Quote Originally Posted by kalinka_vinnie
Coversational examples:

- You didn't know that 5+5 = 10???
- I know that 5+5 is 10!!!

- Do you know math?
- Well... I know 5+5 is 10, so yes. (but if you can also add the 'that' to make the example more definite)

lol, am I crazy?
As a native American English speaker, I can tell you that you're crazy.

"I know that 5 + 5 = 10" and "I know 5 + 5 = 10" have no different fealing at all to me.

However, my first sentance did bring up a time when "that" is necesarry. You can say "I can tell you you're crazy", but it looks awkward when written, so people usually add the "that."