CS 61C SU20

Lec 1 Number Representation

Number Bases

General Formula:

n-digit number in base B $$ d_{n-1}d_{n-2}...d_{1}d_{0}=d_{n-1}B^{n-1}+d_{n-2}B^{n-2}+...d_{1}B^{1}+d_{0}B^{0} $$ a digit d can take on value from 0 to B - 1

n digits (base B) => <= B^n things

n bits => 2^n things

Decimal	Binary	Hexadecimal
0	0b0000	0x0
1	0b0001	0x1
2	0b0010	0x2
3	0b0011	0x3
4	0b0100	0x4
5	0b0101	0x5
6	0b0110	0x6
7	0b0111	0x7
8	0b1000	0x8
9	0b1001	0x9
10	0b1010	0xA
11	0b1011	0xB
12	0b1100	0xC
13	0b1101	0xD
14	0b1110	0xE
15	0b1111	0xF

Unsigned Integers

Represent only non-negative (Uunsigned) integers.

Signed Representations

"first" bit gives sign, rest treated as unsigned (magnitude)

The disadvantage of this representation is that can not represent all positive number. For example, in a 4-bit binary signed representation, the largest positive number that can be represented is 0111 (7), and the smallest negative number that can be represented is 1000 (-8), thus the number 8 cannot be represented.

Binary	Decimal
000	0
001	1
010	2
011	3
100	0
101	-1
110	-2
111	-3

Biased Notation

In biased notation, the binary representation of a number is its two's complement relative to the bias. For example, in an 8-bit biased notation system with a bias of 127, the two's complement of the number 0 would be 01111111, because it is to the left of the bias, while the two's complement of the number 255 would be 11111111, because it is to the right of the bias.

We usually use the (2^(n-1) - )1 as the bias to represent the zero.

The other way to represent the bias is using the absolute value of the the most minimum negative value.

One's Complement(反码)

In computer science, one's complement is a simple binary representation of a number that is obtained by flipping all the bits of the number. More specifically, one's complement is obtained by subtracting each bit from 1.

For example, the one's complement of the binary number 101 is 010. This means that each 1 in the original binary number is replaced with a 0, and each 0 is replaced with a 1.

Two's Complement

Two's complement is a mathematical operation that is used to represent signed numbers in binary form. In two's complement, the most significant bit (leftmost bit) is used as a sign bit, where 0 represents a positive number and 1 represents a negative number.

To obtain the two's complement of a binary number, we first take the one's complement of the number (complementing all the bits), and then add 1 to the result.

For example two's complement of -3 in 3-bit representation. First, we need to convert 3 to binary, which is 0011. Then, we need to flip it to get 1100, and add 1 to it to get 1101, which is the binary two's complement of -3.

Overflow

Sign Extension

Conversions

Hex <=> Binary <=> Decimal

Binary	Decimal
$2^0$	1
$2^1$	2
$2^2$	4
$2^3$	8
$2^4$	16
$2^5$	32
$2^6$	64
$2^7$	128
$2^8$	256
$2^9$	512
$2^{10}$	1024

Binary	Unit
$2^{10}$	Kibi
$2^{20}$	Mebi
$2^{30}$	GiBi
$2^{40}$	TeBi
$2^{50}$	PeBi
$2^{60}$	Exbi
$2^{70}$	Zebi
$2^{80}$	Yobi