NUMBER SYSTEM & CONVERSION
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Number Systems and Conversions: Decimal, Binary, Octal, and Hexadecimal, Exams of Number Theory
An introduction to various number systems, including decimal, binary, octal, and hexadecimal. It covers the basics of each system, the symbols used, and how to convert between them. The document also includes exercises for practice.
Typology: Exams
2021/2022
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NUMBER SYSTEM & CONVERSION
Number System
Number System
Decimal system uses 10 symbols (digits) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Octal System uses eight symbols 0, 1, 2, 3, 4, 5, 6, 7 Binary System uses only two symbols 0 and 1 Hexadecimal System uses sixteen symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F to represent any number, no matter how large or how small.
Octal System (Base – 8 ns) Hexa decimal Number System (Base – 16) Decimal Number (Base-10) Binary Equivalent (Base-2) Base 4 number System (Base - 4) 0 0 0 0 0 1 1 1 1 1 2 2 2 10 2 3 3 3 11 3 4 4 4 100 10 5 5 5 101 11 6 6 6 110 12 7 7 7 111 13 10 8 8 1000 20 11 9 9 1001 21 12 A 10 1010 22 13 B 11 1011 23 14 C 12 1100 30 15 D 13 1101 31 16 E 14 1110 32 17 F 15 1111 33 Familiar System
KB, MB, GB …
In computer, the basic unit is byte (B) And, we use KB, MB, GB many many many times 2 10 = 1024 = 2 20 = 1024 x 1024 = 2 30 = 1024 x 1024 x 1024 = How about these? 2 40 = 2 50 = 2 60 = 2 70 = … 1GB (gigabyte) 1MB (megabyte) 1KB (kilobyte) 1TB (terabyte) 1PB (petabyte) 1EB (exabyte) 1ZB (zettabyte)
Decimal System
The decimal system is composed of 10 numerals or
symbols. These 10 symbols are 0,1,2,3,4,5,6,7,8,9;
using these symbols as digits of a number, we can
express any quantity.
Example : 3501.
3 5 0 1. 5 1 digit decimal point Most Significant Digit Least Significant Digit
Decimal Number Quantity (positional number)
3 5 0 1 (base-10)
1 X 10 0 = 1 0 X 10 1 = 0 5 X 10 2 = 500 3 X 10 3 = 3000 3000 + 500 + 0 + 1 = 3501
Binary-to-Decimal Conversion
1 1 0 1 (base-2)
1 X 2 0 = 1 0 X 2 1 = 0 1 X 2 2 = 4 1 X 2 3 = 8 8 + 4 + 0 + 1 = 13 1101 2 = 13 10
Hexadecimal-to-Decimal Conversion
1 A C F (base-16) [ A = 10, B = 11, C = 12, D = 13, E = 14, F = 15 ]
15 X 16 0 =15x1 = 15 12 X 16 1 =12x16 = 192 10 X 16 2 =10x256 = 2560 1 X 16 3 = 5x4096 = 20480 20480 + 2560 + 192 + 15 = 23247 1ACF 16 = 23247 10
Decimal Number Quantity
(fractional number)
= 5x0.1 = 0.
- 5 X 5 8 1 (base-10)
- = 5x0.1 = 0.
- 8 X
- = 8x0.01 = 0.
- 1 X
- = 1x0.001 = 0.
- 0.5 + 0.08 + 0.001 = 0.
Octal-to-Decimal Conversion
. 2 5 (base-8)
2 X 8
- 1 = 2x0.125 = 0. 5 X 8 - 2 = 5x0.015625 = 0. 0.25 + 0.017825 = 0.
8 = 0. 10
Hexadecimal-to-Decimal Conversion
. F 5 (base-16)
15 X
- 1 = 15x0.0625 = 0. 5 X - 2 = 5x0. = 0. 0.9375 + 0.01953125 = 0. 0.F 16 = 0. 10
Decimal-to-Binary Conversion (positional number)
2 250 2 125 Remainder 0 2 62 Remainder 1 2 31 Remainder 0 2 15 Remainder 1 2 7 Remainder 1 2 3 Remainder 1 1 Remainder 1 250 10 = 1 1 1 1 1 0 1 0 2
Decimal-to-Octal Conversion
8 250 8 31 Remainder 2 3 Remainder 7 250 10 = 372 8