1. Prof. Giancarlo Succi, Ph.D., P.Eng. E-mail: Giancarlo.Succi@unibz.it
Binary Numbers
Prof. Giancarlo Succi, Ph.D., P.Eng.
Prof. G. Succi, Ph.D., P.Eng.

2. Content Number and numerals Numeric systems Base conversion
Finite precision numbers
Binary arithmetic
Floating point representations
Prof. G. Succi, Ph.D., P.Eng.

3. Numbers And Numerals Number: abstract entity
Numeral: string of symbols that represent a number in a given system
The same number can be represented by different numerals in different numeric systems
Example
234 in decimal system
CCXXXIV using roman system
Discuss the role of roman numbers
Present the concept of base as “up-to”
Start doing the base 2 as 0, 1, 10, 11, 100, 101, …
Then show the base 3, 4 etc.
Explain why the base 2 is so important
The same numeral stands for different numbers if different systems are used
Example
(10)10=10
(10)2 = ( 2 )10= 2
Prof. G. Succi, Ph.D., P.Eng.

4. Numeric Systems (1) To define a Numeric System we need:
a set of symbols that we will call digits (E.g. 1,2,3, X,K, ….)
some rules to build up numbers
We can define Positional Numeric System or Non Positional Numeric Systems
Non Positional Numeric System: the value of digits in the number is position independent
Example
In Roman Numeric System:
the symbol V means 5 always, but IV  VI ….
Prof. G. Succi, Ph.D., P.Eng.

5. Numeric Systems (2)
Positional Numeric System: digit value depends on its position within the number (weight)
Each digit represents the coefficient of a power of the base
Exponent is given by the position of the digit within the number
position m-1
position -1
position -k
position m
position 0
comma
position -2
Prof. G. Succi, Ph.D., P.Eng.

6. Numeric Systems (3) The Decimal System 125.42
Positional Numeric System: digit value depends on its position within the number (weight)
Example
The Decimal System
125.42
Prof. G. Succi, Ph.D., P.Eng.

7. Numeric Systems (4) The Binary System 101.01 Example
Prof. G. Succi, Ph.D., P.Eng.

8. Numeric Systems (5) Other Systems Used Octal System: Example base = 8
symbols used = 0,1,2,3,4,5,6,7
Hexadecimal System:
base = 16
symbols used = 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Prof. G. Succi, Ph.D., P.Eng.

9. Base Conversion (decimal to binary)
Is obtained by repeatedly dividing the number by 2, the remainder of division at each step is the digit of the binary number (starting from less significant digits) ( ) 2 10
11010 26
digit t
significan
more 1 / 3 6 13
less = ï þ ý ü
rem
The first method is the one used
Ask a few students to do it
Tell to remember to do it the other way around
… or by determining the powers of 2 contained in the number:
Prof. G. Succi, Ph.D., P.Eng.

10. Base Conversion (binary to decimal )
Is obtained by strictly applying the definition:
How do we convert binary to hexadecimal? Show the long process and then the shortcut for it.
Same for octal.
Prof. G. Succi, Ph.D., P.Eng.

11. Finite precision numbers (1)
Finite Precision numbers: numbers with a finite number of digits.
Some properties are lost:
operators closure
distributive, associative properties
holes in the representation of real numbers
Example
Lets use numbers with 2 decimal numbers with sign
Interval represented: [+99, -99]
Closure respect operator + is lost: = ??? (106 is out of the interval)
Associative property: 25+ (90-30)  (25+ 90)-30
Prof. G. Succi, Ph.D., P.Eng.

12. Finite precision numbers (2)
Example
Lets use consider numbers with just two two decimal digit:
Interval represented: [+0.99, -0.99]
We cannot represent a number between 0.05 and 0.06
Prof. G. Succi, Ph.D., P.Eng.

13. Represented Intervals
By representing positive integers in binary notation, n digits (bits), cover the interval [0,2n-1]
It is easy to note that if the maximum number representable using n bit is
X = 2n -1
then to represent number X are necessary
n = Int( log2 (X)+1)
bits
Example
using n=3 interval [0, 7] is completely represented
Int is the next integer
Prof. G. Succi, Ph.D., P.Eng.

14. Positive and negative integers in base 2 (1)
To represent Integer Relative Numbers, if the same number of digits is used, the interval of absolute is halved
Several notations are used:
Signed Magnitude
the first bit is used for the sign 0 : + ; 1 : -
n-1 bits are used for the magnitude
represented interval: [-2n-1 +1, 2n-1 -1]
Note that 0 has two possible representations
Example
using n=4 interval [-7, 7] is completely represented
Representations: 1000 and 0000
Prof. G. Succi, Ph.D., P.Eng.

