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Binary & Hex

Binary & Hex. The number systems of Computer Science. What you will learn. Decimal Binary Hexadecimal Arithmetic Converting between number bases Counting 2’s Compliment. Number Systems. Prehistory Unary, or marks: / /////// = 7 /////// + ////// = /////////////

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Binary & Hex

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  1. Binary & Hex The number systems of Computer Science

  2. What you will learn • Decimal • Binary • Hexadecimal • Arithmetic • Converting between number bases • Counting • 2’s Compliment

  3. Number Systems Prehistory Unary, or marks: / /////// = 7 /////// + ////// = ///////////// Babylonian (Mesopotamia) around 2000 BC Used 60 numerical symbols – sexadecimal. Used to measure time 5; 25, 30 = 5hr 25min 30 secs. Was positional based by had no concept of zero. 5,4560 = 5 x 60 + 45 x 1 = 34510 Roman Numerals: (non positional, no concept of zero) VII + V = VVII = XII MCMXCIX = 1999 MCMXCVIIII = 1999 MCMLXXXXVIIII = 1999 MDCCCCLXXXXVIIII = 1999

  4. Quote from Pierre Simon Laplace • “The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius. “

  5. Arabic/Indian Numerals (Decimal) • Better, Arabic/Indian based Numerals: 7 + 5 = 12 = 1 x 10 + 2 • 345 is really • 3 x 100 + 4 x 10 + 5 x 1 • 3 x 102 + 4 x 101 + 5 x 100 • 3 is the most significant symbol (carries the most weight) • 5 is the least significant symbol (carries the least weight) • Digits (or symbols) allowed: 0-9 • Base (or radix): 10

  6. Try multiplication in (non-positional) Roman numerals(!): • XXXIII (33 in decimal) • XII (12 in decimal) • --------- • XXXIII • XXXIII • CCCXXX • ----------- • CCCXXXXXXXXXIIIIII • CCCLXXXXVI • CCCXCVI = 396 • The Babylonians wouldn’t have had this problem.

  7. There are many ways to “represent” a number • Representation does not affect computation result • LIX + XXXIII = LXXXXII (Roman) • 59 + 33 = 92 (Decimal) • Representation affects difficulty of computing results • No concept of “Zero” • Computers need a representation that works with fastelectronic circuits

  8. What is the base ? • The decimal numbering system is also known as base 10. The values of the positions are calculated by taking 10 to some power. • Why is the base 10 for decimal numbers ? • Because we use 10 digits. The digits 0 through 9.

  9. Main Memory • Original computers used holes/no hole to represent two state, 1/0. • Modern computers use transistor on/off translates to values 1/0 • Requires use of Binary number system

  10. Binary: positional numbers work great with 2-state devices • Digits (symbols) allowed: 0, 1 • Binary Digits, or bits • Base (radix): 2 • 10012 is really • 1 x 23 + 0 x 22 + 0 X 21 + 1 X 20 • 910 • 110002 is really • ? • ? • Computers usually multiply Arabic numerals by convertingto binary, multiplying and converting back for display

  11. Counting in Binary • Binary • 0 • 1 • 10 • 11 • 100 • 101 • 110 • 111 • Decimal equivalent • 0 • 1 • 2 • 3 • 4 • 5 • 6 • 7

  12. Decimal to Binary • Divide decimal value by 2 until the value is 0 • E.g. • 47 /2 = 23 Remainder 1 • 23/2 = 11 Remainder 1 • 11/2 = 5 Remainder 1 • 5/2 = 2 Remainder 1 • 2/2 = 1 Remainder 0 • 1/2 = 0 Remainder 1 • Read from the bottom up • Ans: 101111

  13. Binary to Decimal • 1000001 1 X 20 = 1 • 26 25 24 23 22 21 20 0 X 21 = 0 • 0 X 22 = 0 • 20 = 1 24 = 16 0 X 23 = 0 • 21 = 225 = 320 X 24 = 0 • 22 = 426 = 640 X 25 = 0 • 23 = 81 X 26 = 6465

  14. Binary to Decimal (alternative) Converting the binary 100101 to 37 Multiply running total by two, add in next digit. Repeat for all digits.

