1. COMS 161 Introduction to Computing
Title: Numeric Processing
Date: November 03, 2004
Lecture Number: 28

2. Announcements
Research paper proposal 2 due
November 8, 2004

3. Review
Integers
Big-endian
Little-endian
Overflow

4. Outline
Real numbers
Representation
Limitations

5. Two’s complement 32 bit signed numbers: maxint minint
= 010
= + 110 =
= + 2,147,483,64610
= + 2,147,483,64710
= – 2,147,483,64810
= – 2,147,483,64710
= – 2,147,483,
= – 310
= – 210
= – 110
maxint
minint

6. Real Numbers Scientific Notation Normalized scientific notation
Number times 10 to a power
= * 103
Normalized scientific notation
Zero to the left of the decimal point
= * 106
Advantageous to store real numbers in 32 bits (word size)

7. Real Numbers 0.1012 = 0 * 20 + 1 * 2-1 + 0 * 2-2 + 1 * 2-3
/ / / /16
= 0 * * * * 2-3
= =

8. Real Numbers Decimal to binary conversion algorithm
1210 = 0 * * 23
1210 – 810 = 410 2 1 1
410 = 1 * 22
410 – 410 = 010

9. Real Numbers
= 0.?????2
20 = 1
2 -1 = 0.5 1
= 1 * 2 -1
2 -2 = 0.25
– 0.5 =
2 -3 = 0.125
2 -4 =
= 1 * 2 -2 1
2 -5 =
– 0.25 =
2 -6 =
= 1 * 2 -3 1
2 -7 =
– = 010
2 -8 = 0. 2

10. Real (Decimal) Number Storage
Real numbers are stored in floating point representation
IEEE Standard 754
Allows using data on different machines
A sign
An exponent
A mantissa also called a significand (normalized decimal fraction)
Single digit to the left of the decimal point