Chapter 13: Number Systems

example 13.1, Page No. 474

In [20]:
# Decimal to binary conversion: 10

import math
#variable declaration
N = 10                      # decimal no to be convered to binary
k = N
#Calculations
i = 0
b =[0,0,0,0,0,0]
while(N!=0):
  b[i] = N%2
  N = N/2
  i = i+1 

#Result
print("Binary equivalent of decimal %d is %d%d%d%d"%(k,b[3],b[2],b[1],b[0]))

Binari equivalent of decimal 10 is 1010

example 13.2, Page No.474

In [23]:
# Decimal to binary conversion: 25

import math
#variable declaration
N = 25                      # decimal no to be convered to binary

#Calculations
k = N
i = 0
b =[0,0,0,0,0,0]
while(N!=0):
  b[i] = N%2
  N = N/2
  i = i+1 

#Result
print("Binary equivalent of decimal %d is %d%d%d%d%d"%(k,b[4],b[3],b[2],b[1],b[0]))

Binari equivalent of decimal 25 is 11001

example 13.3, Page No. 474

In [25]:
# Binary to decimal:101110

import math
b0 = 0         # bit 0
b1 = 1         # bit 1
b2 = 1         # bit 2
b3 = 1         # bit 3
b4 = 0         # bit 4
b5 = 1         # bit 5

#Calculations
D = (2**0)*b0+(2**1)*b1+(2**2)*b2+(2**3)*b3+(2**4)*b4+(2**5)*b5

#Result
print("Decimal no is %d"%D)

Decimal no is 46

example 13.4, Page No. 475

In [33]:
# Decimal to binary conversion: 15,31

import math
#######################################################################################
# for N = 15
#variable declaration
N = 15                      # decimal no to be convered to binary

#Calculations
k = N
i = 0
b =[0,0,0,0,0,0]
while(N!=0):
  b[i] = N%2
  N = N/2
  i = i+1 

#Result
print("(a)\nBinary equivalent of decimal %d is %d%d%d%d%d"%(k,b[4],b[3],b[2],b[1],b[0]))

########################################################################################

# For N =31
#variable declaration
N = 31                      # decimal no to be convered to binary

#Calculations
k = N
i = 0
b =[0,0,0,0,0,0]
while(N!=0):
  b[i] = N%2
  N = N/2
  i = i+1 

#Result
print("Binary equivalent of decimal %d is %d%d%d%d%d"%(k,b[4],b[3],b[2],b[1],b[0]))
##########################################################################################
#Addition
c= bin(15+31)[2:]
print("(b) Addition of 15 and 31 is %s in binary. Decimal equivalent of %s is %d"%(c,c,int(c,2)))

(a)
Binari equivalent of decimal 15 is 01111
Binari equivalent of decimal 31 is 11111
(b) Addition of 15 and 31 is 101110 in binary. Decimal equivalent of 101110 is 46

example 13.5, Page No.475

In [42]:
# Binary subtraction

import math
#Variable declaration
n1 = 11001             # first number
n2 = 10001             # second number

#calculations
n1 = int(str(n1),2)
n2 = int(str(n2),2)
c = bin(n1-n2)[2:]

#Result
print("Subtraction result in binary = %s"%c)

Subtraction result in binary = 1000

example 13.6, Page No.475

In [41]:
# Binary subtraction

import math
#Variable declaration
n1 = 1010             # first number
n2 = 111              # second number

#calculations
n1 = int(str(n1),2)
n2 = int(str(n2),2)
c = bin(n1-n2)[2:]

#Result
print("Subtraction result in binary = %s"%c)

Subtraction result in binary = 11

example 13.7, Page No. 476

In [87]:
# 16- bit signed binary representation

import math
#Variable declaration
a = 8
b = -8
c = 165
d = -165

#Calculations
#c = bin(d)
a = format(a,'#018b')[2:]
b = format(b,'#018b')[3:]
c = format(c,'#018b')[2:]
d = format(d,'#018b')[3:]
str = '1'
#Result
print("In the leading bit we will have 1 to represent '-' sign\n")
print("(a) +8   --> %s"%(a))
print("(b) -8   --> %s%s"%(str,b))
print("(c) +167 --> %s"%(c))
print("(d) -167 --> %s%s"%(str,d))

a = format(-167%(1<<16),'016b')
b = format(167%(1<<16),'016b')
print a
print b

In the leading bit we will have 1 to represent '-' sign

(a) +8   --> 0000000000001000
(b) -8   --> 1000000000001000
(c) +167 --> 0000000010100101
(d) -167 --> 1000000010100101
1111111101011001
0000000010100111

example 13.8, Page No.477

In [95]:
# 2's complement

import math
# variable declaration
a = int('00011111',2)
b = int('11100101',2)
c = int('11110111',2)

