This document explains the basics of number system and code. Number system is important for understanding of digital system or digital electronics.

1. NUMBER SYSTEMS
EXPLAINED WITH
QUATERNARY (BASE-4)
Created by
Vinoth Loganathan
RTL Design Engineer

2. Types
■ Decimal
– Radix 10 (0,1,2,3,4,5,6,7,8,9)
■ Binary
– Radix 2 (0,1)
■ Octal
– Radix 10 (0,1,2,3,4,5,6,7)
■ Hexa-Decimal
– Radix 16 (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F)
■ Quaternary
– Radix 4 (0,1,2,3)
Created by Vinoth Loganathan in interest of VLSI Design Guidance 2

3. Decimal To Quaternary
3 digit: (432)10=(?)4
4 423
4 105 - 3
4 26 - 1
4 6 - 2
1 - 2
(432)10=(12213)4
Reconversion for verification
44 43 42 41 40
1 2 2 1 3
=>
256 +128 +32 +4 +3
= (423)10
Created by Vinoth Loganathan in interest of VLSI Design Guidance 3

4. Cont..
■ 4 digit: (9867)10=(?)4
4 9867
4 2466 - 3
4 616 - 2
4 154 - 0
4 38 - 2
4 9 - 2
2 - 1
■ (9867)10=(2122023)4
Created by Vinoth Loganathan in interest of VLSI Design Guidance 4

5. Cont..
5 digit: (89897)10=(?)4
4 89897
4 22474 - 1
4 5618 - 2
4 1404 - 2
4 351 - 0
4 87 - 3
4 21 - 3
4 5 - 1
1 - 1
(89897)10=(111332221)4
Created by Vinoth Loganathan in interest of VLSI Design Guidance 5

6. Binary to Quaternary
3 Digit: (111)10 = (?)4
Initially, Binary to decimal : 22 21 20
1 1 1
4 + 2 + 1 = (7)10
4 7
1 - 3 verification:
41 40
1 3
(13)10 4 + 3 = (7)10
Created by Vinoth Loganathan in interest of VLSI Design Guidance 6

7. Cont…
4 Digit : (1101)10 = (?)4
Binary to decimal : 8 + 4 + 1 = 13
4 13
3 - 1
Created by Vinoth Loganathan in interest of VLSI Design Guidance 7

8. Cont…
4 Digit : (11101)10 = (?)4
Binary to decimal : 16 + 8 + 4 + 1 = 29
4 29
4 7 - 1
1 - 3
Created by Vinoth Loganathan in interest of VLSI Design Guidance 8

9. Octal to Quaternary
3 bit : (765)10 = (?)4
Octal to decimal : 82 81 80
7 6 5
448 + 48 + 5 = (501)10
4 501
4 125 - 1 Verification:
4 31 - 1 44 43 42 41 4
4 7 - 3 1 3 3 1 1
4 1 - 3 256 192 48 4 1
(13311)4 (501)10
Created by Vinoth Loganathan in interest of VLSI Design Guidance 9

10. Cont..
4 bit : (7656)10 = (?)4
Octal to decimal : 83 82 81 80
7 6 5 6
3584 384 40 6 = (4014)10
4 4014
4 1003 - 2
4 250 - 3
4 62 - 2
4 15 - 2
3 - 3
(332232)4
Created by Vinoth Loganathan in interest of VLSI Design Guidance 10

11. Cont…
5 bit : (67765)10 = (?)4
Octal to decimal : 84 83 82 81 80
6 7 7 6 5
24576 3584 448 48 5 = (2866)10
4 2866
4 7165 - 1
4 1791 - 1
4 447 - 3
4 111 - 3
4 27 - 3
4 6 - 3
1 - 2
(12333311)4
Created by Vinoth Loganathan in interest of VLSI Design Guidance 11

12. Hexa-Decimal to Quaternary
3 bit : (FEA)16 = (?)4
Octal to decimal : 162 161 160
15 14 10
3840 224 10 = (4074)10
4 4074
4 1018 - 2 Verification:
4 254 - 2 44 44 43 42 41 4
4 63 - 2 3 3 3 2 2 2
4 15 - 3 256 192 48 4 1
3 - 3
(333222)4 (FEA)10
Created by Vinoth Loganathan in interest of VLSI Design Guidance 12

13. Cont…
4 bit : (FEBA)10 = (?)4
Octal to decimal : 163 162 161 160
15 14 11 10
61440 3584 176 10 = (65210)10
4 65210
4 16302 - 2
4 4075 - 2
4 1018 - 3
4 254 - 2
4 63 - 2
4 15 - 3
4 3 - 3
(332232)4
Created by Vinoth Loganathan in interest of VLSI Design Guidance 13

14. Cont…
4 bit : (FEDAB)10 = (?)4
Octal to decimal : 164 163 162 161 160
15 14 13 10 11
983040 57344 3328 160 11= (1043883)10
4 1043883
4 260970 - 3
4 65242 - 2
4 16310 - 2
4 4077 - 2
4 1019 - 1
4 254 - 3
4 63 - 2
4 15 - 3
3 - 3
(3332312223)4
Created by Vinoth Loganathan in interest of VLSI Design Guidance 14

15. Quaternary (Radix - 4) Arithmetic
(15)10 + (25)10
4 15 4 25 3 3
3 - 3 4 6 - 1 1 2 1
1 - 2 2 2 0
(33)4 + (121)4
= (220)4
42 41 40
2 2 0
=>
32 8 0
= (40)10
Created by Vinoth Loganathan in interest of VLSI Design Guidance 15

16. Created by Vinoth Loganathan in interest of VLSI Design Guidance 16