2024-03-13

Introduction

Summary

keywords

TODO

HW

Exercise*

Next time

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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.