1. Lecture 2 Bits, Bytes & Number systems

2. Representation of Numbers
Different ways to say “how many”…
Human: decimal number system
Radix-10 or base-10
Base-10 means that a digit can have one of ten possible values
0 through 9.
Computer: binary number system
Radix-2 or base-2
Why binary?
Each digit can have one of two values
0 or 1

3. Bits and Bytes A binary digit is a single numeral in a binary number.
Each 1 and 0 in the number below is a binary digit:
The term “binary digit” is commonly called a “bit.”
Eight bits grouped together is called a “byte.”

4. Relationship between Decimal & Binary
Background:
Number systems are positional
There are 10 symbols that represent decimal quantities
Multi-digit numbers are interpreted as in the following example
79310
= 7 x x
= 7 x x x 100
Each place value in a decimal number is a power of 10.
We can get a general form of this
ABCbase
A x (base)2 + B x (base)1 + C x (base) 0
Remember that the position index starts from 0.
Indicate positions

5. Relationship between Decimal & Binary
Binary numbers are represented using the digits 0 and 1.
Multi-digit numbers are interpreted as in the following example
101112
= 1 x x x x x 20
= 1 x x x x x 1
Each place value in a binary number is a power of 2.

6. Converting binary numbers to decimal
Step 1: Starting with the 1’s place, write the binary place value over each digit in the binary number being converted.
Step 2: Add up all of the place values that have a “1” in them.
Interpret the binary number in decimal
TRY: Interpret the binary number in decimal

7. Converting decimal numbers to binary
We have now learnt how to convert from binary to decimal
Using positional representation
But how about decimal to binary
Repeated division method
Simply keep dividing it by 2 and record the remainder
Repeat above step as many times as necessary until you get a quotient that can’ t be divided by 2
Remainders give the binary digits, starting from the last remainder
Let’s look at some examples…

8. Converting decimal numbers to binary
Let’s convert decimal 23 to binary.
Step 1: 23/2 = 11 remainder 1
Step 2: 11/2 = 5 remainder 1
Step 3: 5/2 = 2 remainder 1
Step 4: 2/2 = 1 remainder 0
The last quotient “1” cannot be divided by 2 any more. So the process ends. The final binary number is read from the very end including the last quotient:
Try: Convert decimal 73, 96, 127, 128 to binary.

9. Hexadecimal
Computers use binary number system because of the electric voltage (high or low voltage)
Very difficult to express for large number representation
Hexadecimal to rescue
Hexadecimal system is interface between human brain and computer brain
4 bits from binary are read together and represented using a single digit
Such 4-bits are known as nibble
This gives a total of 16 different options
The hexadecimal number system is a Base-16 number system:
There are 16 symbols that represent quantities:
Represented by the symbols 0-9 and A-F where the letters represent values: A=10, B=11, C=12, D=13, E=14, and F=15

10. Numbering systems
Decimal
Hexadecimal
Binary
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111
Decimal
Hexadecimal
Binary 8
1000 9
1001 10 A
1010 11 B
1011 12 C
1100 13 D
1101 14 E
1110 15 F
1111

11. Hexadecimal
Thus each byte is two hex digits (shorthand representation for human)
EX: Binary:
How to represent this in hex representation?
Separate them into nibbles
C A
Hexadecimal representation: CA16
Try: convert to Hex representation

12. Converting Hex number to Binary
Converting hexadecimal numbers to binary is just the reverse operation of converting binary to hexadecimal.
Just convert each hexadecimal digit to its four-bit binary pattern. The resulting set of 1s and 0s is the binary equivalent of the hexadecimal number.
Convert A5B916 to Binary.

13. Conversion: Hex and decimal
Hex to decimal
Exact similar to binary to decimal
Use base 16
Try CA16
Decimal to Hex
Exactly similar to decimal to binary
Divide by 16
Try 20210

14. Notes on Bases Subscript is mandatory at least for a while.
We use all three number bases.
When a number is written, you should include the correct subscript.
Pronunciation
Binary and hexadecimal numbers are spoken by naming the digits followed by “binary” or “hexadecimal.”

15. Ranges of Number Systems
Lowest
Highest
Number of values
4-bit binary
(1-digit hex)
00002
010
016
11112
1510
F16
1610
8-bit binary
(2-digit hex)
0016
25510
FF16
25610
16-bit binary
(4-digit hex)
000016
FFFF16
Ranges of Unsigned Number Systems

16. Electronic Prefixes
There is a set of terms used in electronics to represent very large values and very small values.
Kilo, Mega, Giga, Tera – used for representing very large values
E.g., KiloByte, MegaByte etc.
milli, micro, nano, pico – used for representing very small values
E.g., milliseconds, microseconds etc.

17. Reading assignment In Blackboard: reading02_08_29_2012.pdf

18. Practice problems
Let us do some quick example problems …