + ↑q)

theorem irrational.of_neg {x : ℝ} (h : irrational (-x)) :

theorem irrational.neg {x : ℝ} (h : irrational x) :

irrational (-x)

theorem irrational.sub_rat (q : ℚ) {x : ℝ} (h : irrational x) :

irrational (x - ↑q)

theorem irrational.rat_sub (q : ℚ) {x : ℝ} (h : irrational x) :

irrational (↑q - x)

theorem irrational.of_sub_rat (q : ℚ) {x : ℝ} (h : irrational (x - ↑q)) :

theorem irrational.of_rat_sub (q : ℚ) {x : ℝ} (h : irrational (↑q - x)) :

theorem irrational.mul_cases {x y : ℝ} (a : irrational (x * y)) :

irrational x ∨ irrational y

theorem irrational.of_mul_rat (q : ℚ) {x : ℝ} (h : irrational (x * ↑q)) :

theorem irrational.mul_rat {x : ℝ} (h : irrational x) {q : ℚ} (hq : q ≠ 0) :

irrational (x * ↑q)

theorem irrational.of_rat_mul (q : ℚ) {x : ℝ} (a : irrational ((↑q) * x)) :

theorem irrational.rat_mul {x : ℝ} (h : irrational x) {q : ℚ} (hq : q ≠ 0) :

irrational ((↑q) * x)

theorem irrational.of_mul_self {x : ℝ} (h : irrational (x * x)) :

theorem irrational.of_inv {x : ℝ} (h : irrational x⁻¹) :

theorem irrational.inv {x : ℝ} (h : irrational x) :

irrational x⁻¹

theorem irrational.div_cases {x y : ℝ} (h : irrational (x / y)) :

irrational x ∨ irrational y

theorem irrational.of_rat_div (q : ℚ) {x : ℝ} (h : irrational (↑q / x)) :

theorem irrational.of_one_div {x : ℝ} (h : irrational (1 / x)) :

theorem irrational.of_pow {x : ℝ} (n : ℕ) (a : irrational (x ^ n)) :

theorem irrational.of_fpow {x : ℝ} (m : ℤ) (a : irrational (x ^ m)) :

@[simp]

theorem irrational_rat_add_iff {q : ℚ} {x : ℝ} :

irrational (↑q + x) ↔ irrational x

@[simp]

theorem irrational_add_rat_iff {q : ℚ} {x : ℝ} :

irrational (x + ↑q) ↔ irrational x

@[simp]

theorem irrational_rat_sub_iff {q : ℚ} {x : ℝ} :

irrational (↑q - x) ↔ irrational x

@[simp]

theorem irrational_sub_rat_iff {q : ℚ} {x : ℝ} :

irrational (x - ↑q) ↔ irrational x

@[simp]

theorem irrational_neg_iff {x : ℝ} :

irrational (-x) ↔ irrational x

@[simp]

theorem irrational_inv_iff {x : ℝ} :

irrational x⁻¹ ↔ irrational x