provide a dispatchable singleton Zero

SingletonZero.jl

struct ZERO end
const Zero = ZERO()

# to display Zero as Zero
const ShowZero = :Zero
# to display Zero as 0
# const ShowZero = 0
Base.show(io::IO, x::ZERO) = print(io, ShowZero)

Base.zero(::ZERO) = 0
Base.iszero(x::ZERO) = true

Base.convert(::Type{T}, x::ZERO) where {T} = zero(T)
Base.promote_rule(::Type{T}, ::Type{ZERO}) where {T} = T

Base.convert(::Type{ZERO}, x::T) where {T<:Real} = 
    iszero(x) ? Zero : throw(DomainError("0 != $(x)"))



for Op in (:(+), :(-), :(*), :(/))
  @eval begin
     Base.$Op(x::ZERO, y::ZERO) = 0
     Base.$Op(x::T, y::ZERO) where {T<:Real} = $Op(x, 0)
     Base.$Op(x::ZERO, y::T) where {T<:Real} = $Op(0, y)
  end
end

# dispatch

dispatch(x) = false
dispatch(x::ZERO) = true

dispatch(Zero), dispatch(0), dispatch(5)
# (true, false, false)

dispatch0(x) = iszero(x) ? true : false
dispatch0(x::ZERO) = true

dispatch0(Zero), dispatch0(0), dispatch0(5)
# (true, true, false)

# test

iszero(Zero)
# true

# arithmetic

Zero + Zero, Zero + 1, 1 + Zero
# (0, 1, 1)

Zero * Zero, Zero * 2, 2 * Zero
# (0, 0, 0)

# display

Zero
# Zero # 0 if you chose that for ShowZero