diff --git a/arch/x86_64/start_32.S b/arch/x86_64/start_32.S
index 2958500..988d1b4 100644
--- a/arch/x86_64/start_32.S
+++ b/arch/x86_64/start_32.S
@@ -5,10 +5,15 @@
 .set MAGIC,	0x1BADB002
 .set CHECKSUM,	-(MAGIC + FLAGS)
 
-/* the amount of memory to allocate for bootstrap page directories. */
-/* with buffer size 0x1000 (enough for exactly one page directory)
-/* we can map 1 GiB of virtual memory */
-#define BOOTSTRAP_PDIR_SIZE 0x1000
+/* the amount of memory to allocate for bootstrap page directories.
+   with buffer size 0x2000 (enough for exactly two page directories)
+   we can map 2 GiB of virtual memory.
+
+   we use this to map the virtual address range 0xffffffff'80000000 -> 0xffffffff'fffff000
+   to the physical address range 0x00000000 -> 0x80000000,
+   to map the kernel's code and data, and give the kernel plenty of space to initialise itself.
+   */
+#define BOOTSTRAP_PDIR_SIZE 0x2000
 
 .extern start_64	# defined in start_64.S
 
@@ -166,8 +171,9 @@ init_page_tables:
 	xchg %eax, %ecx
 	add $0x1000, %edi
 	sub %ecx, %edi
-	sub $0x08, %edi			# subtract one from the index here, otherwise the mapping will be at 0xffffffffc0000000
-					# (each entry in the array is 8 bytes long)
+
+	#sub $0x08, %edi		# subtract from the index here to adjust where the mapping starts
+					# NOTE that you must subtract in multiples of 0x8
 
 	/* %ESI is a pointer into the pagedir buffer.
 	   %EDI is a pointer into the PDPT.