# The complex numbers #

The complex numbers are modelled as ℝ^2 in the obvious way and it is shown that they form a field of characteristic zero. The result that the complex numbers are algebraically closed, see FieldTheory.AlgebraicClosure.

### Definition and basic arithmetic #

structure Complex :

Complex numbers consist of two Reals: a real part re and an imaginary part im.

• re :

The real part of a complex number.

• im :

The imaginary part of a complex number.

Instances For

Complex numbers consist of two Reals: a real part re and an imaginary part im.

Equations
Instances For
noncomputable instance Complex.instDecidableEq :
Equations
@[simp]
theorem Complex.equivRealProd_apply (z : ) :
Complex.equivRealProd z = (z.re, z.im)

The equivalence between the complex numbers and ℝ × ℝ.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem Complex.eta (z : ) :
{ re := z.re, im := z.im } = z
theorem Complex.ext {z : } {w : } :
z.re = w.rez.im = w.imz = w
theorem Complex.ext_iff {z : } {w : } :
z = w z.re = w.re z.im = w.im
@[simp]
theorem Complex.range_re :
= Set.univ
@[simp]
theorem Complex.range_im :
= Set.univ

The natural inclusion of the real numbers into the complex numbers. The name Complex.ofReal is reserved for the bundled homomorphism.

Equations
• r = { re := r, im := 0 }
Instances For
@[simp]
theorem Complex.ofReal_re (r : ) :
(r).re = r
@[simp]
theorem Complex.ofReal_im (r : ) :
(r).im = 0
theorem Complex.ofReal_def (r : ) :
r = { re := r, im := 0 }
@[simp]
theorem Complex.ofReal_inj {z : } {w : } :
z = w z = w
instance Complex.canLift :
CanLift Complex.ofReal' fun (z : ) => z.im = 0
Equations
def Complex.Set.reProdIm (s : ) (t : ) :

The product of a set on the real axis and a set on the imaginary axis of the complex plane, denoted by s ×ℂ t.

Equations
Instances For

The product of a set on the real axis and a set on the imaginary axis of the complex plane, denoted by s ×ℂ t.

Equations
Instances For
theorem Complex.mem_reProdIm {z : } {s : } {t : } :
z s ×ℂ t z.re s z.im t
instance Complex.instZero :
Equations
Equations
@[simp]
theorem Complex.zero_re :
= 0
@[simp]
theorem Complex.zero_im :
= 0
@[simp]
theorem Complex.ofReal_zero :
0 = 0
@[simp]
theorem Complex.ofReal_eq_zero {z : } :
z = 0 z = 0
theorem Complex.ofReal_ne_zero {z : } :
z 0 z 0
instance Complex.instOne :
Equations
@[simp]
theorem Complex.one_re :
= 1
@[simp]
theorem Complex.one_im :
= 0
@[simp]
theorem Complex.ofReal_one :
1 = 1
@[simp]
theorem Complex.ofReal_eq_one {z : } :
z = 1 z = 1
theorem Complex.ofReal_ne_one {z : } :
z 1 z 1
Equations
@[simp]
theorem Complex.add_re (z : ) (w : ) :
(z + w).re = z.re + w.re
@[simp]
theorem Complex.add_im (z : ) (w : ) :
(z + w).im = z.im + w.im
@[simp]
theorem Complex.ofReal_add (r : ) (s : ) :
(r + s) = r + s
instance Complex.instNeg :
Equations
@[simp]
theorem Complex.neg_re (z : ) :
(-z).re = -z.re
@[simp]
theorem Complex.neg_im (z : ) :
(-z).im = -z.im
@[simp]
theorem Complex.ofReal_neg (r : ) :
(-r) = -r
instance Complex.instSub :
Equations
instance Complex.instMul :
Equations
@[simp]
theorem Complex.mul_re (z : ) (w : ) :
(z * w).re = z.re * w.re - z.im * w.im
@[simp]
theorem Complex.mul_im (z : ) (w : ) :
(z * w).im = z.re * w.im + z.im * w.re
@[simp]
theorem Complex.ofReal_mul (r : ) (s : ) :
(r * s) = r * s
theorem Complex.re_ofReal_mul (r : ) (z : ) :
(r * z).re = r * z.re
theorem Complex.im_ofReal_mul (r : ) (z : ) :
(r * z).im = r * z.im
theorem Complex.re_mul_ofReal (z : ) (r : ) :
(z * r).re = z.re * r
theorem Complex.im_mul_ofReal (z : ) (r : ) :
(z * r).im = z.im * r
theorem Complex.ofReal_mul' (r : ) (z : ) :
r * z = { re := r * z.re, im := r * z.im }