  15. Some buzz words • Single Binary Digit =Bit • 4 bits = nibble • 8 bits = byte • 16 bits = word • 32 bits = longword • 2 nibbles = 1 byte • 2 bytes = 1 word The word size depends on the processor used, above is shown the word size for the 68000

  16. Addition of Binary Numbers • Examples: • 1 0 0 1 0 0 0 1 1 1 0 0 • + 0 1 1 0 + 1 0 0 1 + 0 1 0 1 • 1 1 1 1 1 0 1 0 1 0 0 0 1 Carry a one

  17. Addition (large numbers) • Example showing larger numbers: • 1 0 1 0 0 0 1 1 1 0 1 1 0 0 0 1 • + 0 1 1 1 0 1 0 0 0 0 0 1 1 0 0 1 • 1 0 0 0 1 0 1 1 1 1 1 0 0 1 0 1 0

  18. Working with large numbers • 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1 • Humans can’t work well with binary numbers. We will make errors. • Shorthand for binary that’s easier for us to work with - Hexadecimal

  19. Hexadecimal • Binary 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1 • Hex 5 0 9 7 • Written: 509716

  20. What is Hexadecimal really ? • Binary 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1 • Hex 5 0 9 7 • A number expressed in base 16. It’s easy to convert binary to hex and hex to binary because 16 is 24.

  21. Counting in Hex • Binary Hex Binary Hex • 0 0 0 0 0 1 0 0 0 8 • 0 0 0 1 1 1 0 0 1 9 • 0 0 1 0 2 1 0 1 0 A • 0 0 1 1 3 1 0 1 1 B • 0 1 0 0 4 1 1 0 0 C • 0 1 0 1 5 1 1 0 1 D • 0 1 1 0 6 1 1 1 0 E • 0 1 1 1 7 1 1 1 1 F

  22. Decimal to Hex • Repeat dividing by 16 and noting the remainder until the quotient = 0 E.g. 54401d to Hex 54401/16 = 3400 Remainder 1 3400/16 = 212 Remainder 8 212/16 = 16 Remainder 4 13/16 = 0 Remainder 13 Remainders are expressed Hex numbers Answer: $D481

  23. Hex to Decimal • Write out the 1st 16 numbers • Same as for binary to decimal E.g. F9E116 1x160 =1 Ex161=14x16 =224 9x162 =9x256 =2304 Fx163=15x4096= 61440 = 63969

  24. Another Binary to Hex Conversion • Binary 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 • Hex 7 C 3 F • 7C3F16

  25. Binary Coding Schemes • To represent characters we use Binary Coding Schemes • Two of the most popular are • ASCII (American Standard Code for Information Interchange • EBCDIC (Extended Binary Coded Decimal Interchange Code) • Here are some ASCII codes • CHARACTER ASCII • A 01000001 – 41 in hex • B 01000010 - 42 in hex • C 01000011 - 43 in hex • D 01000100 - 44 in hex • BELL 00000111 – 7 in hex • ESC 00011101 – 1B in hex

  26. Binary Coded Decimal (BCD) • Each digit is converted individually to it’s binary equivalent • 2634 = 0010 0110 0011 0100 • 3825 = 0011 1000 0010 0101 • Disadvantages are that arithmetic is more complicated and storage is inefficent. • Used only in simple device where memory is not a problem e.g. Calculators, clocks, etc.

  27. Negative Integers • Most humans precede number with “-” (e.g., -2000) • Accountants, however, use parentheses: (2000) • Sign-magnitude • Example: -1000 in hex? • 100010 = 3 x 162 + e x 161 + 8 x 160 • -3E816

  28. 2’s Compliment • Used for negative binary numbers • Procedure: Invert all the bits: 0 becomes 1 and 1 becomes 0 Add 1 • E.g. To get –1 110 = 000000012 Invert all bits 11111110 Add 1 11111111 Ans: -110 = 111111112

  29. 2’s Compliment • Using the TC notation, the Most significant indicates the sign. 1 means a negative number • With 8 bits, can represent signed numbers in the range: +127 -1 0 -128 01111111 11111111 00000000 10000000 • With N bits, can represent signed numbers in the range: (2N-1)-1 0 -(2N-1) • TC is widely used because it is easy to manipulate. The sign bit is treated the same as the others

  30. 2s Compliment Clock Stepping clockwise increases the binary number Stepping anti-clockwise decreases the binary number Notice how when we add one to a negative number it’s values move closer to zero

  31. Addition & Subtraction • Addition of TC’s numbers same as normal binary addition • Subtraction of TC’s numbers can use either A-B or A + TC(B) E.g. 01112 – 00112 = ??? Change second number to TC 0111 1101+ 0100 Answer: 01112 – 00112 = 01002

  32. Decimal fractions to binary • More info on p158 Clements. Question: Convert 0.6250 (decimal) to a binary fraction 0.3125 x 2 = 0.6250 yielding 0 0.625 x 2 = 1.2500 yielding 1 0.25 = 0.5000 yielding 0 0.5 x 2 = 1.0000 yielding 1 Ans:0.0101

  33. Binary to Decimal fractions • More info on p158 Clements.

  34. Summary • Decimal • Binary • Hexadecimal • Arithmetic • Converting between number bases • Counting • Coding schemes • 2’s Compliment

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