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Number conversion



                As we have already learned in the previous topic, theoretically there can exist and infinite number of bases as any positive number can be made into a base. But for today we’ll only be focusing on 3 types of bases which are binary, decimal and hexadecimal. As the title stated, we are going to learn about the conversion of a number betweean bases. Lets say you are given a decimal number with value 10, you can turn it into a binary number or  a hexadecimal number. Therefore, we’re here to share with you on the how to? that you have been wondering  about.


Part 1: Converting a decimal number into a binary and vice versa

Example 1:
Given a decimal number with value 15, convert the given number into binary digit of 8-bits.

Step 1: Do a table which reads from right to left with number 2 as the base number.

2^7
2^6
2^5
2^4
2^3
2^2
2^1
2^0
128
64
32
16
8
4
2
1










Side Note: The “^” sign is the power sign for those of you who didn’t know about it. Thus, it is read as
      2 to the power of 0 and ect. 

Step 2: For each of the column insert ones and multiply them with the upper row number to get a total of 15. 

            The rest of the column is filled zeros

2^7
2^6
2^5
2^4
2^3
2^2
2^1
2^0
128
64
32
16
8
4
2
1
0
0
0
0
1
1
1
1
 



Step 3: Write the answer according to the number of bits asked in the question.
               
            Therefore, the answer to example 1 is  00001111.




Example2:

Given a binary number of value 10010010. Convert the following number to a decimal number.

Step 1: Build a table just like in example 1.           

2^7
2^6
2^5
2^4
2^3
2^2
2^1
2^0
128
64
32
16
8
4
2
1












Step 2: For each bit from left to right, fill it in the table in the same order(left to right).  
 
2^7
2^6
2^5
2^4
2^3
2^2
2^1
2^0
128
64
32
16
8
4
2
1
1
0
0
1
0
0
1
0




Step 3: Multiply the bits with the upper row and add all of the numbers to get the answer.

                Answer = (1*128)+(0*64)+(0*32)+(1*16)+(0*8)+(0*4)+(0*2)+(0*1)
                                 = 146

Thus, that concludes part 1 of today’s topic. Next, we’re going to convert numbers involving the hexadecimal system.

Part 2: Converting decimal to hexadecimal number and vice versa.

Example 1:

Given a decimal number with value 273. Convert the following number into a hexadecimal number.

Step 1: Build a table.

16^4
16^3
16^2
16^1
16^0
65536
4096
256
16
1










Step2: Insert the numbers in their proper column.

16^4
16^3
16^2
16^1
16^0
65536
4096
256
16
1
0
0
1
1
1




Step 3: Answer = ???
                
            Answer = 111


Example 2:

Given 3FEA. Convert the following number into a decimal number.

Step 1: Define the alphabets.
                A = 10, E = 14, F = 15, 3 = 3

Step 2: Build a table with base 16.

16^3
16^2
16^1
16^0
4096
256
16
1
3
15
14
10


Step 3: Multiply the numbers with the upper row and add them together to get the value.
               
            Answer = (3*4096) + (15*256) + (14*16) + (10*1)
                         = 16362 
 



Part 3: Converting a hexadecimal number to binary digits and vice versa

Example 1:

Given FEA31. Convert the following hexadecimal number to binary digits.

Step 1: Convert each characters to their respective binary digits.

            F = 15, E = 13, A = 10, 3 = 3, 1 = 1

2^4
2^3
2^2
2^1
2^0





               
  


Step 2: Write the answer.

F = 15 = 1111, E = 14 = 1110, A = 10 = 1010, 3 = 3 = 0011, 1 = 1 = 0001

Answer = 11111110101000110001

PS: You can refer to the table given for easier conversion.

Example 2:

Given 1110111100011010. Convert the following to its hexadecimal number system.

Step 1: Split the digits to 4-bit each.
                1110 1111 0001 1010
      (a)     (b)    (c)     (d)


Step 2: For each (a), (b), (c) and (d) refer to the table given or convert them into hexa numbers using
the table formula.

16^3
16^2
16^1
16^0




               

               

Step 3: Write the answer.
1110 = 14 = E,
1111 = 15 = F,
0001 = 1,
1010 = 10 =

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