1. ECE456: Number Systems (review)
Xing - Fall09
ECE456: Number Systems (review)
Instructor: Dr. Honggang Wang
II-209B,
ECE Dept., Fall 2013
Background Review Lecture

2. Administrative Issues (9/16/13)
Xing - Fall09
Administrative Issues (9/16/13)
Project team set-up due Wednesday, Sept. 25
If you missed the first class, go to the course website for syllabus and 1st lecture.
Dr. Wang
Background Review Lecture

3. 1 Mega bytes = 2? bytes 26 bytes (x) 217 bytes (x) 1024 bytes (x)
Xing - Fall09
1 Mega bytes = 2? bytes
26 bytes (x)
217 bytes (x)
1024 bytes (x)
216 bytes (x)
1E7 bytes (x)
210 bytes (x)
232 bytes (x)
No answer (x)
21 responses
From Background Survey
Dr. Wang
Background Review Lecture

4. Number Systems
Do you know the equivalent hexadecimal, octal, and decimal values of the binary number ?
What is the equivalent binary number of the decimal number 63?
From Background Survey
Dr. Wang

5. Xing - Fall09
Conventions
Term
Normal Usage
Usage as a Power of 2
Kilo (K)
103
210 =1,024
Mega (M)
106
220 =1,048,576
Giga (G)
109
230 =1,073,741,824
Tera (T)
1012
240 =1,099,511,627,776
Mili (m)
10-3
Micro (m)
10-6
Nano (n)
10-9
Pico (p )
10-12
Powers of 2 are most often used in describing memory capacity.
Ex: 1Kilobyte (KB) =1024 bytes= 210 bytes
Powers of 10 are used to describe the CPU clock frequencies: cycles per second (Hz)
Ex: Pentium GHz = 1.8x109 Hz
Dr. Wang
Background Review Lecture

6. Definitions Term Definition bit 0 or 1 byte (B) a group of 8 bits
Xing - Fall09
Definitions
Term
Definition
bit
0 or 1
byte (B)
a group of 8 bits
nibble (nybble)
half a byte (4 bits)
word (w)
a group of bits that is processed simultaneously.
a word may consist of 8/16/32/other number of bits
machine dependent
(ex: 8086 – 16 bits; 80386/80486/Pentium – 32 bits)
double word
2 words
msb (most significant bit)
the leftmost bit in a word
lsb (least significant bit)
the rightmost bit in a word
Hz (hertz)
reciprocal of second
Dr. Wang
Background Review Lecture

7. Review of Number Systems
Xing - Fall09
Review of Number Systems
Overview
Number systems conversions
Chapter 19 (online chapter)
Or Appendix A in 7th edition
We study the number systems that C programmers use.
Dr. Wang
Background Review Lecture

8. Number Systems Two basic types of number systems: Non-positional
Xing - Fall09
Number Systems
Two basic types of number systems:
Non-positional
Ex: Roman numerals: I, II, III, IV, V … X, XI … C
Normally only useful for small numbers
Positional
Ex: the decimal systems
Each position in which a digit/symbol is written has a different positional value, which is a power of the base
Dr. Wang
Background Review Lecture

9. Positional Number Systems (Example)
Xing - Fall09
Positional Number Systems (Example)
Decimal number systems
a base of 10 (determines the magnitude of a place).
is restricted to 10 re-usable digits/symbols (0,1,2,3,4,5,6,7,8,9)
the value of a digit depends on its position (digit x positional value = digit x baseposition)
sum of the value of all digits gives the value of the number.
58710 = 5 x x x 100
= 5 x x x 1 =
= 587
The positional value is a power of the base and the powers begin at 0 and increase by 1 as we move left in the number.
Explain, 10 is the base, power is determined by the position of the symbol or digit, the digit value is obtained by multiply the digit and the positional value!
Dr. Wang
Background Review Lecture

10. Positional Number Systems
Xing - Fall09
Positional Number Systems
Decimal (base is 10):
= 3 x x x x x 10-2
= 3 x x x x x 0.01 = =
In general (base is b),
N = ...P3P2P1P0 . P-1P-2P-3...
= P3b3 + P2b2 + P1b1 + P0b0 + P-1b-1 + P-2b-2 + P-3b
Increase by 1
Decrease by 1
Dr. Wang
Background Review Lecture

11. Xing - Fall09
Exercise (1)
Specify the value of the digit 5 in the following decimal numbers: 25 51
4538
The positional value is a power of the base and the powers begin at 0 and increase by 1 as we move left in the number.
Dr. Wang
Background Review Lecture

