1. Lecture 2 Number Systems
Introduction to Information Technology
Lecture 2 Number Systems
Dr. Ken Tsang 曾镜涛
Room E408 R9

2. Get slides from:
The glossary and PDF version of the slides are here:

3. Outline Decimal Number System Binary Number System
Hexadecimal Number System
Positional Numbering System
Conversions Between Number Systems
Conversions Between Power-of-Two Radices
Bits, Bytes, and Words
Basic Arithmetic Operations with Binary Numbers

4. Natural Numbers
Natural numbers
Zero and any number obtained by repeatedly adding one to it
Negative Numbers
A value less than 0, with a – sign
Integers
A natural number, a negative number, zero
Rational Numbers
An integer or the quotient of two integers
We will only discuss the binary representation of non-negative integers

5. Decimal Number System
A human usually has four fingers and a thumb on each hand, giving a total of ten digits over both hands
10 digits:
0,1,2,3,4,5,6,7,8,9
Also called base-10 number system,
Or Hindu-Arabic, or Arabic system
Counting in base-10
1,2,…,9,10,11,…,19,20,21,…,99,100,…
Decimal number in expanded notation
234 = 2 * * * 1

6. Binary Number System Binary number system has only two digits
0, 1
Also called base-2 system
Counting in binary system
0, 1, 10, 11, 100, 101, 110, 111, 1000,….
Binary number in expanded notation
(1011)2 = 1*23 + 0*22 + 1*21 + 1*20
(1011)2 = 1* *4 + 1*2 + 1*1 = (11)10

7. Gottfried Leibniz ( )
Leibniz, the last universal genius, invented at least two things that are essential for the modern world: calculus, and the binary system.
He invented the binary system around 1679, and published in This became the basis of virtually all modern computers.

8. Leibniz's Step Reckoner
Leibniz designed a machine to carry out multiplication, the 'Stepped Reckoner'. It can multiple number of up to 5 and 12 digits to give a 16 digit operand. The machine was later lost in an attic until 1879.

9. An ancient Chinese binary number system in Yi-Jing (易经）
Two symbols to represent 2 digits
Zero: represented by a broken line
One: represented by an unbroken line
“—” yan 阳爻，“--” yin 阴爻。

10. Hexadecimal Hexadecimal number system has 16 digits
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Also called base-16 system
Counting in Hexadecimal
0,1,…,F,10,11,…,1F,20,…FF,100,…
Hexadecimal number in expanded notation
(FF)16 = 15* *160 = (255)10

11. Some Numbers to Remember

12. Positional Numbering System
The value of a digit in a number depends on:
The digit itself
The position of the digit within the number
So 123 is different from 321
123: 1 hundred, 2 tens, and 3 units
321: 3 hundred, 2 tens, and 1 units

13. Base r Number System r symbols
Value is based on the sum of a power series in powers of r
r is called the base, or radix

14. The Octal System (base 8)
Valid symbols: 0,1,2,3,4,5,6,7
= ?
Questions:
2. How to count in Octal?

15. Why Binary? A computer is a Binary machine
It knows only ones and zeroes
Easy to implement in electronic circuits
Reliable
Cheap

16. Bit and Byte BIT = Binary digIT, “0” or “1”
State of on or off ( high or low) of a computer circuit
Kilo 1K = 210 = 1024 ≈ 103
Mega 1M = 220 = 1,048,576 ≈ 106
Giga 1G = 230 = 1,073,741,824 ≈ 109

17. Bit and Byte Byte is the basic unit of addressable memory
1 Byte = 8 Bits
The right-most bit is called the LSB
Least Significant Bit
The Left-most bit is called the MSB
Most Significant Bit

18. Why Hexadecimal?
Hexadecimal is meaningful to humans, and easy to work with for a computer
Compact
A BYTE is composed of 8 bits
One byte can thus be expressed by 2 digits in hexadecimal
 EF
b  EFh
Simple to convert them to binary

