Floating Point Representation premium.pptx

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1. FLOATING POINT REPRESENTATION OF NUMBERS

2. WHAT IS COMPUTER ARITHMETIC ? COMPUTER ARITHMETIC is a field of COMPUTER SCIENCE that investigates how computers should represent numbers & perform operations on them…

3. COMPUTER ARITHMETIC TWO TYPES OF COMPUTER ARITHMETIC INTEGER ARITHMETIC REAL ARITHMETIC

4. INTEGER ARITHMETIC Arithmetic without fractions. A computer performing integer arithmetic ignores any fractions that are derived…. REAL ARITHMETIC Arithmetic which uses numbers with fractional parts & is used in most computations…

5. REAL ARITHMETIC TWO TYPES OF REAL ARITHMETIC FIXED POINT ARITHMETIC FLOATING POINT ARITHMETIC

6. FIXED POINT ARITHMETIC In computing, FIXED-POINT number representation is a real data type for a number. With the help of fixed number representation ,data is converted into binary form, and then data is processed, stored & used by the system Sign bit Integral part Fractional part 1 bit(0 or 1) 9 bits 6 bits

7. The fixed-point numbers in binary uses a sign bit. A positive number has a sign bit 0,while a negative number has a sign bit 1 Sign bit Integral part The integral part is of different lengths at different places. It depends on the register’s size, like in an 8-bit register, integral part is 4 bits… Fractional part Fractional part is also of different lengths at different places. It depends on the register’s size, like in an 8-bit register, integral part is of 3 bits

8. 8 bits = 1 sign bit + 4 bits (integral) + 3bits (fractional part) 16 bits = 1 sign bit + 9 bits (integral) + 6 bits (fractional part) • • = assigned as sign bit = assigned as integral part = assigned as fractional part • = assigned as assumed binary point

9. Number is 4.5 Convert the number into binary form , 4.5 = 100.1 Represent binary number in Fixed point notation HOW TO WRITE THE NUMBER IN FIXED-POINT NOTATION 1 1

10. Magnitude the maximum and minimum (in magnitude) numbers that may be stored are: 111111111.111111 2 = (2 9 - 1) + (1 - 2 -6 ) (Maximum) = 511.984375 10 000000000.000001 2 = 2 -6 (Minimum) = 0.015625 10 Magnitude of fixed point representation : 0.015625 10 to 511.984375 10 This range is quite insufficient in practice and so a different rule is adopted to represent real numbers.

11. FLOATING POINT REPRESENTATION FLOATING POINT NOTATION SCIENTIFIC NOTATION NORMALIZED NOTATION

12. Scientific Notation Method of representing numbers into a form. Scientific notation is further converted into floating point notation because floating notation only accepts scientific notation. For example- Number = 376.423 ( its not scientific notation ) Number in scientific notation = 37.6423 or 3.76423  

13. Normalized Notation Where m means MANTISSA , b means BASE , e means EXPONENTIAL m *   It is a special case of scientific notation. Normalized means The shifting of the mantissa to the left till its most significant bit is non-zero. Normalized notation-

14. Sign bit The fixed-point numbers in binary uses a sign bit. A positive number has a sign bit 0,while a negative number has a sign bit 1. In floating point representation, sign of a number always depends on mantissa, not on exponent. Hence sign bit in the format is always for mantissa and not for the exponent. Mantissa Mantissa part is of different length at a difference place . It depends on the size of the register like in 16 bit register, mantissa part is of 9 bits. Exponent Exponent is the power of the number. It depends on the registers' size; like in the 16 bit register , exponent part is 7 bits. Excess 16,64,128,512 are used to store exponent in this format.

15. Four things are used to represent a floating point number Sign of Mantissa Sign of Exponent Magnitude of Mantissa Magnitude of Exponent

16. The number 1011.0101 x is represented in this notation as-   0.10110101 x = 0.10110101E01011   Where mantissa is 0.10110101 & the exponent is 1011 1 1 1 1 1 1 1 1 Mantissa (9 bits) Exponent(7bits) Sign of mantissa Sign of exponent Implied binary point

17. Magnitude T he range of numbers (magnitude) that may be stored will be: Maximum = 0.11111111E0111111 = (1 – 2 −8 ) x 2 63 Minimum = 0.10000000E1111111 = 2 −1 x = 2 −64 Magnitude of floating point representation: 2 −64 to 2 63   This range is much larger than the range 2 9 to 2 -6 obtained with the fixed point representation     Calculation(mantissa) n=-8 ∑ (2 n )=2 -1 +2 -2 +…+2 -8 n=-1 =(1-2 -8 ) Calculation(exponent) n=5 ∑ (2 n )=2 +2 1 +…+2 5 n=5 =( 2 6 -1)

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