diff --git a/00-Preface.ipynb b/00-Preface.ipynb
index 35adc2f..44a2719 100644
--- a/00-Preface.ipynb
+++ b/00-Preface.ipynb
@@ -312,8 +312,9 @@
     "\n",
     "There are classic textbooks in the field, such as Grewal and Andrew's *Kalman Filtering*. But sitting down and trying to read many of these books is a dismal and trying experience if you do not have the necessary background. Typically the first few chapters fly through several years of undergraduate math, blithely referring you to textbooks on Itō calculus, and presenting an entire semester's worth of statistics in a few brief paragraphs. These books are good textbooks for an upper undergraduate  or graduate level course, and an invaluable reference to researchers and professionals, but the going is truly difficult for the more casual reader. Notation is introduced without explanation, different texts use different words and variables names for the same concept, and the books are almost devoid of examples or worked problems. I often found myself able to parse the words and comprehend the mathematics of a definition, but had no idea as to what real world phenomena these words and math were attempting to describe. \"But what does that *mean?*\" was my repeated thought. Here are typical examples which once puzzled me:\n",
     "\n",
-    "$$\\hat{x}_{k} = \\Phi_{k}\\hat{x}_{k-1} + G_ku_{k-1} + K_k[z_k - H\\Phi_{k}\\hat{x}_{k-1} - HG_ku_{k-1} ] \\\\\n",
-    "\\mathbf{P}_{k\\mid k} = (I - \\mathbf{K}_k \\mathbf{H}_{k})\\textrm{cov}(\\mathbf{x}_k - \\hat{\\mathbf{x}}_{k\\mid k-1})(I - \\mathbf{K}_k \\mathbf{H}_{k})^{\\text{T}}  + \\mathbf{K}_k\\textrm{cov}(\\mathbf{v}_k )\\mathbf{K}_k^{\\text{T}}$$\n",
+    "$$\\begin{aligned}\\hat{x}_{k} = \\Phi_{k}\\hat{x}_{k-1} + G_k u_{k-1} + K_k [z_k - H \\Phi_{k} \\hat{x}_{k-1} - H G_k u_{k-1}]\n",
+    "\\\\ \n",
+    "\\mathbf{P}_{k\\mid k} = (I - \\mathbf{K}_k \\mathbf{H}_{k})\\textrm{cov}(\\mathbf{x}_k - \\hat{\\mathbf{x}}_{k\\mid k-1})(I - \\mathbf{K}_k \\mathbf{H}_{k})^{\\text{T}}  + \\mathbf{K}_k\\textrm{cov}(\\mathbf{v}_k )\\mathbf{K}_k^{\\text{T}}\\end{aligned}$$\n",
     "\n",
     "However, as I began to finally understand the Kalman filter I realized the underlying concepts are quite straightforward. A few simple probability rules, some intuition about how we integrate disparate knowledge to explain events and the core concepts of the Kalman filter are accessible. Kalman filters have a reputation for difficulty, but shorn of much of the formal terminology the beauty of the subject and of their math became clear to me, and I fell in love with the topic. \n",
     "\n",