ece 301

1. NUMBER SYSTEMS
ECE 301 – Digital Electronics
Dr. Xiang Chen
Electrical and Computer Engineering
Lecture #2
Spring 2019 ECE 301 - Digital Electronics 1

2. Topics Covered
• Binary, Hexadecimal, and Octal number systems.
• Number system conversion
• Binary to decimal.
• Decimal to binary.
• Binary to hexadecimal.
• Hexadecimal to binary.
• Binary Codes
• 8-4-2-1 Weighted
• Binary Coded Decimal (BCD), Excess-3, Gray
• ASCII
• Parity
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3. Assigned Reading
• M/C: 1.2 – 1.4
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4. NUMBER SYSTEMS
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5. Positional Number Systems
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[ kn-1kn-2…k1k0 . k-1k-2…k-(m-1)k-m ]r
n-digit integer m-digit fraction
radix point radix (base)
V(K) = kn-1 x rn-1 + kn-2 x rn-2 + … + k1 x r1 + k0 x r0
+ k-1 x r-1 + k-2 x r-2 + … + k-m x r-m
The weight of each digit is implied by its position in the number.

6. Number Systems
• The number systems of significance to this course are:
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Number System Base Digits
Binary 2 0, 1
Octal 8 0 – 7
Decimal 10 0 – 9
Hexadecimal 16 0 – 9, A – F

7. Number Systems: Decimal
• Radix (base) = 10
• Digits: D = { 0, 1, … , 8, 9 }
• D = dn-1dn-2…d0.d-1d- 2…d-m
• D = 492.3710
• V(D) = 4 x 102
+ 9 x 101
+ 2 x 100
+ 3 x 10-1
+ 7 x 10-2
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V(D) = dn-1 x 10n-1 + dn-2 x 10n-2 + … + d1 x 101 + d0 x 100
+ d-1 x 10-1 + d-2 x 10-2 + … + d-m x 10-m
integer fraction
decimal point

8. Number System: Binary
• Radix (base) = 2
• Digits: B = { 0, 1 }
• B = bn-1bn-2…b0.b-1b- 2…b-m
• B = 10110.0112
• V(B) = 1 x 24
+ 0 x 23
+ 1 x 22
+ 1 x 21
+ 0 x 20
+ 0 x 2-1
+ 1 x 2-2
+ 1 x 2-3
V(B) = 22.37510
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integer fraction
binary point
V(B) = bn-1 x 2n-1 + bn-2 x 2n-2 + …
+ b1 x 21 + b0 x 20
+ b-1 x 2-1 + b-2 x 2-2 + … + b-m x 2-m
base
An n-bit integer can represent a value in the range: 0 to 2n-1.
LSbMSb

9. Number Systems: Hexadecimal
• Radix (base) = 16
• Digits: H = { 0, 1, … , 9, A, …, F }
• A = 10, B = 11, C = 12
• D = 13, E = 14, F = 15
• H = hn-1hn-2…h0.h-1h- 2…h-m
• H = B7E.3C16
• V(H) = 11 x 162
+ 7 x 161
+ 14 x 160
+ 3 x 16-1
+ 12 x 16-2
V(H) = 2942.2310
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V(H) = hn-1 x 16n-1 + hn-2 x 16n-2 + … + h1 x 161 + h0 x 160
+ h-1 x 16-1 + h-2 x 16-2 + … + h-m x 16-m
integer fraction
hexadecimal point
base

10. Conversion: Decimal to Binary
• To convert the integer part of a fixed point decimal number.
• Use repeated division.
• Conversion from decimal (r = 10) to binary (r = 2):
(DINT)10 = bn-1 x 2n-1 + bn-2 x 2n-2 + … + b1 x 21 + b0 x 20
(DINT)10 / 2 = Q0 + remainder
= Q0 + b0
(Q0) / 2 = Q1 + b1
(Q1) / 2 = Q2 + b2
...
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LSb

11. Conversion: Decimal to Binary
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(125)10
125 / 2 = 62 1
62 / 2 = 31 0
31 / 2 = 15 1
15 / 2 = 7 1
7 / 2 = 3 1
3 / 2 = 1 1
1 / 2 = 0 1
= (1111101)2
MSb
LSb

12. Conversion: Decimal to Binary
• To convert the integer part of a fixed-point decimal number.
• Alternately, use repeated subtraction.
• Conversion from decimal (r = 10) to binary (r = 2):
(DINT)10 = bn-1 x 2n-1 + bn-2 x 2n-2 + … + b1 x 21 + b0 x 20
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Largest power of 2 that is less than DINT.

