Divided differences

"Divided differences" is just a fancy way of playing with numbers to figure out new numbers. It's like a big game of number puzzles!

Let's say we have a bunch of numbers, like 2, 4, 6, 8, and 10. We want to figure out what other numbers we can make using these numbers. That's where divided differences come in!

First, we take the difference between each pair of numbers. So, we subtract 2 from 4 to get 2, subtract 4 from 6 to get 2, and so on. Then, we keep doing that same thing over and over again, until we end up with a list of just one number.

Here's what that might look like:

2, 4, 6, 8, 10 (our starting list of numbers)
2, 2, 2, 2 (the first set of differences)
0, 0, 0 (the second set of differences)
0 (the third set of differences)

See how we started with five numbers, but we ended up with just one? That last number, 0, is important. We call it the "divided difference."

Now, we can use that divided difference to figure out new numbers. For example, if we know the divided difference between 2 and 4 is 2, and we know the starting number is 4, we can use that information to figure out what the next number in the list might be.

It's kind of like a secret code that helps us figure out new numbers using the numbers we already have. And that's divided differences ELI5!