Number Systems

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Number SystemsNumber Systems

Lecture 10Digital Design and Computer Architecture

Harris & HarrisMorgan Kaufmann / Elsevier, 2007

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Binary RepresentationsBinary Representations

• What kind of numbers do you know how to represent using binary representations?

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Fractional NumbersFractional Numbers

• Fixed-Point• Floating-Point

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Fixed-Point NumbersFixed-Point Numbers

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

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Fixed-Point NumbersFixed-Point Numbers

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

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Floating-Point NumbersFloating-Point Numbers

• Like scientific notation

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Floating-Point NumbersFloating-Point Numbers

• How do you represent the value 22810 using a 32-bit floating point representation?

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Floating-Point NumbersFloating-Point Numbers

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Floating-Point ExampleFloating-Point Example

• How do you represent the value 5810 using a 32-bit floating point representation?

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Floating-Point Numbers: Special CasesFloating-Point Numbers: Special Cases

• How do you represent the value 0 using IEEE 754 32-bit floating-point notation?

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Floating-Point Numbers: PrecisionFloating-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

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Floating-Point Numbers: RoundingFloating-Point Numbers: Rounding

• Overflow, Underflow• Rounding modes:

– down– up– toward zero– to nearest

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Floating-Point AdditionFloating-Point Addition

1. Extract exponent and fraction bits

2. Prepend leading 1 to form mantissa

3. Compare exponents

4. Shift smaller mantissa if necessary

5. Add mantissas

6. Normalize mantissa and adjust exponent if necessary

7. Round result

8. Assemble exponent and fraction back into floating-point format

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Floating-Point Addition: ExampleFloating-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.

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Floating-Point Addition: ExampleFloating-Point Addition: Example

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Next TimeNext Time

• Sequential Building Blocks

• Memory Arrays