### The imaginary unit, I#

The imaginary unit.

Equations
Instances For
@[simp]
theorem Complex.I_re :
@[simp]
theorem Complex.I_im :
@[simp]
theorem Complex.I_mul_I :
theorem Complex.I_mul (z : ) :
= { re := -z.im, im := z.re }
@[simp]
theorem Complex.mk_eq_add_mul_I (a : ) (b : ) :
{ re := a, im := b } = a +
@[simp]
theorem Complex.re_add_im (z : ) :
z.re + z.im * Complex.I = z
theorem Complex.mul_I_re (z : ) :
().re = -z.im
theorem Complex.mul_I_im (z : ) :
().im = z.re
theorem Complex.I_mul_re (z : ) :
().re = -z.im
theorem Complex.I_mul_im (z : ) :
().im = z.re
@[simp]
theorem Complex.equivRealProd_symm_apply (p : ) :
p = p.1 + p.2 * Complex.I
@[simp]
theorem Complex.equivRealProdAddHom_apply (z : ) :
@[simp]
@[simp]

The natural AddEquiv from ℂ to ℝ × ℝ.

Equations
Instances For

### Commutative ring instance and lemmas #

Equations

Scalar multiplication by R on ℝ extends to ℂ. This is used here and in Matlib.Data.Complex.Module to transfer instances from ℝ to ℂ, but is not needed outside, so we make it scoped.

Equations
• Complex.SMul.instSMulRealComplex = { smul := fun (r : R) (x : ) => { re := r x.re - 0 * x.im, im := r x.im + 0 * x.re } }
Instances For
theorem Complex.smul_re {R : Type u_1} [] (r : R) (z : ) :
(r z).re = r z.re
theorem Complex.smul_im {R : Type u_1} [] (r : R) (z : ) :
(r z).im = r z.im
@[simp]
theorem Complex.real_smul {x : } {z : } :
x z = x * z
Equations
• One or more equations did not get rendered due to their size.
instance Complex.commRing :
Equations
instance Complex.instRing :

This shortcut instance ensures we do not find Ring via the noncomputable Complex.field instance.

Equations

This shortcut instance ensures we do not find CommSemiring via the noncomputable Complex.field instance.

Equations

This shortcut instance ensures we do not find Semiring via the noncomputable Complex.field instance.

Equations

The "real part" map, considered as an additive group homomorphism.

Equations
Instances For
@[simp]

The "imaginary part" map, considered as an additive group homomorphism.

Equations
Instances For
@[simp]

### Cast lemmas #

noncomputable instance Complex.instNNRatCast :
Equations
noncomputable instance Complex.instRatCast :
Equations
@[simp]
theorem Complex.ofReal_ofNat (n : ) [n.AtLeastTwo] :
() =
@[simp]
theorem Complex.ofReal_natCast (n : ) :
n = n
@[simp]
theorem Complex.ofReal_intCast (n : ) :
n = n
@[simp]
theorem Complex.ofReal_nnratCast (q : ℚ≥0) :
q = q
@[simp]
theorem Complex.ofReal_ratCast (q : ) :
q = q
@[deprecated Complex.ofReal_ratCast]
theorem Complex.ofReal_rat_cast (q : ) :
q = q

Alias of Complex.ofReal_ratCast.