12. Binary Octal Hexadecimal Decimal Base 2 2 symbols:0,1 Base 8
Xing - Fall09
Decimal
(base 10)
Binary
(base 2) 1 2 10 3 11 4
100 5
101 6
110 7
111 8
1000 9
1001
1010
1011 12
1100 13
1101 14
1110 15
1111 16
10000 17
10001
Octal
(base 8)
Hexadecimal
(base 16) 1 2 3 4 5 6 7 10 8 11 9 12 A 13 B 14 C 15 D 16 E 17 F 20 21
Binary
Base 2
2 symbols:0,1
Octal
Base 8
8 symbols: 0,1,2,3,4,5,6,7
,3,4,5,6,7,8,9
Hexadecimal
Base 16
16 symbols: 0,1,2,3,4,5,6,7,8,9, A,B,C,D,E,F
More compact representation of the binary system
Decimal
Base 10
10 symbols: 0,1,2,3,4,5,6,7,8,9
Dr. Wang
Background Review Lecture

13. Example of Equivalent Numbers
Xing - Fall09
Example of Equivalent Numbers
Binary:
Octal:
Decimal:
Hexadecimal: 50A716
Notice how the number of digits gets smaller as the base increases.
Dr. Wang
Background Review Lecture

14. Agenda Overview of number systems Number systems conversions
Xing - Fall09
Agenda
Overview of number systems
Positional and non-positional
Base, positional value, symbol value
Binary, decimal, octal, hexadecimal
Number systems conversions
Dr. Wang
Background Review Lecture

15. Number Systems Conversions
Xing - Fall09
Number Systems Conversions
Binary, Octal, and Hex to Decimal
Decimal to Hex, Octal, and Binary
Binary Hex
Binary Octal
Hex Octal
Dr. Wang
Background Review Lecture

16. 1. Binary, Octal, Hex  Decimal
Xing - Fall09
1. Binary, Octal, Hex  Decimal
Multiply the decimal equivalent of each digit by its positional/place value (a power of the base) and sum these products
In general (base is b: 2 for binary, 8 for Octal, 16 for Hex),
N = ...P3P2P1P0 . P-1P-2P-3...
= P3b3 + P2b2 + P1b1 + P0b0 + P-1b-1 + P-2b-2 + P-3b
Dr. Wang
Background Review Lecture

17. Xing - Fall09
Exercise (2)
Convert the following numbers to their decimal equivalents
1AB.616
Dr. Wang
Background Review Lecture

18. 2. Decimal  Binary, Octal, or Hex
Xing - Fall09
2. Decimal  Binary, Octal, or Hex
To convert decimal numbers to any base we divide with the corresponding base until the quotient is zero and write the remainders in reverse order.
Dr. Wang
Background Review Lecture

19. Decimal  Octal, Binary, Hex
Xing - Fall09
Decimal  Octal, Binary, Hex
Divide the decimal number successively by 8 (for Octal), 2 (for Binary), 16 (for Hex)
After each division record the remainder
Octal: 0,1,…, or, 7
Binary: either a 1 or 0
Hex: 1, 2,…, or,9, or A, B, …, or F
Continue until the result of the division (quotient) is 0
Write the remainders in reverse order
Quoetint vs remainder.
Dr. Wang
Background Review Lecture

20. Exercise (3) Convert 123|10 to Base 8 Convert 59|10 to Base 2
Xing - Fall09
Exercise (3)
Convert 123|10 to Base 8
Convert 59|10 to Base 2
Convert 42|10 to Base 16
Dr. Wang
Background Review Lecture

21. Number Systems Conversions (Agenda)
Xing - Fall09
Number Systems Conversions (Agenda)
Binary, Octal, and Hex to Decimal
Decimal to Hex, Octal, and Binary
Binary Hex
Binary Octal
Hex Octal
Dr. Wang
Background Review Lecture

22. Xing - Fall09
Binary Hex
Dr. Wang
Background Review Lecture

23. Binary to Hexadecimal Conversion
Xing - Fall09
Binary to Hexadecimal Conversion
Decimal
Binary
Hexadecimal
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000 9
1001 10
1010 A 11
1011 B 12
1100 C 13
1101 D 14
1110 E 15
1111 F
|2 = ?|16
Work from right to left
Divide into 4-bit groups ## B
NOTE: # is a place holder for zero!
Dr. Wang
Background Review Lecture

