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Binary Representation

This guide explores how binary representation works for both integers and floating-point numbers using a 4-bit system. We delve into the two's complement method for signed integers, explaining how negative values are represented and the range of possible values. Additionally, we cover binary real numbers, scientific notation, and floating-point representation, highlighting challenges such as precision and approximation. Lastly, we touch on the representation of different data types in computing, including signed, unsigned, and floating-point numbers.

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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: 0111 (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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