@[simp]
theorem Complex.re_ofNat (n : ) [n.AtLeastTwo] :
().re =
@[simp]
theorem Complex.im_ofNat (n : ) [n.AtLeastTwo] :
().im = 0
@[simp]
theorem Complex.natCast_re (n : ) :
(n).re = n
@[simp]
theorem Complex.natCast_im (n : ) :
(n).im = 0
@[simp]
theorem Complex.intCast_re (n : ) :
(n).re = n
@[simp]
theorem Complex.intCast_im (n : ) :
(n).im = 0
@[simp]
theorem Complex.re_nnratCast (q : ℚ≥0) :
(q).re = q
@[simp]
theorem Complex.im_nnratCast (q : ℚ≥0) :
(q).im = 0
@[simp]
theorem Complex.ratCast_re (q : ) :
(q).re = q
@[simp]
theorem Complex.ratCast_im (q : ) :
(q).im = 0
@[deprecated Complex.ratCast_im]
theorem Complex.rat_cast_im (q : ) :
(q).im = 0

Alias of Complex.ratCast_im.

theorem Complex.ofReal_nsmul (n : ) (r : ) :
(n r) = n r
theorem Complex.ofReal_zsmul (n : ) (r : ) :
(n r) = n r

### Complex conjugation #

This defines the complex conjugate as the star operation of the StarRing ℂ. It is recommended to use the ring endomorphism version starRingEnd, available under the notation conj in the locale ComplexConjugate.

Equations
@[simp]
theorem Complex.conj_re (z : ) :
(() z).re = z.re
@[simp]
theorem Complex.conj_im (z : ) :
(() z).im = -z.im
@[simp]
theorem Complex.conj_ofReal (r : ) :
() r = r
@[simp]
theorem Complex.conj_I :
theorem Complex.conj_natCast (n : ) :
() n = n
@[deprecated Complex.conj_natCast]
theorem Complex.conj_nat_cast (n : ) :
() n = n

Alias of Complex.conj_natCast.

theorem Complex.conj_ofNat (n : ) [n.AtLeastTwo] :
() () =
theorem Complex.conj_eq_iff_real {z : } :
() z = z ∃ (r : ), z = r
theorem Complex.conj_eq_iff_re {z : } :
() z = z z.re = z
theorem Complex.conj_eq_iff_im {z : } :
() z = z z.im = 0
@[simp]
theorem Complex.star_def :
star = ()

### Norm squared #

The norm squared function.

Equations
Instances For
theorem Complex.normSq_apply (z : ) :
Complex.normSq z = z.re * z.re + z.im * z.im
@[simp]
theorem Complex.normSq_ofReal (r : ) :
Complex.normSq r = r * r
@[simp]
theorem Complex.normSq_natCast (n : ) :
Complex.normSq n = n * n
@[deprecated Complex.normSq_natCast]
theorem Complex.normSq_nat_cast (n : ) :
Complex.normSq n = n * n

Alias of Complex.normSq_natCast.

@[simp]
theorem Complex.normSq_intCast (z : ) :
Complex.normSq z = z * z
@[deprecated Complex.normSq_intCast]
theorem Complex.normSq_int_cast (z : ) :
Complex.normSq z = z * z

Alias of Complex.normSq_intCast.

@[simp]
theorem Complex.normSq_ratCast (q : ) :
Complex.normSq q = q * q
@[deprecated Complex.normSq_ratCast]
theorem Complex.normSq_rat_cast (q : ) :
Complex.normSq q = q * q

Alias of Complex.normSq_ratCast.

