Convert the following numbers to binary numbers DADA16 70072

Convert the following numbers to binary numbers (DA.DA)_16 (7007.2)_8 (3210.01)_10 (12345.4375)_10

Solution

a)(DA.DA)16

Method-1(hexa-decimal to decimal - decimal to binary)

To decimal
d*16^1+a*16^0.d*16^-1+a*16^-2=208+10.(0.8125+0.0390625)
           =218.8462837---->its decimal

Decimal to Binary(remainders are noted by continously dividing quotient)
218/2
quotient/2--->remainder
109/2--->0
54/2--->1
27/2--->0
13/2--->1
6/2--->1
3/2--->0
1/2--->1
1/2--->1-----> take remainders in reverse order i.e 11011010

For decimal point value(take decimal value and multiply the decimal point with16)
.8462837*2=1.69256756------>1
.69256756*2=1.3851351------->1
.3851351*2=0.7702702---------->0
0.7702702*2=1.5405405----------->1
0.5405405*2=1.081081----------->1
0.081081*2=0.1621621----------->0
0.162162*2=0.324324----------->1----------->1101101

the final binary converted number is 11011010.1101101

Method-2 (direct hexa-decimal to binary)
-------------

(da.da)16 => 11011010.11011010

Step 1 - Convert each hexa-decimal digit to a 4 digit binary number (the hexa-decimal digits may be treated as decimal for this conversion).
Step 2 - Combine all the resulting binary groups

d a .d a  
1101 1010 .1101 1010

b)

(7002.2)8 => 111000000010010

Step 1 - Convert each octal digit to a 3 digit binary number (the octal digits may be treated as decimal for this conversion).
Step 2 - Combine all the resulting binary groups (of 3 digits each) into a single binary number.

7   0   0   2. 2  
111   000   000   010.   010

c) (3210.01)4 .

3210.01
Generallu radix 4 not used converted it simalarly to hexa and octal
(3210.01) => 11100100.0001

Step 1 - Convert each 4 radix digit to a 2 digit binary number (the 4 radix digits may be treated as decimal for this conversion).
Step 2 - Combine all the resulting binary groups

3 2 1 0 .0 1
11 10 01 00 .00 01

d)

(12345.4375)10 ==> (11000000111001.0111)2

Decimal to Binary Conversion:
Step1: Consider the left side part of the decimal point
step2: Now divide the number by 2 and consider the remainder
step3: Repeat the step 2 until the gets divides(please see the below for reference)
Now note down the binary digits from bottom to top
step4: Now consider the digits after the decimal point(right side)
Step5: Now multiply the number with 2 and consider the binary value before the decimal point.
Step6: Repeat step5 until we are getting the repeated binary digits
Step7: Combine the both binary values which we calculated earlier from left part of the decimal as well as right side.

12345=>
12345/2=
Number   Quotient   Remainder  
12345/2   6172   1  
6172/2   3086   0  
3086/2   1543   0  
1543/2   771   1  
771/2   385   1  
385/2   192   1  
192/2   96   0  
96/2   48   0  
48/2   24   0  
24/2   12   0  
   6   0  
   3   0  
   1   1  

Conversion of decimal Part
0.4375 =>
0.4375*2 = > 0.8750 - > 0
0.8750*2 = > 1.75 - > 1
0.75*2 = > 1.5 - > 1