15. Positive and negative integers in base 2 (2)
One’s Complement
a 0 is added on left of positive numbers
to change sign the numeral is complemented bit by bit
positive numerals start with 0, negative ones with 1
n-1 bits are used for the magnitude
represented interval: [-2n-1 +1, 2n-1 -1]
Note that 0 has two representations
Example
using n=4 interval [-7, 7] is completely represented
( -7+2)
Representations for 0: 1000 and 0000
Prof. G. Succi, Ph.D., P.Eng.

16. Positive and negative integers in base 2 (3)
Two’s Complement
a 0 is added on left of positive numbers
negative numbers are obtained by complementing bit by bit the corresponding positive a and then adding 1
positive numerals start with 0, negative ones with 1
n-1 bits are used for the magnitude
represented interval: [-2n-1, 2n-1 -1]
We earned a number by giving 0 just one representation
Example
using n=4 interval [-8, 7] is completely represented
Prof. G. Succi, Ph.D., P.Eng.

17. Positive and negative integers in base 2 (4)
Excess Notation
numbers are represented as a sum of the number with a power of 2
Using n bits excess 2n-1 is used
represented interval: [-2n-1, 2n-1 -1]
practical rule: numerals are obtained by complementing the highest digit in 2’s complement notation
Example
using n=4 interval [-8, 7] excess=8
= 
= 
Prof. G. Succi, Ph.D., P.Eng.

18. Binary Arithmetic (1) the sum
Sum in binary notation is performed bit by bit carrying the rest to next digit
Example
Prof. G. Succi, Ph.D., P.Eng.

19. Binary Arithmetic (2) the sum in 2’s C
In 2’sC sum and subtraction are managed in the same way, just throw away Carry
Carry is thrown away
If the two terms have different sign the result is always correct.
If the two terms have the same sign but the result has a different one ….
… we have an ERROR
Example
What about the sum using 8 bit representation in 2’sC
Overflow Error !
Prof. G. Succi, Ph.D., P.Eng.

20. Binary Arithmetic (3) multiplication
Easy ... repeated addition
….. or the usual rules learned for the product of integers or real numbers can be used, recalling the following multiplication table:
0 1
Prof. G. Succi, Ph.D., P.Eng.

21. Floating Point Notation
Example
Using Base 10
Using numerals with 5 digits of the kind  .XXX  EE
Mantissa =  .XXX three signed digits 0.1 | m |  1
Exponent =  EE two signed digits  e  99
Represented numbers are:
Represented interval is
Negative Overflow
Positive Overflow
Positive Underflow
Negative Underflow
Prof. G. Succi, Ph.D., P.Eng.

22. Standard IEEE 754 (1985)
Standard definition (i.e. architecture independent)
Simple precision ( uses 32 bits to represent sign, exponent and mantissa
Double precision (uses 64 bits)
There are notation using normalized mantissa or not
Some configurations of the exponent are reserved (i.e. not standardized)
+/- (1 bit)
Exponent (8 bit)
Mantissa (23 bit)
+/- (1 bit)
Exponent (11 bit)
Mantissa (52 bit)
Prof. G. Succi, Ph.D., P.Eng.

23. Floating Point Sum
To sum (or subtract) floating point numbers it is necessary to scale mantissa to have the same exponents in all terms
Example
n1 + n2 : using standard IEEE 754 notation in 32 bits
Note that lower terms is loosing digits
Prof. G. Succi, Ph.D., P.Eng.

24. Floating Point product
Multiply mantissa and sum exponents
If necessary mantissa is scaled and exponent is incremented/decremented
Example
n1 x n2 : using standard IEEE 754 notation in 32 bits
Prof. G. Succi, Ph.D., P.Eng.

25. Binary Arithmetic (4) division
For your knowledge
Binary Arithmetic (4) division
Need to represent fractions
Prof. G. Succi, Ph.D., P.Eng.

26. Fractional Representation
For your knowledge
Fractional Representation
Fixed Point Notation:
stores numerals as:
Floating Point Notation:
A n digit floating point number in base b is represented by.
Where is a numeral in base b called mantissa normalized with
and e is an integer called the exponent.
Prof. G. Succi, Ph.D., P.Eng.