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Number Systems

Number Systems. Lecture 10 Digital Design and Computer Architecture Harris & Harris Morgan Kaufmann / Elsevier, 2007. Binary Representations. What kind of numbers do you know how to represent using binary representations?. Fractional Numbers. Fixed-Point Floating-Point.

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Number Systems

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  1. Number Systems Lecture 10 Digital Design and Computer Architecture Harris & Harris Morgan Kaufmann / Elsevier, 2007

  2. Binary Representations • What kind of numbers do you know how to represent using binary representations?

  3. Fractional Numbers • Fixed-Point • Floating-Point

  4. Fixed-Point Numbers • How do you represent 6.510 using an 8-bit binary representation with 4 integer bits and 4 fraction bits?

  5. Fixed-Point Numbers • How do you represent -6.510 using an 8-bit binary representation with 4 integer bits and 4 fraction bits?

  6. Floating-Point Numbers • Like scientific notation

  7. Floating-Point Numbers • How do you represent the value 22810 using a 32-bit floating point representation?

  8. Floating-Point Numbers

  9. Floating-Point Example • How do you represent the value 5810 using a 32-bit floating point representation?

  10. Floating-Point Numbers: Special Cases • How do you represent the value 0 using IEEE 754 32-bit floating-point notation?

  11. Floating-Point Numbers: Precision • Single-Precision: • 32-bit notation • 1 sign bit, 8 exponent bits, 23 fraction bits • bias = 127 • Double-Precision: • 64-bit notation • 1 sign bit, 11 exponent bits, 52 fraction bits • bias = 1023

  12. Floating-Point Numbers: Rounding • Overflow, Underflow • Rounding modes: • down • up • toward zero • to nearest

  13. Floating-Point Addition • Extract exponent and fraction bits • Prepend leading 1 to form mantissa • Compare exponents • Shift smaller mantissa if necessary • Add mantissas • Normalize mantissa and adjust exponent if necessary • Round result • Assemble exponent and fraction back into floating-point format

  14. Floating-Point Addition: Example Add the following floating-point numbers: 1.510 3.2510 Start by representing the numbers in IEEE 754 single-precision floating-point notation.

  15. Floating-Point Addition: Example

  16. Next Time • Sequential Building Blocks • Memory Arrays

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