From b674997274d49cb4c5e2eb28f8bcccca6139b85c Mon Sep 17 00:00:00 2001
From: Colin Leach <colin.leach@comcast.net>
Date: Tue, 28 Nov 2023 15:12:16 -0700
Subject: [PATCH] Moved sections around, in the search for a clearer flow of
 ideas.

---
 concepts/complex-numbers/about.md | 33 ++++++++++++++++++++-----------
 1 file changed, 21 insertions(+), 12 deletions(-)

diff --git a/concepts/complex-numbers/about.md b/concepts/complex-numbers/about.md
index 481863513e..9a97eeebaa 100644
--- a/concepts/complex-numbers/about.md
+++ b/concepts/complex-numbers/about.md
@@ -14,7 +14,7 @@ Again, it is best to focus on the underlying simplicity and not the strange name
 
 There are two common ways to create complex numbers.
 
-The [`complex(real, imag)`][complex] constructor takes two `float` parameters:
+1) The [`complex(real, imag)`][complex] constructor takes two `float` parameters:
 
 ```python
 >>> z1 = complex(1.5, 2.0)
@@ -22,13 +22,18 @@ The [`complex(real, imag)`][complex] constructor takes two `float` parameters:
 (1.5+2j)
 ```
 
-Most engineers are happy with `j`.
-Most scientists and mathematicians prefer the mathematical notation `i`, but in designing Python the engineers won.
+2) The complex number can be specified as `<real part> + <complex part>j`, or just `<complex part>j` if the real part is zero:
 
+```python
+>>> z2 = 2.0 + 1.5j
+>>> z2
+(2+1.5j)
+```
+The end result is identical to using a constructor.
 
-So there are two rules for an imaginary part in Python:
+There are two rules for an imaginary part in Python:
 
-- It is designated with `j` not `i`.
+- It is designated with `j` (not `i` as you may see in textbooks).
 - The `j` must immediately follow a number, to prevent Python seeing it as a variable name. If necessary, use `1j`.
 
 ```python
@@ -44,13 +49,8 @@ NameError: name 'j' is not defined
 <class 'complex'>
 ```
 
-With this, we have a second and perhaps clearer way to create a complex number:
-```python
->>> z2 = 2.0 + 1.5j
->>> z2
-(2+1.5j)
-```
-The end result is identical to using a constructor.
+Most engineers are happy with `j`.
+Most scientists and mathematicians prefer the mathematical notation `i`, but in designing Python the engineers won.
 
 To access the parts individually:
 ```python
@@ -73,6 +73,15 @@ However, it is still a complex number in Python.
 (1+0j)
 ```
 
+You may have heard that "`i` (or `j`) is the square root of -1".
+
+For now, all this means is that the imaginary part _by definition_ satisfies the equality
+```python
+1j * 1j == -1  # => True
+```
+
+This is a simple idea, but it leads to interesting consequences.
+
 ## Arithmetic
 
 Most of the [`operators`][operators] used with floats and ints also work with complex numbers: