[ruby-core:109325] [Ruby master Bug#18937] Inconsistent definitions of Complex#<=> and Numeric#<=> with others

Issue #18937 has been updated by msnm (Masahiro Nomoto).


@mame

I have a motivation to create a "quaternion" class which is highly interoperable with other built-in numeric classes. (gem: [quaternion_c2](https://rubygems.org/gems/quaternion_c2))

I recently noticed that there was `Complex#<=>` in Ruby >= 2.7 .  I want to implement `Quaternion#<=>` though I believe few people compare complex numbers by `#<=>`.  However, `Complex#<=>` feels odd and I can't define `Quaternion#<=>` well.

---

> `Complex#<=>` is a different method from `Object#<=>`, so I don't see that as a problem in itself. Some Ruby core methods don't necessarily follow the Liskov Substitution Princple. I think this is one example of such cases that we favors mathematical intuition over the principle. If you are facing any practical problem, please elaborate the situation.

Thanks.  I changed my understanding: "(non-real) complex numbers are **not comparable**, then `Complex#<=>` for them always returns nil even if `self.equal?(other)`".

```ruby
Complex::I <=> Complex::I  #=> nil
```

How about `Numeric#<=>` ?  I think `num1 == num2` (equivalency) is more intuitive than `num1.equal?(num2)` (identity) in mathematical contexts.  Another idea is that the method always returns nil in order to tell "the custom numeric class may not be comparable".

---

>> This prevents users from adding yet another complex class having #<=>.
>
> I understand this as follows.

That's right.  Moreover, `MyInteger#<=>` will produce asymmetric (not antisymmetric) behaviors.

```ruby
class MyInteger
  def initialize(n)
    @n = n
  end

  def coerce(obj)
    [obj, @n]
  end

  def <=>(obj)
    if obj.kind_of?(MyInteger)
      @n <=> obj.instance_variable_get(:@n)
    else
      @n <=> obj
    end
  end
end

my_int = MyInteger.new(2)
p 1 + 0i <=> my_int  #=> nil (expected: -1)
p my_int <=> 1 + 0i  #=> 1
```


----------------------------------------
Bug #18937: Inconsistent definitions of Complex#<=> and Numeric#<=> with others
https://bugs.ruby-lang.org/issues/18937#change-98469

* Author: msnm (Masahiro Nomoto)
* Status: Open
* Priority: Normal
* ruby -v: 3.1.2
* Backport: 2.7: UNKNOWN, 3.0: UNKNOWN, 3.1: UNKNOWN
----------------------------------------
Object#<=> says "Returns 0 if `obj` and `other` are the same object or `obj == other`" .
https://ruby-doc.org/core-3.1.2/Object.html#method-i-3C-3D-3E

However, neither Complex#<=> nor Numeric#<=> satisfies that definition.

```ruby
num1 = Complex(0, 42)
num2 = Complex(0, 42)
p num1.equal?(num2)  #=> false
p num1 == num2       #=> true

# using Complex#<=>
p num1 <=> num2  #=> nil

# using Numeric#<=>
Complex.remove_method(:<=>)
p num1 <=> num2  #=> nil

# using Object#<=> (Kernel#<=>)
Numeric.remove_method(:<=>)
p num1 <=> num2  #=> 0
```

Complex#<=> has another problem that it does not coerce numeric objects while Integer#<=> and Float#<=> do.
This prevents users from adding yet another complex class having #<=>.

---

Here is my proposal of Complex#<=> behavior (in Ruby).
This considers #15857, complex numbers are comparable when their imaginary parts are 0.

```ruby
class Complex
  def <=>(other)
    return (self == other ? 0 : nil) if self.imag != 0

    if other.kind_of?(Complex)
      if other.imag == 0
        return self.real <=> other.real
      else
        return nil
      end
    elsif other.kind_of?(Numeric) && other.real?
      return self.real <=> other
    elsif other.respond_to?(:coerce)
      num1, num2 = other.coerce(self)
      return num1 <=> num2
    else
      return nil
    end
  end
end
``` 




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