To Lecture Notes

ISP 121 -- 5/14/08

Questions

1. Match the positional number system base on the right to the item on the left.

   DNA Code	8
   Yuki	4
   Modern Computers	3
   Babylonians	64
   Mayans	5
   Cryptography	20
   Yoruba	2
   Islamic Prayers	60

   Ans: See the Binary and Hex Tutorial.

2. Expand 327 in powers of ten. Recall that 7 is in the ones (10⁰) place, 2 is in the tens (10¹) place and 3 is in the hundreds (10³) place.
   Ans: 3 × 10² + 2 × 10¹ + 7 × 10⁰

3. Convert the following to decimal (base-10):
   1201₃    4131₅    53EX₁₂

   Recall that the digit X represents 10 and the digit E represents 11 in the duodecimal (base-12) system. Ans:
   1201₃ = 1 × 3³ + 2 × 3² + 0 × 3¹ + 1 × 3⁰ = 27 + 18 + 0 + 1 = 46
   4131₅ = 4 × 5³ + 1 × 5² + 3 × 5¹ + 1 × 5⁰ = 500 + 25 + 15 + 1 = 541
   53EX₁₂ = 5 × 12³ + 3 × 12² + 11 × 12¹ + 10 × 12⁰ = 8,640 + 432 + 132 + 10 = 9,214

Binary to Decimal

• A bit is a binary digit. It can be either 0 or 1.

• A byte is eight bits.

• Convert the one byte-binary numbers 00001011 and 00111001 to decimal.
  Ans: 00001011₂ = 8 + 2 + 1 = 11; 00111001₂ = 32 + 16 + 8 + 1 = 57.

Decimal to Binary

• Convert the decimal numbers 14, 75 and 113 to one-byte binary. Ans:
  14 - 8 = 6; 6 - 4 = 2; 2 - 2 = 0; 14 = 8 + 4 + 2 = 00001110₂
  75 - 64 = 11; 11 - 8 = 3; 3 - 2 = 1; 1 - 1 = 0; 64 + 8 + 2 + 1 = 01001011₂
  113 - 64 = 49; 49 - 32 = 17; 17 - 16 = 1; 1 - 1 = 0; 64 + 32 + 16 + 1 = 01110001₂

Hex <--> Binary

• Hex means hexadecimal or base-16.

• Hex requires sixteen digits:
  0   1   2   3   4   5   6   7   8   9   A   B   C   D   E   F

• Here is a table of Hex Digits, which includes their translation to decimal and binary.

• To convert from hex to binary, translate digit by digit into binary.

• Example:   Convert the following from hex to binary: 2AE9
  Ans: 2AE9₁₆ = 00101010 11101001₂

• To convert from binary to hex, translate each nybble (four bit group).

• Example:   Translate the two-byte binary number 10111010 00111111 into hex.
  Ans: 10111010 00111111₂ = BA 3F₁₆

ASCII Codes

• ASCII means American Standard Code for Information Interchange.

• Example:   Show how the letter A is encoded as binary ASCII.
  Look up A in the Hex column of the ASCII Table of Printable Characters to get 41₁₆. Translate this into binary to obtain 01000001₁₆.

• Example:   Translate the message !%TUNA9*>~ into binary ASCII.
  Ans: !%TUNA9*>~ = 21 25 54 55 4E 41 39 2A 3E 7E =
  00100001 00100101 01010100 01010101 01001110
  01000001 00111001 00101010 00111110 01111110

• Example:   Translate the binary ASCII message
  00110010 00111010 00110011 00110000

  into characters.
  Ans: 00110010 00111010 00110011 00110000 = 32 3A 33 30 = 2:30

Signed Binary Numbers

• Signed binary numbers do not use a sign that is separate from the number.

• Signed binary numbers are divided into two groups:
  00000000 to 01111111 represent 0 to 127.

  11111111 to 10000000 represent -1 to -128.

• Here is a table showing the relationship between binary and decimal signed numbers.

• Note that numbers with a leading 0 are positive; numbers with a leading 1 are negative.

• Signed binary numbers work like the odometer on a car. Starting with a new car with odometer set at 00000000 and backing up one mile out of the dealership parking lot results in an odometer reading of 11111111, which corresponds to -1.

• Example: How is -9 represented as signed binary? Ans:

  Signed Binary	Decimal
  00000000	0
  11111111	-1
  11111110	-2
  11111101	-3
  11111100	-4
  11111011	-5
  11111010	-6
  11111001	-7
  11111000	-8
  11110111	-9

   

• Converting Decimal to Signed Binary & Signed Binary to Decimal
  The algorithm is quite easy, it goes like this: Suppose we use 8 bits to represent numbers.

  If x is an integer, -128 <= x <= -1, then x represented as a signed binary number is flip(|x|₂) + 1.

  In the above, |x| means the absolute value of x and to find the flip of any binary number just reverse each bit of the number (i.e., if a bit is 0 make it 1 and if it's 1 make it 0).

• Examples:
  1. How is -9 represented as signed binary?   Ans.
     Since |-9| = 9 and 9₂ = 00001001, flip(|-9|₂) = 11110110 and therefore flip(|-9|₂) + 1 = 11110111
  2. How is -103 represented as signed binary?   Ans.
     103₂ = 01100111 and flip(01100111) + 1 = 10011001

• The above process is reversible. That is, if x is a negative signed binary number, then to find it's decimal equivalent flip the bits of x, add 1, convert the result to decimal and prefix it with a negative sign.

• Examples:
  1. Convert 11010100 to decimal.   Ans.
     flip(11010100) + 1 = 00101011 + 1 = 00101100. This is 44 in decimal. So our answer is -44.
   

Activity A10

• Work on Activity A10.