Question

Convert the following binary numbers to decimals.
a) 110011
b) 101010
c) 10000
d) 11001100
e) 100000
f) 111000
g) 100100
h) 10001

Answer 1

Hint: As the base number for binary is 2, for the conversion to decimal, we will write the given number from right side (instead of left) and calculate the sum when the subsequent numbers are multiplied with increasing power of 2.
We will use the base numbers for representation i.e. 2 for binary and 10 for decimal.

Complete step-by-step answer:
a) \[{(110011)_2} = {\left( x \right)_{10}}\]
$
  1 \times {2^0} = 1 \\
  1 \times {2^1} = 2 \\
  0 \times {2^2} = 0 \\
  0 \times {2^3} = 0 \\
  1 \times {2^4} = 16 \\
  1 \times {2^5} = 32 \\
 $
To get the value of x, all the numbers are added:
1 + 2 + 0 + 0 + 16 + 32 = 51
${\left( x \right)_{10}} = {\left( {51} \right)_{10}}$
Therefore, 110011 can be written as 51 in decimal.

b) \[{(101010)_2} = {\left( x \right)_{10}}\]
$
  0 \times {2^0} = 0 \\
  1 \times {2^1} = 2 \\
  0 \times {2^2} = 0 \\
  1 \times {2^3} = 8 \\
  0 \times {2^4} = 0 \\
  1 \times {2^5} = 32 \\
 $
To get the value of x, all the numbers are added:
0 + 2 + 0 + 8 + 0 + 32 = 42
${\left( x \right)_{10}} = {\left( {42} \right)_{10}}$
Therefore, 101010 can be written as 42 in decimal.

c) \[{(10000)_2} = {\left( x \right)_{10}}\]
\[
  0 \times {2^0} = 0 \\
  0 \times {2^1} = 0 \\
  0 \times {2^2} = 0 \\
  0 \times {2^3} = 0 \\
  1 \times {2^4} = 16 \\
 \]
To get the value of x, all the numbers are added:
0 + 0 + 0 + 0 + 16 = 16
${\left( x \right)_{10}} = {\left( {16} \right)_{10}}$
Therefore, 10000 can be written as 16 in decimal.

d) ${\left( {11001100} \right)_2} = {\left( x \right)_{10}}$
$
  0 \times {2^0} = 0 \\
  0 \times {2^1} = 0 \\
  1 \times {2^2} = 4 \\
  1 \times {2^3} = 8 \\
  0 \times {2^4} = 0 \\
  0 \times {2^5} = 0 \\
  1 \times {2^6} = 64 \\
  1 \times {2^7} = 128 \\
 $
To get the value of x, all the numbers are added:
0 + 0 + 4 + 8 + 0 + 0 + 64 + 128 = 204
${\left( x \right)_{10}} = {\left( {204} \right)_{10}}$
Therefore, 11001100 can be written as 204 in decimal.

e) ${\left( {100000} \right)_2} = {\left( x \right)_{10}}$
\[
  0 \times {2^0} = 0 \\
  0 \times {2^1} = 0 \\
  0 \times {2^2} = 0 \\
  0 \times {2^3} = 0 \\
  0 \times {2^4} = 0 \\
  1 \times {2^5} = 32 \\
 \]
To get the value of x, all the numbers are added:
0 + 0 + 0 + 0 + 0 + 32 = 32
${\left( x \right)_{10}} = {\left( {32} \right)_{10}}$
Therefore, 100000 can be written as 32 in decimal.

f) ${\left( {111000} \right)_2} = {\left( x \right)_{10}}$
\[
  0 \times {2^0} = 0 \\
  0 \times {2^1} = 0 \\
  0 \times {2^2} = 0 \\
  1 \times {2^3} = 8 \\
  1 \times {2^4} = 16 \\
  1 \times {2^5} = 32 \\
 \]
To get the value of x, all the numbers are added:
0 + 0 + 0 + 8 + 16 + 32 = 56
${\left( x \right)_{10}} = {\left( {56} \right)_{10}}$
Therefore, 111000 can be written as 56 in decimal.

g) ${\left( {100100} \right)_2} = {\left( x \right)_{10}}$
\[
  0 \times {2^0} = 0 \\
  0 \times {2^1} = 0 \\
  1 \times {2^2} = 4 \\
  0 \times {2^3} = 0 \\
  0 \times {2^4} = 0 \\
  1 \times {2^5} = 32 \\
 \]
To get the value of x, all the numbers are added:
0 + 0 + 4 + 0 + 0 + 32 = 36
${\left( x \right)_{10}} = {\left( {36} \right)_{10}}$
Therefore, 100100 can be written as 36 in decimal.
h) ${\left( {10001} \right)_2} = {\left( x \right)_{10}}$
\[
 1 \times {2^0} = 1 \\
 0 \times {2^1} = 0 \\
 0 \times {2^2} = 0 \\
 0 \times {2^3} = 0 \\
 1 \times {2^4} = 16 \\
 \]
To get the value of x, all the numbers are added:
1 + 0 + 0 + 0 + 16 = 17
${\left( x \right)_{10}} = {\left( {17} \right)_{10}}$
Therefore, 10001 can be written as 17 in decimal.

Note: In the binary numerals we only use two values i.e. 1 and 2 whereas in decimal numerals we can use any digit from 0 to 9.
Since only 2 digits are used in binary, it has base number 2 and 10 digits are used in decimal, it has base number 10