1 / 22

Number Systems

Number Systems. Decimal number system Binary number system Hexadecimal number system Converting Negative numbers Character representation. Decimal (base 10). Uses positional representation Each digit corresponds to a power of 10 based on its position in the number

aletta
Télécharger la présentation

Number Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Number Systems • Decimal number system • Binary number system • Hexadecimal number system • Converting • Negative numbers • Character representation

  2. Decimal (base 10) • Uses positional representation • Each digit corresponds to a power of 10 based on its position in the number • The powers of 10 increment from 0, 1, 2, etc. as you move right to left • 1,479 = 1 * 103 + 4 * 102 + 7 * 101 + 9 * 100

  3. Binary (base 2) • Two digits: 0, 1 • To make the binary numbers more readable, the digits are often put in groups of 4 • 1010 = 1 * 23 + 0 * 22 + 1 * 21 + 0 * 20 = 8 + 2 = 10 • 1100 1001 = 1 * 27 + 1 * 26 + 1 * 23 + 1 * 20 = 128 + 64 + 8 + 1 = 201

  4. Hexadecimal (base 16) • Shorter & easier to read than binary • 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F • “0x” often precedes hexadecimal numbers • 0x123 = 1 * 162 + 2 * 161 + 3 * 160 = 1 * 256 + 2 * 16 + 3 * 1 = 256 + 32 + 3= 291

  5. Hexadecimal (base 16) • Another example • 0xABC = A * 162 + B * 161 + C * 160 = 10 * 256 + 11 * 16 + 12 * 1 = 2560 + 176 + 12= 2748

  6. Converting • See “Links” on webpage for link to converter between binary, decimal, and hexadecimal values

  7. Converting • From binary or hex to decimal • Use positional representation as shown previously • From decimal to binary or hex • Keep dividing by 2 (or 16) • Remainders give the digits, starting from lowest power • From binary to hex (or vice versa) • Replace each set of four binary digits by the corresponding hexadecimal digit (or vice versa)

  8. Convert Decimal to Binary • 22 decimal = 10110 binary • Calculations: 2 |22r. 0 2 |11r. 12 | 5r. 12 | 2r. 02 | 1r. 1 0

  9. Convert Decimal to Binary • 22 decimal = 10110 binary • Calculations: • Write down powers of 2 • Subtract the largest power of 2 that is less than the number • Add a 1 if can subtract, and a 0 if cannot 22 6 2 32 16 8 4 2 11 0 1 1 0

  10. Class Exercise • Bin to dec: 1001 0011 • Dec to bin: 105 • Bin to hex: 0010 1110 1000 1011 • Hex to bin: 0xFEDC • Hex to dec: 0x10A • Dec to hex: 165

  11. Other Bases • Base 8 (octal number system) • 123 = 1 * 82 + 2 * 81 + 3 * 160 = 1 * 64 + 2 * 8 + 3 * 1 = 64 + 16 + 3 = 83 • Base 13 • 123 = 1 * 132 + 2 * 131 + 3 * 130 = 1 * 169 + 2 * 13 + 3 * 1 = 169 + 26 + 3= 198

  12. Unsigned Integers • Represents positive integers only • Not necessary to indicate a sign, so all bits can be used for the magnitude: • 1 byte = 8 bits = 28 = 256 (0 to 255) • 255 decimal = 1111 1111 binary • 2 bytes = 16 bits = 216 = 65,536 (0 to 65,535) • 4 bytes = 32 bits = 232 = 4,294,967,296 (0 to 4,294,967,295)

  13. Signed Integers • Represents positive and negative integers • MSB (Most Significant Bit – leftmost bit) used to indicate sign • 0 = positive, 1 = negative • One less bit is used for the magnitude, with one extra negative value • 1 byte = 8-1 bits = 27 (-128 to +127) • -128 decimal = 1000 0000 binary • -1 decimal = 1111 1111 binary • 127 decimal = 0111 1111 binary • 2 bytes = 16-1 bits = 215 (-32,768 to +32,767) • 4 bytes = 32-1 bits = 231 (-2,147,483,648 to +2,147,483,647 )

  14. 1’s & 2’s Complement • 1’s complement form • Formed by reversing (complementing) each bit • 2’s complement form • Formed by adding 1 to 1's complement • Negative numbers are stored this way • Additive inverse of a number • Computer never has to subtract • A – B = A + (-B)

  15. Binary to Decimal Conversion • Unsigned Integers • Convert binary directly to decimal form • Signed Integers • If MSB = 0, convert directly to decimal • If MSB = 1, convert to 2's complement form (reverse the bits & add 1), then to decimal form

  16. Signed Integers (8-bit) • Example: -9 0000 1001b = +9 1111 0110b +1b 1111 0111b = -9 • Example: -32 0010 0000b = +32 1101 1111b +1b 1110 0000b = -32 • Convert to decimal 1111 0111b = -9 0000 1000b +1b0000 1001b = +9 • Convert to decimal 1110 0000b = -32 0001 1111b +1b 0010 0000b = +32

  17. Class Exercise • Convert the following negative decimal numbers to 8 bit binary using the 2’s complement • -1, -3, -8, -127

  18. Class Exercise • Evaluate 32 + (- 5) in 8 bit binary • Convert 32 to binary • Convert -5 to 2’s complement • Add together • Check your answer by converting back to decimal

  19. Character Representation • ASCII (See link to ASCII Codes) • American Standard Code for Information Interchange • Standard encoding scheme used to represent characters in binary format on computers • 7-bit encoding, so 128 characters can be represented • 0 to 31 (& 127) are "control characters" (cannot print) • Ctrl-A or ^A is 1, ^B is 2, etc. • Used for screen formatting & data communication • 32 to 126 are printable

  20. Binary Data • Decimal, hex & character representations are easier for humans to understand; however… • All the data in the computer is binary • An int is typically 32 binary digits • int y = 5; (y = 0x00000005;) • In computer y = 00000000 00000000 00000000 00000101 • int z = -5; (y = 0xFFFFFFFB;) • In computer, z = 11111111 11111111 11111111 11111011

  21. Binary Data • A char is typically 8 binary digits • char x = '5'; (or char x = 53, or char x = 0x35) • In computer, x = 00110101 • char x = 5; (or char x = 0x05;) • In computer, x = 00000101 • Note that the ASCII character 5 has a different binary value than the numeral 5 • Also note that 1 ASCII character = 2 hex numbers

More Related