Why is binary search valuable if it only works on sorted lists?

There are a couple of reasons to use binary search even though it works with sorted lists. For one, when you add an item to an already sorted list, you can find the right spot quickly, instead of sorting from scratch afterward. Plus, you usually search through lists more often than you modify them, making binary search really efficient overall.

You actually don't need to sort your list every time you add an item. When you insert, you can find the right position using binary search, which keeps your list sorted without the need for a full sort. This way, the list stays organized as you go.

You're right that sorting can seem slow, but with binary search, you optimize the insertion process. This means you can keep the list sorted without heavy lifting each time. Think of it like a dictionary; you don't want to shuffle the entire thing every time you add a new word!

It's all about trade-offs! If you add items more than you search them, a linear search might actually be better since it’s quick to add items. But if you're mostly searching, the efficiency of binary search pays off, making sorted data worth it in the long run.