@[simp]
theorem Complex.normSq_ofNat (n : ) [n.AtLeastTwo] :
Complex.normSq () =
@[simp]
theorem Complex.normSq_mk (x : ) (y : ) :
Complex.normSq { re := x, im := y } = x * x + y * y
theorem Complex.normSq_add_mul_I (x : ) (y : ) :
Complex.normSq (x + ) = x ^ 2 + y ^ 2
theorem Complex.normSq_eq_conj_mul_self {z : } :
(Complex.normSq z) = () z * z
theorem Complex.normSq_zero :
Complex.normSq 0 = 0
theorem Complex.normSq_one :
Complex.normSq 1 = 1
@[simp]
theorem Complex.normSq_I :
Complex.normSq Complex.I = 1
theorem Complex.normSq_nonneg (z : ) :
0 Complex.normSq z
theorem Complex.normSq_eq_zero {z : } :
Complex.normSq z = 0 z = 0
@[simp]
theorem Complex.normSq_pos {z : } :
0 < Complex.normSq z z 0
@[simp]
theorem Complex.normSq_neg (z : ) :
Complex.normSq (-z) = Complex.normSq z
@[simp]
theorem Complex.normSq_conj (z : ) :
Complex.normSq (() z) = Complex.normSq z
theorem Complex.normSq_mul (z : ) (w : ) :
Complex.normSq (z * w) = Complex.normSq z * Complex.normSq w
theorem Complex.normSq_add (z : ) (w : ) :
Complex.normSq (z + w) = Complex.normSq z + Complex.normSq w + 2 * (z * () w).re
theorem Complex.re_sq_le_normSq (z : ) :
z.re * z.re Complex.normSq z
theorem Complex.im_sq_le_normSq (z : ) :
z.im * z.im Complex.normSq z
theorem Complex.mul_conj (z : ) :
z * () z = (Complex.normSq z)
theorem Complex.add_conj (z : ) :
z + () z = (2 * z.re)

The coercion ℝ → ℂ as a RingHom.

Equations
Instances For
@[simp]
theorem Complex.ofReal_eq_coe (r : ) :
Complex.ofReal r = r
@[simp]
theorem Complex.I_sq :
= -1
@[simp]
theorem Complex.I_pow_four :
= 1
@[simp]
theorem Complex.sub_re (z : ) (w : ) :
(z - w).re = z.re - w.re
@[simp]
theorem Complex.sub_im (z : ) (w : ) :
(z - w).im = z.im - w.im
@[simp]
theorem Complex.ofReal_sub (r : ) (s : ) :
(r - s) = r - s
@[simp]
theorem Complex.ofReal_pow (r : ) (n : ) :
(r ^ n) = r ^ n
theorem Complex.sub_conj (z : ) :
z - () z = (2 * z.im) * Complex.I
theorem Complex.normSq_sub (z : ) (w : ) :
Complex.normSq (z - w) = Complex.normSq z + Complex.normSq w - 2 * (z * () w).re

### Inversion #

noncomputable instance Complex.instInv :
Equations
theorem Complex.inv_def (z : ) :
z⁻¹ = () z * (Complex.normSq z)⁻¹
@[simp]
theorem Complex.inv_re (z : ) :
z⁻¹.re = z.re / Complex.normSq z
@[simp]
theorem Complex.inv_im (z : ) :
z⁻¹.im = -z.im / Complex.normSq z
@[simp]
theorem Complex.ofReal_inv (r : ) :
r⁻¹ = (r)⁻¹
theorem Complex.mul_inv_cancel {z : } (h : z 0) :
z * z⁻¹ = 1
noncomputable instance Complex.instDivInvMonoid :
Equations
theorem Complex.div_re (z : ) (w : ) :
(z / w).re = z.re * w.re / Complex.normSq w + z.im * w.im / Complex.normSq w
theorem Complex.div_im (z : ) (w : ) :
(z / w).im = z.im * w.re / Complex.normSq w - z.re * w.im / Complex.normSq w