13. Conversion: Decimal to Binary
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(157)10
157 – 128 = 29 1 x 27
29 – 16 = 13 1 x 24
13 – 8 = 5 1 x 23
5 – 4 = 1 1 x 22
1 – 1 = 0 1 x 20
= (10011101)2
LSb
MSb

14. Conversion: Decimal to Binary
• To convert the fraction part of a fixed-point decimal number.
• Use repeated multiplication.
• Conversion from decimal (r = 10) to binary (r = 2):
(DFRAC)10 = b-1 x 2-1 + b-2 x 2-2 + … + b-(m-1) x 2-(m-1) + b-m x 2-m
(DFRAC)10 * 2 = b-1 + R0
R0 * 2 = b-2 + R1
R1 * 2 = b-3 + R2
…
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15. 0.6875 * 2 = 1.375 1
Conversion: Decimal to Binary
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(0.6875)10
0.375 * 2 = 0.750 0
0.75 * 2 = 1.500 1
= (0.1011)2
LSb
MSb
0.5 * 2 = 1.000 1

16. Number Systems: Hexadecimal
• Hexadecimal is a useful shorthand notation for binary.
• Each hexadecimal digit corresponds to four bits.
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Hexadecimal Decimal Binary
0 0 0000
1 1 0001
2 2 0010
3 3 0011
4 4 0100
5 5 0101
6 6 0110
7 7 0111
Hexadecimal Decimal Binary
8 8 1000
9 9 1001
A 10 1010
B 11 1011
C 12 1100
D 13 1101
E 14 1110
F 15 1111

17. Conversion: Hexadecimal to Binary
• Convert each hex digit to an equivalent 4-bit binary number.
• Use the 8-4-2-1 binary weighted code.
• Repeat for all digits in the hexadecimal number.
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4 C 9 . 1 B 16
0100 1100 1001 0001 1011
4C9.1B16 = 0100 1100 1001 . 0001 1011 2

18. Conversion: Binary to Hexadecimal
• Separate the binary number into 4-bit groups.
• Start at the binary point and work outwards.
• Pad with 0’s on the left (integer) and on the right (fraction).
• Convert each 4-bit group to the equivalent hex digit.
• Use the 8-4-2-1 weighted binary code.
• Repeat for all 4-bit groups in the binary number.
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110 1000 1110 . 0111 101 2
6 8 E 7 A
Pad with 0’s Pad with 0’s
110 1000 1110 . 0111 101 2 = 68E.7A16

19. CODES
Binary codes (BCD, Excess-3, Gray, One-hot)
ASCII
Parity
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20. Binary Coded Decimal (BCD)
• Each decimal digit is represented by a 4-bit pattern (or code).
• Only 10 of the 16 patterns are used.
• Remaining 6 patterns should never occur.
• Treated as “don’t cares”.
• Weighted Code:
• Each bit position has a specific weight (8-4-2-1).
• Decimal value of each code can be determined.
• Very different from binary representation.
• Advantage: easily displayed on simple
digit-oriented (7-Segment) displays.
• Disadvantage: complex arithmetic circuits.
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Decimal BCD
0 0000
1 0001
2 0010
3 0011
4 0100
5 0101
6 0110
7 0111
8 1000
9 1001
8-4-2-1 Code

21. Conversion: Decimal to BCD
• Convert each decimal digit to its 4-bit binary code.
• Use the 8-4-2-1 binary weighted code.
• Repeat for all digits in the decimal number.
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9 3 7 . 2 5 10
1001 0011 0111 0010 0101
937.2510 = 1001 0011 0111 . 0010 0101 BCD
Binary Representation: 937.2510 = 1110101001.012