24. Hexadecimal to Binary Conversion
Xing - Fall09
Hexadecimal to Binary Conversion
Decimal
Binary
Hexadecimal
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000 9
1001 10
1010 A 11
1011 B 12
1100 C 13
1101 D 14
1110 E 15
1111 F
FACE|16 = ?|2
F A C E
FACE|16= |2
Dr. Wang
Background Review Lecture

25. Xing - Fall09
Binary Octal
Dr. Wang
Background Review Lecture

26. Binary to Octal Conversion
Xing - Fall09
Binary to Octal Conversion
|2=?|8 #
|2=25615|8
Binary
Octal
000
001 1
010 2
011 3
100 4
101 5
110 6
111 7
Work from right to left
Divide into 3 bit groups
Dr. Wang
Background Review Lecture

27. Octal to Binary Conversion
Xing - Fall09
Octal to Binary Conversion
Binary
Octal
000
001 1
010 2
011 3
100 4
101 5
110 6
111 7
1247|8=?|2
1247|8= |2
= |2
Note: one need not write the leading zeros
Dr. Wang
Background Review Lecture

28. Xing - Fall09
Hexadecimal Octal
Dr. Wang
Background Review Lecture

29. How do we convert from hexadecimal to octal and vice versa?
Xing - Fall09
How do we convert from hexadecimal to octal and vice versa?
Convert to binary first!
Dr. Wang
Background Review Lecture

30. Xing - Fall09
Exercise (4)
Do you know the equivalent hexadecimal, octal, and decimal values of the binary number ? ______Yes _______No If you answered Yes, please indicate them below:
Equivalent hexadecimal number:________________
Equivalent octal number: __________________
Equivalent decimal number: __________________
What is the equivalent binary number of the decimal number 63? _____________________________
From Background Survey
Dr. Wang
Background Review Lecture

31. Exercise (5) Convert 18110 to binary and hex Convert 121F16 to decimal
Xing - Fall09
Exercise (5)
Convert to binary and hex
Convert 121F16 to decimal
Convert to hex
Convert A17F16 to octal
Dr. Wang
Background Review Lecture

32. Xing - Fall09
Summary
Basic number systems concepts (base, positional/place value, symbol value)
Convert back and forth between decimal numbers and their binary, octal, and hexadecimal equivalents
Abbreviate binary numbers as octal or hexadecimal numbers
Convert octal and hexadecimal numbers to binary numbers
Dr. Wang
Background Review Lecture

33. Xing - Fall09
Solution (1)
Specify the value of the digit 5 in the following decimal numbers:
the 5 in 25 = 5 x 100 = 5 the 5 in 51 = 5 x 101 = 50 the 5 in 4538 = 5 x 102 = 500
The positional value is a power of the base and the powers begin at 0 and increase by 1 as we move left in the number.
Dr. Wang
Background Review Lecture

34. Xing - Fall09
Solution (2)
= 1 x x x x x x 21+1 x 20 =
= 7710
= 1 x x x x x x 2-2
= /2 + 1/4 =
1AB.616 = 1 x A x B x x 16-1
= 1 x x x x 16 = =
= 1 x x x x x 8-2
= 1 x x x 1 + 2/8 + 5/64 = =
Dr. Wang
Background Review Lecture

35. Base you are converting to
Xing - Fall09
Solution (3-1)
Convert 123|10 to Base 8:
8 )123
8 )15 R 3
8 )1 R 7
0 R 1
Therefore, 123|10 = 173|8
Base you are converting to
Read Up!
Dr. Wang
Background Review Lecture

36. Solution (3-2) Convert 59|10 to Base 2: 59|10 =1110112
Xing - Fall09
Solution (3-2)
Convert 59|10 to Base 2:
59|10 =
Convert 42|10 to Base 16:
16 )42
16 )2 R A
0 R 2
Read Up!
Therefore, 42|10 = 2A|16
Dr. Wang
Background Review Lecture

37. Xing - Fall09
Solution (4)
Do you know the equivalent hexadecimal, octal, and decimal values of the binary number ? ______Yes _______No If you answered Yes, please indicate them below:
Equivalent hexadecimal number:__CA___________
Equivalent octal number: _______312___________
Equivalent decimal number: _____202_____________
What is the equivalent binary number of the decimal number 63? ________111111_____________
From Background Survey
Dr. Wang
Background Review Lecture

38. Solution (5) Convert 18110 to binary (10110101) and hex (B5)
Xing - Fall09
Solution (5)
Convert to binary ( ) and hex (B5)
Convert 121F16 to decimal ( )
Convert to hex (2AC16)
Convert A17F16 to octal ( )
Dr. Wang
Background Review Lecture