#Calculations
a = format(-a%(1<<8),'08b')
b = format(-b%(1<<8),'08b')
c = format(-c%(1<<8),'08b')
print("(a) 2's complement of 00011111 --> %s " %a)
print("(b) 2's complement of 11100101 --> %s " %b)
print("(c) 2's complement of 11110111 --> %s " %c)

(a) 2's complement of 00011111 --> 11100001 
(b) 2's complement of 11100101 --> 00011011 
(c) 2's complement of 11110111 --> 00001001

example 13.9, Page No. 13.9

In [97]:
# 8-bit number range

import math
#calculations
a = int('01111111',2)
b = int('10000000',2)

#Result
print("largest positive 8-bit no in %d and smallest negative number is -%d"%(a,b))

largest positive 8-bit no in 127 and smallest negative number is -128

example 13.10, Page No.10

In [116]:
# 2's complement and addition,subtraction

import math
#Variable declaration
A = -24
B = 16
#Calculations
#2's complement
a = format(A%(1<<8),'08b')
b = format(-B%(1<<8),'08b')
# addition
c = bin(A+B)[3:]
c = format(-int(c,2)%(1<<8),'08b')
#Subtraction
d = bin(A-B)[3:]
d = format(-int(d,2)%(1<<8),'08b')
#Result
print("(a)\n2's complement of -24 --> %s " %a)
print("2's complement of 16 --> %s " %b)
print("(b) A+B = %s"%c) 
print("(c) A-B = %s"%d)

(a)
2's complement of -24 --> 11101000 
2's complement of 16 --> 11110000 
(b) A+B = 11111000
(c) A-B = 11011000

example 13.11, Page No. 479

In [130]:
# 2's complement and addition,subtraction

import math
#Variable declaration
A = -60
B = -28
#Calculations
#2's complement
a = format(A%(1<<8),'08b')
b = format(B%(1<<8),'08b')
# addition
c = bin(A+B)[3:]
c = format(-int(c,2)%(1<<8),'08b')
#Subtraction
x = B-A
d = bin(B-A)[2:]
d = format(int(d,2),'08b')
#Result
print("(a)\n2's complement of A --> %s " %a)
print("2's complement of B --> %s " %b)
print("(b) B+A = %s"%c) 
print("(c) B-A = %s"%d)

(a)
2's complement of A --> 11000100 
2's complement of B --> 11100100 
(b) B+A = 10101000
(c) B-A = 00100000

example 13.12, Page No. 479

In [230]:
# decimal to binary

# For N = 0.6875
#variable declaration
N = 0.6875                      # decimal no to be convered to binary

#Calculations
k = N
i = 0
b =[0,0,0,0,0,0,0,0,0,0,0,0]
while(N!=0):
    N = N*2
    #print N
    b[i]=0
    #print b[i]
    if (N>1) or(N==1):
        N =N-1
        b[i]=1
    i=i+1
    
#Result
print("Binary equivalent of decimal %f is 0.%d%d%d%d "%(k,b[0],b[1],b[2],b[3]))

Binary equivalent of decimal 0.687500 is 0.1011

example 13.13, Page No.480

In [222]:
# decimal to binary

# For N = 0.634
#variable declaration
N = 0.634                      # decimal no to be convered to binary

#Calculations
k = N
i = 0
b =[0,0,0,0,0,0,0,0,0,0,0,0]
while(i!=7):
    N = N*2
    #print N
    b[i]=0
    #print b[i]
    if (N>1) or(N==1):
        N =N-1
        b[i]=1
    i=i+1

#Result    
#Result
print("Binary equivalent of decimal %f is 0.%d%d%d%d%d%d%d "%(k,b[0],b[1],b[2],b[3],b[4],b[5],b[6]))

Binary equivalent of decimal 0.634000 is 0.1010001

example 13.14, Page No.480

In [225]:
# decimal to binary

# For N = 0.634
#variable declaration
N = 39.12                      # decimal no to be convered to binary

N1 = 39
k = N1
#Calculations
i = 0
b =[0,0,0,0,0,0,0,0]
while(N1!=0):
  b[i] = N1%2
  N1 = N1/2
  i = i+1 

#Result
print("Binary equivalent of decimal %d is %d%d%d%d%d%d%d"%(k,b[6],b[5],b[4],b[3],b[2],b[1],b[0]))