22. Conversion: BCD to Decimal
• Separate the BCD number into 4-bit groups.
• Start at the binary point and work outwards.
• Pad with 0’s on the left and on the right.
• Convert each 4-bit group to the equivalent decimal digit.
• Use the 8-4-2-1 weighted binary code.
• Repeat for all 4-bit groups in the BCD number.
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0100 1001 0010 . 0101 0011 BCD
4 9 2 5 3
0100 1001 0010 . 0101 0011 BCD = 492.5310

23. Excess-3 Code
• Each decimal digit is represented by a 4-bit pattern (or code).
• Only 10 of the 16 patterns are used.
• Remaining 6 patterns should never occur.
• Treated as “don’t cares”.
• Based on the 8-4-2-1 weighted code.
• 8421 weighted code + 3
• Very different from binary representation.
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Decimal Excess-3
0 0011
1 0100
2 0101
3 0110
4 0111
5 1000
6 1001
7 1010
8 1011
9 1100

24. Gray Code
• Each decimal digit is represented by a 4-bit pattern (or code).
• Only 10 of the 16 patterns are used.
• Remaining 6 patterns should never occur.
• Treated as “don’t cares”.
• Unweighted Code:
• Bit positions do not have specific weights.
• Decimal value assigned to each pattern/code.
• Very different from binary representation.
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Decimal Gray Code
0 0000
1 0001
2 0011
3 0010
4 0110
5 1110
6 1010
7 1011
8 1001
9 1000

25. ASCII
• American Standard Code for Information Interchange
• Commonly used for storage and transfer of alphanumeric info.
• 7-bit weighted code used to represent 128 characters.
• Used for letters, numbers, and other characters.
• An eighth bit is often used for parity.
• Any word or number can be represented using ASCII.
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HeLp!
Binary:
Hexadecimal: 0x48 0x65 0x4C 0x70
0100 1000 0110 0101 0100 1100 0111 0000
ASCII is a 7-bit Code.
MSb is assumed to be 0.

26. ASCII Table
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27. Parity
• Additional bit(s) to provide error-checking capability.
• Even parity:
• Even number of 1’s in code.
• Odd parity:
• Odd number of 1’s in code.
• Each decimal digit represented
by a 5-bit pattern.
• 4 bits for the code
• 1 bit for parity (error checking).
• For example: decimal value = 6
• Even parity: 0 0110
• Odd parity: 1 0110
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Decimal BCD Even Odd
0 0000 0 1
1 0001 1 0
2 0010 1 0
3 0011 0 1
4 0100 1 0
5 0101 0 1
6 0110 0 1
7 0111 1 0
8 1000 1 0
9 1001 0 1

28. Questions?
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29. ADDITIONAL SLIDES
Number Systems
Conversion
Binary Codes
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30. Binary: Powers of Two
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Power of Two Decimal Value
20 1
21 2
22 4
23 8
24 16
25 32
26 64
27 128
28 256
29 512
210 1024

31. Conversion: Decimal to Binary
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(346)10
346 / 2 = 173 0
173 / 2 = 86 1
86 / 2 = 43 0
43 / 2 = 21 1
21 / 2 = 10 1
10 / 2 = 5 0
5 / 2 = 2 1
2 / 2 = 1 0
1 / 2 = 0 1
= (101011010)2
MSb
LSb

32. Conversion: Decimal to Binary
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(0.3125)10
0.3125 * 2 = 0.625 0
0.625 * 2 = 1.250 1
0.25 * 2 = 0.500 0
0.5 * 2 = 1.000 1
= (0.0101)2
LSb
MSb

33. Conversion: Decimal to Binary
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(0.3)10
0.3 * 2 = 0.6 0
0.6 * 2 = 1.2 1
0.2 * 2 = 0.4 0
0.4 * 2 = 0.8 0
0.8 * 2 = 1.6 1
0.6 * 2 = 1.2 1
0.2 * 2 = 0.4 0
0.4 * 2 = 0.8 0
0.8 * 2 = 1.6 1
…
MSb
= (0.0 1001)2

