Universal Gate

Universal gate

Universal gate is a gate that can implement to any other Boolean function without use any other gate, like AND, OR, NOT, NAND, NOR. In this section, we will focus on NAND gate and NOR gate.

Why NAND/ NOR gate also are called universal gate?

This is because they can be implemented to others gate/function, so they called universal gate. Example NAND gate can used as NOT gate.

NAND gate

The name NAND is the contraction of NOT-AND.  The graphical symbol of NAND gate consists of AND gate symbol and a burble. 

The name NAND is the contraction of NOT-AND.(**High output=1,low output=0)

The truth table and symbol of the NAND gate is shown as below.

A	B	X
0 0 1 1	0 1 0 1	1 1 1 0

                     

                 

Implementing NOT using only NAND gate

Implementing AND gate using NAND gate

Implementing OR gate using NAND gate

Implementing NOR gate using only NAND gate

NOR gate

The name NOR is the abbreviation of  NOT-OR.

The grahical symbol for NOR gate consist of OR gate symbol and a burble .

The truth table and graphical symbol of NOR gate is shown below

A	B	X
0 0 1 1	0 1 0 1	1 0 0 0

Below have some example implemented

Implementing NOT gate using only NOR gate

Implementing AND gate using only NOR gate

Implementing OR gate using only NOR gate

Implementing NAND gate using NOR gate

Equivalent way to draw NAND/NOR gate

 A) NAND gate

       B) NOR gate

       Additon:

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Number System Conversion

Number system conversion

In this section, we will only focus in number system conversion of decimal, binary, octal and hexadecimal.

Decimal to binary

Example :Convert decimal number 33.875 to binary number.

weight	25	24	23	22	21	20	2-1	2-2	2-3
value	32	16	8	4	2	1	0.5	0.25	.0125
binary	1	0	0	0	0	1	1	1	1

                33.875                1.875               0.875               0.375               0.125

               -32.0                   -1.000             -0.500              -0.250              -0.125

                 1.875                  0.875              0.375               0.125                      0

So, 33,875₁₀ = 100001.111₂

Decimal to octal

Convert decimal 65.140625 to octal

weight	82	81	80	8-1	8-2
value	64	8	1	0.125	0.015625
hexadecimal	1	0	0	1	1

         65.140625                   1.140625                  0.140625             0.015625

        -64.000000                  -1.000000                -0.125000            -0.015625

           1.140625                   0.140625                  0.015625                    0      

Decimal to Hexadecimal

Convert decimal 272.0625 to hexadecimal

weight	162	161	160	16-1
value	256	16	1	0.0625
hexadecimal	1	1	0	1

           272.0625                     16.0625                       0.0625

          -256.0000                    -16.0000                     -0.0625

             16.0625                       0.0625                           0     

Binary to octal

To convert to octal use 3 digit,

            Example:

                            1110101110.101₂ = 001  110  010  110 . 101₂ =  1626.5₈

                                                             1          6      2     6       5

Binary to hexadecimal

To convert to hexadecimal use 4 digit,

            Example:

                            1110101110.101₂ = 0011   1001   0110  .  1010₂ =  396.A₁₆

                                                               3          9          6          10

As the same way to convert from octal, hexadecimal to binary.

Octal to binary

            Example:

                              470.53₈   =       4          7          0    .   5        3    =100111000.101011₂

                                                     100     111     000     101     011

2’s complement

Step to convert to 2’s complement:

           Step 1: change octal, decimal or hexadecimal to binary.

           Step 2:change 0 become 1 and 1 become 0.At this step, number is 1’s complement.

                           Example: 00001100(12)    à   11110011

                                           11110001(114)  à 00001110

                                           00000010(2)      à 11111101

           Step 3: 1’ complement +1. The final value is 2’ complement  

                                           11110011+1=11110100   (-12)

                                           00001110 + 1 = 00001111  (-144)

                                           11111101+1=11111110   (-2)

Example: convert (-27) to 2’s complement

          For using 8-bits, the number 27 is represented by

                                             00011011₂

           Then convert 1 to 0, 0 to 1,

                                             11100100

          Adding 1 at the final step,

                                       11100101 =  -27

                     

How to differentiate positive / negative number in binary???

Use the most left bit as the sign bit, 0 will represent to positive number,1 will represent to negative number.

            0010 = 2                        1110 = - 2

                                             

Two's complement	Decimal
0111	7
0110	6
0101	5
0100	4
0011	3
0010	2
0001	1
0000	0
1111	−1
1110	−2
1101	−3
1100	−4
1011	−5
1010	−6
1001	−7
1000	−8

Sign bit repetition in 7-bit and 8-bit integers using 2’s complement

Decimal	7-bit notation	8-bit notation
8	0001000	00001000
-8	1111000	11111000
32	0100000	00100000
-32	1100000	11100000

8-bit two's-complement integers

Bits	Unsigned value	2's complement value
00000000	0	0
00000001	1	1
00000010	2	2
01111110	126	126
01111111	127	127
10000000	128	−128
10000001	129	−127
10000010	130	−126
11111110	254	−2
11111111	255	−1

Why 1111(8) does not involve in 4 bit?

Because it is 4-bit number, the most positive 4-bit number is 0111(7), the most negative is 1000(-8).

8 is too large to represent in 4-bit . Therefore, the most positive is 7 (0111).

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