diff --git a/ch02/05_bpe-from-scratch/bpe-from-scratch.ipynb b/ch02/05_bpe-from-scratch/bpe-from-scratch.ipynb
index 6dc17e0..2e8981d 100644
--- a/ch02/05_bpe-from-scratch/bpe-from-scratch.ipynb
+++ b/ch02/05_bpe-from-scratch/bpe-from-scratch.ipynb
@@ -246,7 +246,7 @@
    "metadata": {},
    "source": [
     "- The BPE algorithm was originally described in 1994: \"[A New Algorithm for Data Compression](http://www.pennelynn.com/Documents/CUJ/HTML/94HTML/19940045.HTM)\" by Philip Gage\n",
-    "- Before we get to the actual code implementation, the form that is used for LLM tokenizers today can be summarized as follows:"
+    "- Before we get to the actual code implementation, the form that is used for LLM tokenizers today can be summarized as described in the following sections."
    ]
   },
   {
@@ -286,7 +286,7 @@
     " \n",
     "## 1.4 BPE algorithm example\n",
     "\n",
-    "### 1.4.1 Concrete example of the encoding part (steps 1 & 2)\n",
+    "### 1.4.1 Concrete example of the encoding part (steps 1 & 2 in section 1.3)\n",
     "\n",
     "- Suppose we have the text (training dataset) `the cat in the hat` from which we want to build the vocabulary for a BPE tokenizer\n",
     "\n",
@@ -348,7 +348,7 @@
     "- and so forth\n",
     "\n",
     " \n",
-    "### 1.4.2 Concrete example of the decoding part (steps 3)\n",
+    "### 1.4.2 Concrete example of the decoding part (step 3 in section 1.3)\n",
     "\n",
     "- To restore the original text, we reverse the process by substituting each token ID with its corresponding pair in the reverse order they were introduced\n",
     "- Start with the final compressed text: `<258>cat in <258>hat`\n",
@@ -604,10 +604,10 @@
     "                break\n",
     "\n",
     "            # Find the pair with the best (lowest) rank\n",
-    "            min_rank = 1_000_000_000\n",
+    "            min_rank = float(\"inf\")\n",
     "            bigram = None\n",
     "            for p in pairs:\n",
-    "                r = self.bpe_ranks.get(p, 1_000_000_000)\n",
+    "                r = self.bpe_ranks.get(p, float(\"inf\"))\n",
     "                if r < min_rank:\n",
     "                    min_rank = r\n",
     "                    bigram = p\n",