5.2 Numbering systems
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This document discusses numbering systems and conversions between systems. It defines base as the number of symbols used in a system and provides examples. The most and least significant digits in a number are defined based on their position. Binary addition and subtraction rules are covered. Methods for converting between decimal, binary, octal and hexadecimal systems are outlined, which involve repeatedly dividing by the base value and tracking the remainders or place values.
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5.2 Numbering systems
- 1. Module 5: Digital Techniques and Electronic Instrument Systems 5.2 Numbering Systems
- 2. Number and Base Base: How many symbols (digits) are used in the current numbering system? e.g. if the base of a number is 10, we use (maximum) 10 symbols to display this number. if the base of a number is 8, we use 8 symbols to display this number. (0, 1, 2, 3, 4, 5, 6, 7). Examples: 1578, 1808210, 1001012.
- 3. MSD and LSD In every numbering system, the value of a digit is determined not only by its symbol, but also by its position. The first digit of a number is called Most Significant digit (MSD). The last digit of a number is called Less Significant digit (LSD). Example: 184470287 MSD LSD
- 4. Binary System Rules 0 + 0 = 0 1 + 0 = 1 0 + 1 = 1 1 + 1 = 0 and 1 carry. 0 – 0 = 0 1 – 0 = 1 0 – 1 = 1 with 1 carry 1 – 1 = 0 Complementary subtraction: Get the complementary of the subtrahend. Add 1 Make addition Discard any carries that exceed the size of the minuend.
- 5. Numbering System Conversions Decimal to digital: I divide the decimal number by 2 I repeatedly divide the result by 2, until the dividend becomes zero. The residue of each division in reverse order is the binary number. Digital to decimal: In the first digit d0 of the binary number I assign d0 *20. In the nth digit of the binary number I assign dn *2n. The sum: d0*20+ d1*21 + … + dn*2n is the decimal number. Octal, Hex number conversion: The same methodology, but “2” is replaced with “8” and “16” respectively.