Binary and Arithmetic (Unit 2.1)
Adding two numbers in binary representation is pretty simple. Here we are gonna build an Adder for binary numbers.
Why does the Adder matter most?
Adder + Negative representation = Subtraction
Multiplication and Division are not hardware’s work, but software’s.
Three steps to build an Adder
- Half Adder: adds 2 bits
- Full Adder: adds 3 bits (used in “carry a 1”)
- Adder: adds 2 numbers
Half Adder: adds 2 bits
When we focus on a single slice of bits adding, like “1 + 1”, it has 2 outputs: a carry bit “1” and a result bit “0”.
(truth table)
(chip code)
CHIP HalfAdder is the first thing we need to implement in Project 2.
Full Adder: adds 3 bits
The last Half Adder is only for the case when carry is “0”, while Full Adder includes adding the carry and 2 bits together.
(truth table)
(chip code)
Adder: adds 2 numbers
Assuming the numbers are 16-bit long, we can implement it by connecting 16 Full Adders or 15 Full Adders and 1 Half Adder for the right-most bits.
Negative Numbers
2’s complement is elegant, and it enables us to do subtraction (negative addition) exactly the same way as addition.
Computing “-x”
By solving this, we can deal with subtraction! So, no more hardware needed for it!~
$$
-x = 2^n-x = 1 + (2^n - 1) - x
$$
Why converting this way?
Because $(2^n-1) $ is “all ones”, which makes subtraction very easy (no need to borrow anything!). —> JUST FLIP THE BITS
Then, add one. —> easy.
Negation steps in one word:
- Flip all the bits;
- Add one.