Outline

• Information Theory
• Binary
• Hexadecimal

Q) How does a computer store data?

Information Theory : studies the quantification of information

• A Mathematical Theory of Communication, (Shannon,1948)

  • Term bit was introduced as something that can have the representation of 1 or 0 most commonly an electrical signal
  • Foundations of the electrical computer was built off of this knowledge.

• Sequences of bits have no intrinsic meaning except for the representation that we assign to them, both by convention and by building particular operations into the hardware.

Binary

• Computers store, transmit, manipulate, and calculate data using the binary numbering system.

• It is represented in many forms as:

  • Positive or negative magnetic states
  • lands or pits, light passing or not, pulses of light, pulses of electricity, etc.

• Binary can be used to represent any value you can imagine (numbers, images, etc. ).

• The smallest unit is bit, either 0 or 1.

Binary ()

• Mathematically, we are working with base of 2 ().

• The number of combinations:

• For every bit you add, you double the possible combinations

  n	0	1	2	3	4	5	6
  2^n	1	2	4	8	16	Answer 128	Answer 265

• You can calculate the maximum number you can have:

Binary to Decimal

Your turn

• Answer

• Answer

Decimal to Binary: Value =

Another example: to Binary

So, 61 in decimal is 111101 in binary.

Your Turn

• Answer

• Answer

Hexadecimal ()

• Hexadecimal is base 16
• Alphabet : 0 1 2 3 4 5 6 7 8 9 A B C D E F
• You can calculate the maximum number you can have:

Hexadecimal, Binary, Decimal

Hexadecimal to Binary

• Break the binary number into groups of 4 binary digits, why

  Answer

• Each group represnt values from 0-15 (F)

Hexadecimal to Decimal

In hex each position's weight should be 16 times the previous.

Hexadecimal to Decimal

The conversion can be done between 1 and 4 hexadecimal positions, much like Hexadecimal Binary.

Endianness: Little or Big Endian

• Order in which a sequence of bytes is stored/read in a computer’s memory
  • Big endian: first byte is the most significant (Left)
  • Little endian: first byte is the least significant

• Intel processors use little endian
• ISO standard for network address is big endian

Exmaple:

= 12 34 56 78

Big Endian =

Answer

Little Endian =

Answer

Representing Characters

• Computers use numeric encodings to represent character data inside the memory of the machine, in which each character is assigned an integral value.

• Character codes, however, are not very useful unless they are standardised. When different computer manufacturers use different coding sequence (as was indeed the case in the early years), it is harder to share such data across machines.

• The first widely adopted character encoding was ASCII (American Standard Code for Information Interchange).

American Standard Code for Information Interchange

ASCII Table

Can 256-character codes accommodate all the characters in other languages, e.g., Chinese, Japanese, Greek, etc.?

Unicode

• It has therefore been superseded by Unicode, which allows for a much larger number of characters.
• Worldwide standard for processing, displaying and interchanging all types of language texts
• Uses 2 bytes per character 16 bits 65,536 code points each point has 17 planes, 1,114,112 characters

Exmaple

Binary as a signal

Fundamentally, binary is electrical signals represented as 1 or 0, a sequence of eight is mapped to the decimal number which in turn is mapped to a ASCII symbol.

00100001 = 33 = !

*take note, these last 3 slides are helpful for the practical part of the assignment.

Labs

In addition to your lectures in Mathematics in Computer Science (For Business students, please get in touch with me if you need more help and support, and I will be happy to set up tutorials), I'm providing an extra lab for you to work on in your own time

1. Numbering systems conversion: Lab here