Chapter 2 Bits, Data Types, and Operations

1. Bit, Binary digit, codes
2. Data type
   1. unsigned integer
   2. signed integer
   3. logical variable
   4. floating point number
   5. ASCII code

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BIT

• 0 and 1 are binary digit
• BIT stand for Binary Digit :::

Signed Ineger

2’s complement 二补数，补码

• Given n bits, 2’s complement can only represents 2^n numbers, $-2^{n-1} \le x \le 2^{n-1} - 1$
• Make sure, say that -3 + 3 = 0 | 0000 | 0 | | —- | —- | | 0001 | 1 | | 0010 | 2 | | … | … | | 0111 | 7 | | 1000 | -8 | | 1001 | -7 | | … | … | | 1110 | -2 | | 1111 | -1 |

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OVERFLOW

• the case like -4 + -5 = +7, but not -9

  SIGN EXTENSION

  Different bits number operate together, which isn’t work.
  For example, a 4-bit number plus a 8-bit number. So we should turn a 4-bit number to 8-bit number.

• Method to turn, which is called SIGN EXTENTION

  • if positive, add 0 at the front
  • if negative, add 1 at the front

01001100 | 76
11111011 | -5
01000111 | 71 :::

Conversion between Binary and Decimal

Logical Variable

• can be represented by one bit
• BIT VECTOR, a batch of logical variable

Component

• NOT

truth table

• OR

truth table

• AND

truth table

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De Morgan’s Law

:::

• XOR / Exclusive-OR

truth table

• XNOR / Exclusive -NOR Equivalence

truth table