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Lecture 2 Number Systems

Lecture 2 Number Systems. Introduction to Information Technology. Dr. Ken Tsang 曾镜涛 Email: kentsang@uic.edu.hk http://www.uic.edu.hk/~kentsang/IT/IT3.htm Room E408 R9. Get slides from:.

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Lecture 2 Number Systems

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  1. Lecture 2Number Systems Introduction to Information Technology Dr. Ken Tsang 曾镜涛 Email:kentsang@uic.edu.hk http://www.uic.edu.hk/~kentsang/IT/IT3.htm Room E408 R9

  2. Get slides from: The glossary and PDF version of the slides are here: http://www.uic.edu.hk/~davetowey/teaching/CS/it1010/lectures/2.Glossary.pdfhttp://www.uic.edu.hk/~davetowey/teaching/CS/it1010/lectures/2.Number.Systems.pdfhttp://www.uic.edu.hk/~davetowey/teaching/CS/it1010/lectures/2x2_2.Number.Systems.pdf

  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 * 100 + 3 * 10 + 4 * 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*8 + 0*4 + 1*2 + 1*1 = (11)10

  7. Gottfried Leibniz (1646-1716) 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 1701. 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-16system • Counting in Hexadecimal • 0,1,…,F,10,11,…,1F,20,…FF,100,… • Hexadecimal number in expanded notation • (FF)16 = 15*161+ 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 1. 268 = ? 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 ( highorlow) 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 • 11101111  EF • 11101111b 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. quotient remainder 321 / 2 = 160 1 160 / 2 = 80 0 80 / 2 = 40 0 40 / 2 = 20 0 20 / 2 = 10 0 10 / 2 = 5 0 5 / 2 = 2 1 2 / 2 = 1 0 1 / 2 = 0 1 Conversions Between Number Systems • Decimal to Binary 32110 = ?2 Reading the remainders from bottom to top, we have 32110 = 1010000012

  23. One More Example Convert 14710 to binary So, 14710 = 100100112

  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 Please show your steps of conversion clearly. • Convert 19910 to binary • Convert 25510 to binary • Convert 25510 to hexadecimal • Convert 2558 to decimal • Convert 12316 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) • 0001 0001  1110 1110 • Add 1 • 1110 1110  1110 1111 • Add this to the minuend • 1110 1111 + 0010 0101 = 1 0001 0100 • Drop/Ignore the MSB • 0001 0100

  34. Why “Two’s Complement” works? • Suppose A = 1001001 a 7-bit binary minuend • B = 0011011 a 7-bit binary subtrahend • Want to calculate the difference C = A – B • Rewrite C = A + (1111111 – B ) +1 – 1000000 • D = 1111111 – B = 1100100 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 + 1 - 1000000

  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 + 1 - 10000

  36. Binary Multiplication

  37. Exercises • 00011010 + 00001100 = ? • 00110011 - 00010110 = ? • 00101001 × 00000110 =?

  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. • http://en.wikipedia.org/wiki/High_dynamic_range_imaging

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