define numerator and denominator for Complex (#55694)

Fixes #55693

base/rational.jl

@@ -293,8 +293,14 @@ julia> numerator(4)
 4
 ```
 """
-numerator(x::Integer) = x
+numerator(x::Union{Integer,Complex{<:Integer}}) = x
 numerator(x::Rational) = x.num
+function numerator(z::Complex{<:Rational})
+    den = denominator(z)
+    reim = (real(z), imag(z))
+    result = checked_mul.(numerator.(reim), div.(den, denominator.(reim)))
+    complex(result...)
+end
 
 """
     denominator(x)
@@ -310,8 +316,9 @@ julia> denominator(4)
 1
 ```
 """
-denominator(x::Integer) = one(x)
+denominator(x::Union{Integer,Complex{<:Integer}}) = one(x)
 denominator(x::Rational) = x.den
+denominator(z::Complex{<:Rational}) = lcm(denominator(real(z)), denominator(imag(z)))
 
 sign(x::Rational) = oftype(x, sign(x.num))
 signbit(x::Rational) = signbit(x.num)

test/rational.jl

@@ -801,3 +801,20 @@ end
     @test rationalize(Int64, nextfloat(0.1) * im; tol=0) == precise_next * im
     @test rationalize(0.1im; tol=eps(0.1)) == rationalize(0.1im)
 end
+
+@testset "complex numerator, denominator" begin
+    z = complex(3*3, 2*3*5)
+    @test z === numerator(z) === numerator(z // 2) === numerator(z // 5)
+    @test complex(3, 2*5) === numerator(z // 3)
+    @test isone(denominator(z))
+    @test 2 === denominator(z // 2)
+    @test 1 === denominator(z // 3)
+    @test 5 === denominator(z // 5)
+    for den ∈ 1:10
+        q = z // den
+        @test q === (numerator(q)//denominator(q))
+    end
+    @testset "do not overflow silently" begin
+        @test_throws OverflowError numerator(Int8(1)//Int8(31) + Int8(8)im//Int8(3))
+    end
+end