### Field instance and lemmas #

noncomputable instance Complex.instField :
Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Complex.ofReal_nnqsmul (q : ℚ≥0) (r : ) :
(q r) = q r
@[simp]
theorem Complex.ofReal_qsmul (q : ) (r : ) :
(q r) = q r
theorem Complex.conj_inv (x : ) :
() x⁻¹ = (() x)⁻¹
@[simp]
theorem Complex.ofReal_div (r : ) (s : ) :
(r / s) = r / s
@[simp]
theorem Complex.ofReal_zpow (r : ) (n : ) :
(r ^ n) = r ^ n
@[simp]
theorem Complex.div_I (z : ) :
= -()
@[simp]
theorem Complex.inv_I :
theorem Complex.normSq_inv (z : ) :
Complex.normSq z⁻¹ = (Complex.normSq z)⁻¹
theorem Complex.normSq_div (z : ) (w : ) :
Complex.normSq (z / w) = Complex.normSq z / Complex.normSq w
theorem Complex.div_ofReal (z : ) (x : ) :
z / x = { re := z.re / x, im := z.im / x }
theorem Complex.div_natCast (z : ) (n : ) :
z / n = { re := z.re / n, im := z.im / n }
@[deprecated Complex.div_natCast]
theorem Complex.div_nat_cast (z : ) (n : ) :
z / n = { re := z.re / n, im := z.im / n }

Alias of Complex.div_natCast.

theorem Complex.div_intCast (z : ) (n : ) :
z / n = { re := z.re / n, im := z.im / n }
@[deprecated Complex.div_intCast]
theorem Complex.div_int_cast (z : ) (n : ) :
z / n = { re := z.re / n, im := z.im / n }

Alias of Complex.div_intCast.

theorem Complex.div_ratCast (z : ) (x : ) :
z / x = { re := z.re / x, im := z.im / x }
@[deprecated Complex.div_ratCast]
theorem Complex.div_rat_cast (z : ) (x : ) :
z / x = { re := z.re / x, im := z.im / x }

Alias of Complex.div_ratCast.

theorem Complex.div_ofNat (z : ) (n : ) [n.AtLeastTwo] :
z / = { re := z.re / , im := z.im / }
@[simp]
theorem Complex.div_ofReal_re (z : ) (x : ) :
(z / x).re = z.re / x
@[simp]
theorem Complex.div_ofReal_im (z : ) (x : ) :
(z / x).im = z.im / x
@[simp]
theorem Complex.div_natCast_re (z : ) (n : ) :
(z / n).re = z.re / n
@[simp]
theorem Complex.div_natCast_im (z : ) (n : ) :
(z / n).im = z.im / n
@[simp]
theorem Complex.div_intCast_re (z : ) (n : ) :
(z / n).re = z.re / n
@[simp]
theorem Complex.div_intCast_im (z : ) (n : ) :
(z / n).im = z.im / n
@[simp]
theorem Complex.div_ratCast_re (z : ) (x : ) :
(z / x).re = z.re / x
@[simp]
theorem Complex.div_ratCast_im (z : ) (x : ) :
(z / x).im = z.im / x
@[deprecated Complex.div_ratCast_im]
theorem Complex.div_rat_cast_im (z : ) (x : ) :
(z / x).im = z.im / x

Alias of Complex.div_ratCast_im.

@[simp]
theorem Complex.div_ofNat_re (z : ) (n : ) [n.AtLeastTwo] :
(z / ).re = z.re /
@[simp]
theorem Complex.div_ofNat_im (z : ) (n : ) [n.AtLeastTwo] :
(z / ).im = z.im /

### Characteristic zero #

Equations
theorem Complex.re_eq_add_conj (z : ) :
z.re = (z + () z) / 2

A complex number z plus its conjugate conj z is 2 times its real part.

theorem Complex.im_eq_sub_conj (z : ) :
z.im = (z - () z) / ()

A complex number z minus its conjugate conj z is 2i times its imaginary part.

unsafe instance Complex.instRepr :

Show the imaginary number ⟨x, y⟩ as an "x + y*I" string

Note that the Real numbers used for x and y will show as cauchy sequences due to the way Real numbers are represented.

Equations
• One or more equations did not get rendered due to their size.