34. Binary: Number of Values and Range
• For an N-bit unsigned binary number,
• Number of values that can be represented: 2N
• Range of values: 0 <= V(B) <= 2N – 1
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N # of Values Range
4 16 0 <= V(B) <= 15
5 32 0 <= V(B) <= 31
8 256 0 <= V(B) <= 255
16 65536 0 <= V(B) <= 65535
32 232 0 <= V(B) <= 232 – 1

35. Number Systems: Octal
• Radix (base) = 8
• Digits: O = { 0, 1, … , 6, 7 }
• O = on-1on-2…o0.o-1o- 2…o-m
• O = 2374.158
• V(O) = 2 x 83
+ 3 x 82
+ 7 x 81
+ 4 x 80
+ 1 x 8-1
+ 5 x 8-2
V(O) = 1276.2010
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V(O) = on-1 x 8n-1 + on-2 x 8n-2 + … + o1 x 81 + o0 x 80
+ o-1 x 8-1 + o-2 x 8-2 + … + o-m x 8-m
integer fraction
octal point
base

36. Number Systems: Octal
• Octal was a useful shorthand notation for binary.
• Each octal digit corresponds to three bits.
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Octal Decimal Binary
0 0 000
1 1 001
2 2 010
3 3 011
4 4 100
5 5 101
6 6 110
7 7 111

37. Conversion: Binary to/from Octal
• Convert the given octal numbers to binary:
101100010001.1110112
101 100 010 001 . 111 011 2 = 5421.738
1111010110101.01112
001 111 010 110 101 . 011 100 2 = 17265.348
• Convert the given binary number to octal:
341.528 = 011 100 001 . 101 010 2
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38. Conversion: Binary to Octal
• Separate the binary number into 3-bit groups.
• Start at the binary point and work outwards.
• Pad with 0’s on the left (integer) and on the right (fraction).
• Convert each 3-bit group to the equivalent octal digit.
• Use the 4-2-1 weighted binary code.
• Repeat for all 3-bit groups in the binary number.
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1 111 010 110 101 . 011 1 2
1 7 2 5 3
Pad with 0’s Pad with 0’s
1111010110101 . 0111 2 = 17265.348
6 4

39. Conversion: Octal to Binary
• Convert each octal digit to an equivalent 3-bit binary number.
• Use the 4-2-1 binary weighted code.
• Repeat for all digits in the octal number.
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3 4 1 . 5 2 8
011 100 001 101 010
341.528 = 011 100 001 . 101 010 2

40. Number Systems
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Decimal Binary Hex Octal
00 00000 00 00
01 00001 01 01
02 00010 02 02
03 00011 03 03
04 00100 04 04
05 00101 05 05
06 00110 06 06
07 00111 07 07
08 01000 08 10
09 01001 09 11
Decimal Binary Hex Octal
10 01010 0A 12
11 01011 0B 13
12 01100 0C 14
13 01101 0D 15
14 01110 0E 16
15 01111 0F 17
16 10000 10 20
17 10001 11 21
18 10010 12 22
19 10011 13 23

41. Conversion: Decimal to Base-r
• To convert a fixed-point decimal number to any other base:
1. Use repeated division to convert integer part.
• Divide by new base.
2. Use repeated multiplication to convert fraction part.
• Multiply by new base.
3. Combine integer and fraction parts in new base.
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[ kn-1kn-2…k1k0 . k-1k-2…k-(m-1)k-m ]10
n-digit integer m-digit fraction

42. Binary Codes
• Each decimal digit is represented by a 4-bit pattern (or code).
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Decimal Binary-Coded Decimal Excess-3 Gray
0 0000 0011 0000
1 0001 0100 0001
2 0010 0101 0011
3 0011 0110 0010
4 0100 0111 0110
5 0101 1000 1110
6 0110 1001 1010
7 0111 1010 1011
8 1000 1011 1001
9 1001 1100 1000