1. Number Systems

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

3. The decimal system is a base 10 (modulo 10) number system:
10 digits:
Counting beyond 9 requires additional place values as powers of 10:
______ ______ ______ ______ ______
(104) (103) (102) (101) (100)

4. (in digital terms, logic 0 or logic 1)
The Binary System:
The Binary System is a base 2 (mod 2) number 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:
______ ______ ______ ______ ______
(24) (23) (22) (21) (20)

5. Example: Convert 3710 to binary.
METHOD I: Sum-of-weights:
____ ____ ____ ____ ____ ____
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: 3710 is in binary. 1 1 1

6. Convert to decimal:
Sum-of-weights uses total of each place value:
1x64 + 0x32 + 1x16 + 1x8 + 0x4 + 1x2 + 0x 1 =

7. The Hexadecimal System
The Hexadecimal system is a base 16 (mod 16) number system:
“Hexa” = 6 “Decimal” = 10
16 digits: 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)

8. Convert 5810 to hexadecimal:
Sum-of-weights:
____ ____ ____
Check: 3x x1 = = 5810 = 3A16
***Repeated division-by-base is most effective for larger conversions. 3 A

9. THE SHORTCUT FOR CONVERTING
Tips for Conversions:
THE SHORTCUT FOR CONVERTING
BINARY TO HEXADECIMAL
HEXADECIMAL TO BINARY
Since there is a relationship between 2 and 16
(24 = 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 to hexadecimal:
= b516

10. Tips for Conversions (Continued):
Convert 3F716 to binary:
*Remember to represent each digit as a 4-bit binary word!*
Drop initial 0’s to simplify.
3F716 =

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