19. Conversions Between Number Systems
Binary to Decimal

20. Conversions Between Number Systems
Hexadecimal to Decimal

21. Conversions Between Number Systems
Octal to Decimal
(32)8 = (?)10
What’s wrong?
(187)8 = 1*64 + 8*8 + 7*1

22. Conversions Between Number Systems
Decimal to Binary
32110 = ?2
remainder
quotient
321 / 2 =
160 1
160 / 2 = 80
80 / 2 = 40
40 / 2 = 20
20 / 2 = 10
10 / 2 = 5
5 / = 2
2 / =
1 / =
Reading the remainders from bottom to top, we have
32110 =

23. One More Example
Convert to binary
So, =

24. Conversions Between Number Systems
Decimal to Base r
Same as Decimal to Binary
Divide the number by r
Record the quotient and remainder
Divide the new quotient by r again
…..
Repeat until the newest quotient is 0
Read the remainder from bottom to top

25. Exercises Convert 19910 to binary Convert 25510 to binary
Please show your steps of conversion clearly.
Convert to binary
Convert to binary
Convert to hexadecimal
Convert 2558 to decimal
Convert to decimal

26. Conversions Between Power-of-2 Radices
Because 16 = 24, a group of 4 bits is easily recognized as a Hexadecimal digit
And a group of 3 bits is easily recognized as one Octal digit
To convert a Hex or Octal number to a binary number
Represent each Hex or Octal digit with 4 or 3 bits in binary

27. Conversions Between Power-of-2 Radices
Convert a binary number to Hex or Oct number

28. Basic Arithmetic Operations with Binary Numbers
Rules for Binary Addition
1+1=0, with one to carry to the next place

29. Example

30. Example

31. Basic Arithmetic Operations with Binary Numbers
Rules for Binary Subtraction
1 - 0 = 1
1 - 1 = 0
0 - 0 = 0
0 - 1 = 1 … borrow 1 from the next most significant bit

32. Example
minuend
subtrahend
difference

33. Two’s Complement Alternative way of doing Binary Subtraction
Invert the digits (of the subtrahend) 
Add 1 
Add this to the minuend =
Drop/Ignore the MSB

34. Why “Two’s Complement” works?
Suppose A = a 7-bit binary minuend
B = a 7-bit binary subtrahend
Want to calculate the difference C = A – B
Rewrite C = A + ( – B ) +1 –
D = – B = same as converting 0 to 1 and 1 to 0 in B (taking 2’s complement of each bit in B)
So C = A + D

35. A “ten’s complement” scheme for decimal subtraction
A = 1234 a 4-digit decimal minuend
B = 0567 a 4-digit decimal subtrahend
Want to calculate the difference C = A – B
Rewrite C = A + (9999 – B ) +1 – 10000
D = 9999 – B = 9432 (taking 10’s complement of each digit in B)
So C = A + D

36. Binary Multiplication

37. Exercises
= ?
= ?
× =?

38. Summary Decimal, Binary, and Hexadecimal Systems
Positional Numbering Systems
Conversions Between Number Systems
Conversions Between Power-of-Two Radices
Bits and Bytes
Basic Arithmetic Operations with Binary Numbers

39. Resolution: Scanner and digital camera
Scanner and digital camera manufacturers often refer to two different types of resolution when listing product specs: optical resolution and interpolated (or digital) resolution. The optical resolution is the true measurement of resolution that the output device can capture. Interpolated, or digital, resolution is acquired artificially.
SPI (samples per inch) refers to scanning resolution.

40. Summary- In this lecture, we have discussed:
Digitizing images
Pixels & resolution
Some common graphic file formats
Digital cameras & how to purchase one
Dynamic range, white balance, and color temperature
Graphic softwares

41. High dynamic range imaging (HDRI)
The intention of HDRI is to accurately represent the wide range of intensity levels found in real scenes ranging from direct sunlight to the deepest shadows.
HDR images require a higher number of bits per color channel than traditional images, both because of the linear encoding and because they need to represent values from 10−4 to 108 (the range of visible luminance values) or more. 16-bit ("half precision") or 32-bit floating point numbers are often used to represent HDR pixels.