2022/10/31 (更新日: 2022/11/02)

【Lecture2】Computer Systems & Networks

General Infotech

Contents

• 1. Digital Logic
• 2. Binary Calculation

Digital Logic

• All digital computers are built from a set of low level digital logic switches or Logic Gates.
• Gates operate on binary signals that only have one of two values:
  Signals from 0 to 1 volt is used to represent a binary 0 (OFF)
  Signals from 2 to 5 volts is used to represent a binary 1 (ON)
• Three basic logic functions that can be applied to binary signals:
  AND: Output true if ALL inputs are true
  OR: Output true if ANY input is true
  NOT: Output is the reverse of the input
• More complex functions can be build from these three basic gates

Basic Logic Gates

Combinatorial Logic Gates

• NAND: The opposite of the And
• NOR: The opposite of the Or
• XOR: 計算式の中の繰上げ時に使用される

Equivalent of XOR

Math Functions

Half Adders

The simplest math function is the half-adder which can add 2 digits (bits) to give a sum and a carry bit. However, if we want to add more than 1 bit values together we need to deal with the carry. Full-adders accept the two inputs to be added plus the carry from a previous stage.

Full-Adders

The logic to perform “add with carry” combines two half adders together.

Binary Calculation

Contents

• 1. Natural numbers and Integers
• 2. Addition
• 3. Subtraction
• 4. Multiplication
• 5. Division

Natural numbers and Integers

• N: 1, 3, 0, 7 (Natural Numbers)
• Z: -1, -7, -1234, 5, 0, 763 (Integers)

The letter (Z) is the symbol used to represent integers.

Addition

Subtraction

How to represent “-” in binary

• Negative numbers always start with “1”
• Positive numbers always start with “0”

There’s 256 patterns in a bite, but the positive numbers are until 127 and the negative numbers are until -128. This is because 0 is included in “positive”. In total, there are 256 patterns.

Subtraction

Multiplication

Pattern 1

Pattern 2

Division

//Formula (Algorithm)
for n times:
   (P,A) >> 1
   if P-B >= O:
      P = P-B
      A = A+1
   end if
end for
// A: Numerator, B: Denominator, P: Temporary Variable (0*数字Aの桁数分), N: Word Size (数字Aの桁数) 

General Infotech

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