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Information Technology
ITEC 1000
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Introduction to Information Technology ITEC 1000 – Winter 2010 – Peter Khaiter Lecture 5 – Floating Point Numbers – Feb 1 Floating Point Numbers - Real numbers - Used in computer when the number o Is outside the integer range of the computer (too large or too small) o Contains a decimal fraction Format - Mantissa: stored in sign-magnitude format - Assume decimal point located at the beginning of mantissa - Exponent stored in Excess-N notation: Complementary notation o Pick middle value as offset where N is the middle value: 0..99  e.g., excess-50 Floating Point Format: Excess-50 - First digit represents the sign of mantissa o 0 is used as a “+“sign o 5 is used as a “-“sign (arbitrarily) - Two next digits represent exponent in excess-50 - Five last digits represent mantissa o Fixed decimal point located at the beginning Normalization - Shift numbers left by increasing the exponent until leading zeros eliminated - Converting decimal number into standard format o Provide number with exponent (0 if not yet specified) o Increase/decrease exponent to shift decimal point to proper position o Decrease exponent to eliminate leading zeros on mantissa o Correct precision by adding 0’s or discarding/rounding least significant digits Floating Point Calculations - Addition and subtraction o Exponent and mantissa treated separately o Exponents of numbers must agree  Align decimal points  Least significant digits may be lost o Mantissa overflow requires exponent again shifted right Multiplication and Division - Mantissas: multiplied or divided - Exponents: added or subtracted o Normalization necessary to  Restore location of decimal point  Maintain precision of the result - Adjust excess value since added twice o Example: 2 numbers with exponent = 53 represented in excess-50 notation o 53 + 53 =106 o Since 50 added twice, subtract: 106 – 50 =56 - Maintaining precision: o Normalizing and rounding multiplication IEEE
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