Unsigned Integers#

Unsigned Binary Arithmetic

Signed Integers#

Signed Magnitude(原码)
10101B=-5D

1’s Complement(反码)
原码数据位取反
11010B=-5D
11111B=-0
11010B(-5)+00101B(5)=11111B(-0)

2’s Complement(补码)
反码+1, 解决了0出现两种表示的问题, 同时便于加法运算
11011B=-5D
00101B(5)+11011B(-5)=0

To take the two’s complement of a number:

1. copy bits from right to left until (and including) the first ‘1’
   
2. fip remaining bits to the left

Methods of converting Decimal to Binary(2’s C)

1. Division $(104/2=52r0)$
   
2. Subtract Powers of Two $(104-64=40,40-32=8)$

Sign Extension#

0100(4) —> 00000100(4)
1100(-4) —> 11111100(-4)

Floating Point#

Use equivalence of “scientific notation”: $F \times 2^{E}$
Need to represent F(fraction), E(exponent), and Sign

IEEE 754 Floating-Point Standard (32-bit):
Sign: 1 bit
Exponent: 8 bits
Fraction: 23 bits
$N = (-1)^{S}\cdot (1+F) \cdot 2^{E-127}$, $E\in \{1,2,\dots,254\}$
$N = (-1)^{S}\cdot (F) \cdot 2^{-126}$, $E = 0$

Further Reading#

Defining Floating Point Precision

Text#

ASCII Characters

LC3#

Data type: 16 bit 2’s complement signed integer operations: ADD, AND, NOT

Transistor#