1. Number Systems Benchmark Companies Inc PO Box 473768 Aurora CO 80047

2.  Decimal  Binary  Hexadecimal  Octal  Binary Coded Decimal (BCD) Number Systems:

3. The Decimal System 10 digits: 0 1 2 3 4 5 6 7 8 9 Counting beyond 9 requires additional place values to begin. This will go on to infinity: i.e. … 8,9,10,11……98,99,100,101…999,1000,... The decimal system is a base 10 (modulo 10) number system:

4. The Decimal System Counting beyond 9 requires additional place values as powers of 10: ______ ______ ______ ______ ______. _____ 10000 1000 100 10 1.1 (10 4 ) (10 3 ) (10 2 ) (10 1 ) (10 0 ). (10 -1 ) 1 w/4 0’s 1 w/3 0’s 1 w/2 0’s 1 w/1 0’s 1 w/0 0’s.

5. The Binary System: 2 digits: 0 or 1 (in digital terms, logic 0 or logic 1) Counting beyond 1 requires additional place values. This will go on to infinity: i.e. 0,1,10,11,100,101,110,111,1000,1001,... The Binary System is a base 2 (mod 2) number system:

6. The Binary System: 2 digits: 0 or 1 (in digital terms, logic 0 or logic 1) Counting beyond 1 requires additional place values as powers of 2: ______ ______ ______ ______ ______. _____ 16 8 4 2 1.5 (2 4 ) (2 3 ) (2 2 ) (2 1 ) (2 0 ). (2 -1 ) The Binary System is a base 2 (mod 2) number system:

7. Example: Convert 37 10 to binary. It is important to be able to convert binary to decimal and vice-versa. In this example, convert 37 base 10 to it’s binary (base 2) equivalent number. Base 10 >>> Base 2

8. Example: Convert 37 10 to binary. METHOD I: Sum-of-weights: ____ ____ ____ 32 16 8 4 2 1 METHOD II: Repeated-division-by-base (here, base 2) 37/2 = 18 remainder of 1  This is your LSB 18/2 = 9 remainder of 0 9/2 = 4 remainder of 1 4/2 = 2 remainder of 0 2/2 = 1 remainder of 0 ½ = 0 remainder of 1  This is your MSB This process gives you the same result: 37 10 is 100101 in binary. 1 01100

9. Example: Convert 1011010 2 to decimal: In this example, convert 1011010 base 2 to it’s decimal (base 10) equivalent number. Base 2 >>> Base 10

10. Example: Convert 1011010 2 to decimal: Sum-of-weights uses total of each place value: 1x2 6 + 0x2 5 + 1x2 4 + 1x2 3 + 0x2 2 + 1x2 1 +0x2 0 1x64 + 0x32 + 1x16 + 1x8 + 0x4 + 1x2 + 0x 1 64 + 0 + 16 + 8 + 0 + 2 + 0 = 90 10

11. The Hexadecimal System “Hexa” = 6“Decimal” = 10 16 digits: 0123456789 A b C d E F representing decimal 10 through decimal 15 (use of lower case helps differentiate between b and 8 or d and 0 in a digital display) The Hexadecimal system is a base 16 (mod 16) number system:

12. Convert 58 10 to hexadecimal: Sum-of-weights: ____ ____ ____ 256 16 1 Check: 3x16 + 10x1 = 48+10 = 58 10 = 3A 16 ***Repeated division-by-base is most effective for larger conversions. 03A

13. THE SHORTCUT FOR CONVERTING BINARY TO HEXADECIMAL HEXADECIMAL TO BINARY Since there is a relationship between 2 and 16 (2 4 = 16), there is a relationship between the place values in binary and the place values in hexadecimal – look for groups of 4 instead of 3. Example: Convert 10110101 2 to hexadecimal: 10110101= b5 16 Tips for Conversions:

14. Convert 3F7 16 to binary: *Remember to represent each digit as a 4-bit binary word!* 001111110111 Drop initial 0’s to simplify. 3F7 16 = 1111110111 Tips for Conversions (Continued):

15. Binary Coded Decimals (BCD) Uses a 4-bit binary representation of each digit in decimal Example: 672 in BCD would be 0110 0111 0010 Example: 1001 0110 0101 1000 is BCD for 9658 ***In BCD, there will not be values beyond 1001 (decimal 9)