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# 2024-03-13

## Introduction

> **Summary**

> **keywords**

> **TODO**

> **HW**

> **Exercise**\*

> **Next time**

***

## Arithmetic w/ Unsigned Numbers

### Addition

* add bit-wise.
* if there's carry-out, you add 1 to the next digit.

### Subtraction

* borrow values from bigger digits, and subtract.
* This is so complex. Just use signed number arithmetic and convert back.

### Multiplication

* add each multiplication to each digit.
* Just like we learned in elementary school.

## Arithmetic w/ Signed Numbers

### Addition

* The system is well defined, so you just need to calculate bit-wise.

### Problems

* When (positive)+(negative) makes additional digit carry out, you discard it.
* When (positive) +(positive) makes additional digit carry out, it is an overflow.
* When (negative) + (negative) makes sign digit into positive, it is an underflow.
* In either way, overflowed number can't be represented in bit limitation.

### How do you handle overflow/underflow?

* This problem also easily happens in multiplications.
* How to handle? just use additional 1 bit more for error catching.

(ONLY FOR 2's complement) Sign extension might handle it.

* the sign-bit should be replicated for n more digits in front.

### Subtraction

* take the 2's complement of the subtrahend(latter value) and add it.
* discard any final carry bit.

## Binary Codes & its arithemetics

> It is not defined mathematically.

* It is just a set of numerical information.
* It's a pattern of numerical representation of another element of information.
* Packed representation is bit-inefficient.

### Addition

THOUGH it is not defined mathematically, it mimicks a sequence of natural numbers.\
So a psudo-addition is possible.

If the sum is smaller than 10, the result is correct.\
If the sum is bigger than 9, since we don't use 10\~15 reprentations, we need to skip 6 representation and get the next representation. That is numerically equivalent to "adding 0110" more.