N2 =0.12
#Calculations
k = N2
i = 0
b =[0,0,0,0,0,0,0,0,0,0,0,0]
while(i!=7):
    N2 = N2*2
    #print N
    b[i]=0
    #print b[i]
    if (N2>1) or(N2==1):
        N2 =N2-1
        b[i]=1
    i=i+1
 
#Result
print("Binary equivalent of decimal %f is 0.%d%d%d%d%d%d%d "%(k,b[0],b[1],b[2],b[3],b[4],b[5],b[6]))

Binary equivalent of decimal 39 is 0100111
Binary equivalent of decimal 0.120000 is 0.0001111

example 13.15, Page No.481

In [227]:
# binary Addition

import math
#Variable declaration
a1 = int('101101',2)       # integer part of first no
a2 = int('0101',2)         # fractiona part of first no
b1 = int('10001',2)        # integer part of second no
b2 = int('1010',2)         # fractiona part of second no(in fraction we can add any no of 0s)

#Calculations
c1= bin(a1+b1)[2:]
c2 = bin(a2+b2)[2:]

print("Addition --> %s.%s"%(c1,c2))

Addition --> 111110.1111

example 13.16, Page No. 481

In [231]:
# binary to decimal conversion

import math
#Variable declaration
i0 = 1               # LSB of integer
i1 = 0
i2 = 0
i3 = 1
i4 = 1               # MSB of integer
f1 = 0
f2 = 0
f3 = 1
f4 = 0
f5 = 1
f6 = 1

#Calculations
D = i0*(2**0)+i1*(2**1)+i2*(2**2)+i3*(2**3)+i4*(2**4)+f1*(2**-1)+f2*(2**-2)+f3*(2**-3)+f4*(2**-4)+f5*(2**-5)+f6*(2**-6)

#Result
print("Decimal equivalent = %f"%D)

Decimal equivalent = 25.171875

example 13.17, Page No. 482

In [236]:
# Hexadecimal to decimal

import math
#variable declaration
a = 3
b = 10  # decimal equivalent of A
c = 8
#Calculations
# binary equvalent of  8            A                   3 
#                    1000         1010                0011
# a
Da = (1*2**0)+(1*2**1)+(0*2**2)+(0*2**3)+(0*2**4)+(1*2**5)+(0*2**6)+(1*2**7)+(0*2**8)+(0*2**9)+(0*2**10)+(1*2**11)
print("(a) Decimal equivalent of 8A3 is %d"%Da)
# b 
Db = (a*(16**0))+(b*(16**1))+(c*(16**2))
print("(b) Decimal equivalent of 8A3 is %d"%Db)

(a) Decimal equivalent of 8A3 is 2211
(b) Decimal equivalent of 8A3 is 2211

example 13.18, Page No. 482

In [260]:
# decimal to hexadecimal

import math
#Variable declaration
N = 268                # no to be converted into hexadecimal

N1 = 268
k = N1
#Calculations
i = 0
b =[0,0,0,0,0,0,0,0]
while(N1!=0):
  b[i] = N1%16
  N1 = N1/16
  i = i+1 

#Result
print("Remainder1: %d = C\nRemainder2: %d\nRemainder3: %d\n\nHexaecimal equivalent of %d = 10C"%(b[0],b[1],b[2],k))

Remainder1: 12 = C
Remainder2: 0
Remainder3: 1

Hexaecimal equivalent of 268 = 10C

example 13.19, Page No.483

In [259]:
# decimal to hexadecimal

import math
#Variable declaration
N = 5741                # no to be converted into hexadecimal

N1 = 5741
k = N1
#Calculations
i = 0
b =[0,0,0,0,0,0,0,0]
while(N1!=0):
  b[i] = N1%16
  N1 = N1/16
  i = i+1 

#Result
print("Remainder1: %d = D\nRemainder2: %d\nRemainder3: %d\nRemainder4: %d\n\nHexaecimal equivalent of %d =166D"%(b[0],b[1],b[2],b[3],k))

Remainder1: 13 = D
Remainder2: 6
Remainder3: 6
Remainder4: 1

Hexaecimal equivalent of 5741 =166D

example 13.20, Page No.483

In [262]:
# Hexadecimal to decimal

import math
#variable declaration
a = 0
b = 7
c = 13  #  decimal equivalent of D
#Calculations
D = (a*(16**0))+(b*(16**1))+(c*(16**2))

#Result
print("(b) Decimal equivalent of 8A3 is %d"%D)

(b) Decimal equivalent of 8A3 is 3440