1. Computer Science Engineering B.E.(4 th sem) c omputer system organization Topic-Floating and decimal arithmetic S ubmitted to– Prof. Shweta Agrawal Submitted by- Shivani jyotiki Prabha pal Sandhya uikey

2. Contents  History  Basic terms  Real numbers  Floating representation  Floating point multiplication flowchart  Rounding and Errors

3. History  The first floating point representation was firstly used in “V1” machine (1945). It had 7- bit exponent, 16-bit mantissa, and a sign bit.  In 1954, floating point representation was used by IBM for the modern computing system.  In 1962, the UNIVAC 1100/2200 series was introduced. It contains single precision and double precision.

4. Computers are integer machines and are capable of representing real numbers only by using complex codes. The most popula code for reprasenting real numbers is called the IEEE FLOTING –POINT STANDARD. The floting point number is derived from the fact that there is no fixer number of digit before and after the decimal; That is, the decimal point can flot.if the no. of digit before and after the decimal point is set, called fixed point representation. Floting point is slower than fixed point.

5. Basic Terms  Scientific notation: A notation that renders numbers with a single digit to the left of the decimal point.  Normalized: A number in floating-point notation that has no leading 0s.  Floating point: Computer arithmetic that represents numbers in which the binary point is not fixed.  Fraction: The value, between 0 and 1, placed in the fraction field of the floating point.  Exponent: In the numerical representation system of floating-point arithmetic, the value that is placed in the exponent field.

6. Real Numbers  Two’s complement representation deal with signed integer values only.  Without modification, these formats are not useful in scientific or business applications that deal with real number values.  Floating-point representation solves this problem.

7. Floating-Point Representation  We introduce a hypothetical “Simple Model” to explain the concepts  In this model:  A floating-point number is 14 bits in length  The exponent field is 5 bits  The significand field is 8 bits

8.  Floating-point numbers allow an arbitrary number of decimal places to the right of the decimal point.  For example: 0.5  0.25 = 0.125  They are often expressed in scientific notation.  For example: 0.125 = 1.25  10 -1 5,000,000 = 5.0  10 6 Floating Point Representation

9.  Computers use a form of scientific notation for floating-point representation  Numbers written in scientific notation have three components:

10. Floating Point Multiplication flowchart

11. Rounding and Errors  Floating-point overflow and underflow can cause programs to crash.  Overflow occurs when there is no room to store the high-order bits resulting from a calculation.  Underflow occurs when a value is too small to store, possibly resulting in division by zero.