[Math] Explanation of carry in carry out borrow in and borrow out for binary addition and subtraction with examples

Question

Hi I am having a hard time understand what carry in, carry out, borrow in and borrow out mean

can anyone help me out and show me some examples thanks

Answer 1

It is very much like base 10. If you add $11_2+11_2$, the ones place makes $2$, which carries in binary (just like you carry in base $10$ if the sum is $10$ or more). So you write down a $0$ and carry $1$. In the two's place, you have three $1$'s (including the carry), so you write down a $1$ and carry $1$ because $3_{10}=11_2$. In the fours place you have just the one you carried, so you write it down. Putting it all together:$$\ \ \ 11_2\\ \underline{+\ 11_2}\\ \ \ 110_2$$

Borrowing is the same way. If you ever subtract $0-1$ in binary, you borrow a $1$ from the next place up, making it $10-1$ and write down $1$ The borrows have more tendency to continue because more of the digits you might borrow from are zero themselves, but it is the same idea as subtracting $1000-5$ in base $10$