binary addition / truth table?

Question

I'm trying to understand how binary addition works. If I understand it correctly, with two variables e.g. x and y you add a third one called c_i. On the other side, you add the sum and c_o.

The rules as far as I understood them: If you have two 1's, you carry 1 out and the result is 0.

Here is what confuses me:

1) In normal addition, you would carry in the 1 you carry out to the next line. Why don't we do this here, and therefore, what happens to the carry outs?

2) why is the carry in an "input" of its own, shouldn't it be just x and y since the carry ins/outs usually belong to one operation distinct from the values we use?

Thanks in advance!!

Answer 1

what happens to the carry outs?

You can combine n full adder circuits to make an adder for n-bit numbers. When you do this, you connect the carry-out from each adder to the carry ins of the adder for the next most significant bit.

Here's an example adding 2 3-bit numbers, X = A + B:

^{simulate this circuit – Schematic created using CircuitLab}

why is the carry in an "input" of its own, shouldn't it be just x and y since the carry ins/outs usually belong to one operation distinct from the values we use?

It's more efficient to just calculate the sum of 3 bits in one operation than to add x and y, and then add cin in a 2nd step.

Answer 2

You do!

That C_in is your "carry in" for that bit, and that "carry out" goes to the next bit's cin.

Answer 3

1. Not sure I understand your confusion, You add the inputs (x,y) and carry in (c_in) and produce a sum. The upper bit of the sum now becomes the carry (c_out).
2. C_in is infact an input of its own. Its named so because this addition described in the truth table is just a part of multi-bit summation. Here's an example:

$$ x_3x_2x_1\\ y_3y_2y_1\\ ----\\ s_3s_2s_1 $$

So, first step is to add \$x_1 + y_1\$ which produces \$s_1 and c_1\$ now in second step add \$x_2 + y_2 + c_1\$ which gives \$s_2 and c_2\$ So on..

Answer 4

binary math, no different from decimal math in grade school

  11111
+ 10001
=======

one plus one is 0b10, or zero carry the one.

     1
  11111
+ 10001
=======
      0

one plus one plus zero is zero carry the one.

    11
  11111
+ 10001
=======
     00

you can finish it from there the carry out of one column is the carry in of the next...just like paper and pencil math from grade school.