+ NO latex

python · skirpichev · Dec 5, 2023 · Oct 29, 2025 · Oct 31, 2025 · Oct 31, 2025
commit 369a213354e15a8aae32d14dda65f05792c949a3
@@ -38,61 +38,56 @@ Python) in such system is a complex number with zero real component.
 
 Unfortunately, this doesn't work well, if components of the complex number are
 floating-point numbers, used as model of the *extended* real line.  Such as
-ones, specified by the IEC 60559 standard, which beyond normal
-(finite and nonzero) numbers has special values like signed zero, infinities
-and nans.  Lets take simple examples with multiplication (where
-:math:`\rightarrow` denotes type coersion and :math:`\Rightarrow` is
-substitution of :math:`0.0+i` for :math:`i`):
+ones, specified by the IEC 60559 standard, which beyond normal (finite and
+nonzero) numbers has special values like signed zero, infinities and nans.
+Lets take simple examples with multiplication in Python-like pseudocode (where
+``~>`` denotes type coersion and ``~=`` is substitution of ``complex(0, y)``
+for ``yj``):
 
-.. math::
-   :label: ex-mul1
+.. code:: python
+
+   2.0 * (inf+3j) ~> complex(2, 0) * complex(inf, 3)
+   == complex(2.0*inf - 0.0*3.0, 2.0*3.0 + 0.0*inf)
+   == complex(inf, nan)
 
-   2.0 \times (\infty + 3.0 i) & \rightarrow (2.0 + 0.0 i) \times (\infty + 3.0 i) \\
-   & = (2.0\times\infty - 0.0\times 3.0) + i(0.0\times\infty + 2.0\times 3.0) \\
-   & = \infty + i\mathrm{nan}
+.. code:: python
 
-.. math::
-   :label: ex-mul2
+   2j * (inf+3j) ~= complex(0, 2) * complex(inf, 3)
+   == complex(0.0*inf - 2.0*3.0, 2.0*inf + 0.0*3.0)
+   == complex(nan, inf)
 
-   2.0 i \times (\infty + 3.0 i) & \Rightarrow (0.0 + 2.0 i) \times (\infty + 3.0 i) \\
-   & = (0.0\times\infty - 2.0\times 3.0) + i (0.0\times 3.0 + 2.0\times\infty) \\
-   & = \mathrm{nan} + i\infty
+.. code:: python
 
-.. math::
-   :label: ex-mul3
+   2j * complex(-0.0, 3) ~= complex(0, 2) * complex(-0.0, 3)
+   == complex(-0.0*0.0 - 2.0*3.0, -2.0*0.0 + 0.0*3.0)
+   == complex(-6.0, 0.0)
 
-   2.0 i \times (-0.0 + 3.0 i) & \Rightarrow (0.0 + 2.0 i) \times (-0.0 + 3.0 i) \\
-   & = (-0.0\times 0.0 - 2.0\times 3.0) + i (-2.0\times 0.0 + 0.0\times 3.0) \\
-   & = -6.0 + i 0.0
 
 Users with some mathematical background instead would expect, that ``x+yj``
-notation in the Python is just usual rectangular form for a complex number
-with components ``x`` (real) and ``y`` (imaginary) and that above examples
-will work, following rules of elementary algebra (with assumption that
-:math:`i` is a symbol such that :math:`i^2=-1`):
+notation in the Python is equal to ``complex(x, y)`` and is just the usual
+rectangular form for a complex number with components ``x`` (real) and ``y``
+(imaginary) and that above examples will work, following rules of elementary
+algebra (with assumption that ``1j**2`` is ``-1``):
 
-.. math::
+.. code:: python
 
-   2.0 \times (\infty + 3.0 i) & = 2.0\times\infty + i 2.0\times 3.0 = \infty + i 6.0 \\
-   2.0 i \times (\infty + 3.0 i) & = -2.0\times 3.0 + i 2.0\times\infty = -6.0 + i\infty \\
-   2.0 i \times (-0.0 + 3.0 i) & = -2.0\times 3.0 - i 2.0\times 0.0 = -6.0 - i 0.0 \\
+   2.0 * (inf+3j) == complex(2.0*inf, 2.0*3.0) == complex(inf, 6)
+   2j * (inf+3j) == complex(-2.0*3.0, 2.0*inf) == complex(-6, inf)
+   2j * complex(-0.0, 3) == complex(-2.0*3.0, -2.0*0.0) == complex(-6, -0.0)
 
 
 Same affects addition (or subtraction):
 
-.. math::
-   :label: ex-add1
+.. code:: python
+
+   2.0 - 0j ~= 2.0 - complex(0, 0) ~> complex(2.0, 0) - complex(0, 0)
+   == complex(2.0 - 0.0, 0.0 - 0.0) == complex(2, 0)
 
-   2.0 - 0.0 i & \rightarrow (2.0 + 0.0 i) - 0.0 i \\
-               & \Rightarrow (2.0 + 0.0 i) - (0.0 + 0.0 i) \\
-               & = 2.0 + 0.0 i
+.. code:: python
 
-.. math::
-   :label: ex-add2
+   -0.0 + 2j ~= -0.0 + complex(0, 2) ~> complex(-0.0, 0) + complex(0, 2)
+   == complex(-0.0 + 0.0, 0.0 + 2.0) == complex(0, 2)
 
-   -0.0 + 2.0 i & \rightarrow (-0.0 + 0.0 i) + 2.0 i \\
-               & \Rightarrow (-0.0 + 0.0 i) + (0.0 + 2.0 i) \\
-               & = 0.0 + 2.0 i
 
 Simplistic approach for complex arithmetic is underlying reason for numerous
 and reccuring issues in the CPython bugtracker (here is an incomplete list:
@@ -136,10 +131,10 @@ it's less useful to end users.
 
 A more modern approach [2]_, reflecting advances in the IEC standard for real
 floating-point arithmetic, instead avoid coersion of reals to complexes
-(:math:`\rightarrow`) and use a separate data type (imaginary) to represent
-the imaginary unit, *ignoring it's real component in arithmetic* (i.e. no
-implicit cast (:math:`\Rightarrow`) to a complex number with zero real part).
-The ``cmath.asin()`` would be implemented with this approach simply by:
+(``~>``) and use a separate data type (imaginary) to represent the imaginary
+unit, *ignoring it's real component in arithmetic* (i.e. no implicit cast
+(``~=``) to a complex number with zero real part).  The ``cmath.asin()`` would
+be implemented with this approach simply by:
 
 .. code:: python
 
@@ -158,8 +153,8 @@ Though it's more important, that in the IEC floating-point arithmetic results
 here are uniquely determined by usual mathematical formulae.
 
 For a first step, :cpython-pr:`124829` added in the CPython 3.14 mixed-mode
-rules for complex arithmetic, combining real and complex operands.  So,
-examples like :eq:`ex-mul1` or :eq:`ex-add1` now are working correctly:
+rules for complex arithmetic, combining real and complex operands.  So, some
+examples from above now are working correctly:
 
 .. code:: pycon
 
@@ -170,8 +165,8 @@ examples like :eq:`ex-mul1` or :eq:`ex-add1` now are working correctly:
    (2-0j)
 
 
-Unfortunately, this is only a half-way solution.  To fix the rest of above
-examples we need a separate type for pure-imaginary complex numbers.
+Unfortunately, this is only a half-way solution.  To fix the rest of examples
+we need a separate type for pure-imaginary complex numbers.
 
 
 Rationale