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Binary Representation

Binary Representation. Binary Representation for Numbers. Assume 4-bit numbers 5 as an integer 0101 -5 as an integer How? 5.0 as a real number How? What about 5.5?. Sign Bit. R eserve the most-significant bit to indicate sign Consider integers in 4 bits

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Binary Representation

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  1. Binary Representation

  2. Binary Representation for Numbers • Assume 4-bit numbers • 5 as an integer • 0101 • -5 as an integer • How? • 5.0 as a real number • How? • What about 5.5?

  3. Sign Bit • Reserve the most-significant bit to indicate sign • Consider integers in 4 bits • Most-significant bit is sign: 0 is positive, 1 is negative • The 3 remaining bits is magnitude • 0010 = 2 • 1010 = -2 • How many possible combinations for 4 bits? • How many unique integers using this scheme?

  4. Two’s Complement • Advantages • # of combinations of bits = # of unique integers • Addition is “natural” • Convert to two’s complement (and vice versa) • invert the bits • add one • ignore the extra carry bit if present • Consider 4-bit numbers • 0010[2] -> 1101 -> 1110 [-2]

  5. Addition • 0010 [2]+ 1110 [-2] • 0000 [ignoring the final carry—extra bit] • 0011[3] + 1110[-2] • 0001 [1] • 1110 [-2] + 1101[-3] • 1011 [-5]

  6. Range of Two’s Complement • 4-bit numbers • Largest positive: 0001 (binary) => 7 (decimal) • Smallest negative: 1000 (binary) => -8 (decimal) • # of unique integers = # of bit combinations = 16 • n bits • ?

  7. Binary Real Numbers • 5.5 • 101.1 • 5.25 • 101.01 • 5.125 • 101.001 • 5.75 • 101.11

  8. 8 bits only • 5.5 • 101.1 -> 000101 10 • 5.25 • 101.01 -> 000101 01 • 5.125 • 101.001 -> ?? • With only 2 places after the point, the precision is .25 • What if the point is allowed to move around?

  9. Floating-point Numbers • Decimal • 54.3 • 5.43 x 101 [scientific notation] • Binary • 101.001 • 10.1001 x 21 [more correctly: 10.1001 x 101] • 1.01001 x 22 [more correctly: 1.01001 x 1010] • What can we say about the most significant bit?

  10. Floating-point Numbers • General form: sign 1.mantissa x 2exponent • the most significant digit is right before the dot • Always 1 [no need to represent it] • Exponent in Two’s complement • 1.01001 x 22 • Sign: 0 (positive) • Mantissa: 0100 • Exponent: 010 (decimal 2)

  11. Java Floating-point Numbers • Sign: • 1 bit [0 is positive] • Mantissa: • 23 bits in float • 52 bits in double • Exponent: • 8 bits in float • 11 bits in double

  12. Imprecision in Floating-Point Numbers Floating-point numbers often are only approximations since they are stored with a finite number of bits. Hence 1.0/3.0 is slightly less than 1/3. 1.0/3.0 + 1.0/3.0 + 1.0/3.0 could be less than 1. www.cs.fit.edu/~pkc/classes/iComputing/FloatEquality.java

  13. Abstraction Levels • Binary • Data • Numbers (unsigned, signed [Two’s complement], floating point) • Text (ASCII, Unicode) • HTML • Color • Image (JPEG) • Video (MPEG) • Sound (MP3) • Instructions • Machine language (CPU-dependent) • Text (ASCII) • Assembly language (CPU-dependent) • High-level language (CPU -independent: Java, C++, FORTRAN)

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