Question

The base of the decimal number system is ten, meaning, for example, that \[123={{1.10}^{2}}+2.10+3\]. In the binary system, which has base two, the first five positive integers are 1, 10, 11, 100, 101. The numeral 10011 in the binary system would then be written in the decimal system as:
A. 19
B. 40
C. 10011
D. 11
E. 7

Answer 1

Hint:
in the decimal number system, the weight of each digit from right to left increases by a factor of 10. In the binary number system, the weight of each digit increases by a factor of 2. Then the first digit has a weight of 1 (\[{{2}^{0}}\]), the second digit has a weight of 2 (\[{{2}^{1}}\]), the third a weight of 4 (\[{{2}^{2}}\]), the fourth a weight of 8 (\[{{2}^{3}}\]) and so on. By adding together all the decimal number values from right to left at the positions that are represented by a “1” gives us the equivalent decimal number. Then, we can convert binary to decimal by finding the decimal equivalent of the binary array of digits 101100101 and expanding the binary digits into a series with a base of 2 giving an equivalent in decimal or denary.

Complete step by step solution:
Multiply each digit of the binary number with its weight and add them. The weight of the least significant digit that is the rightmost digit is \[{{2}^{0}}\]. The weight of each digit increases by a factor of 2. Since, the first digit has a weight of 1 (\[{{2}^{0}}\]), the second digit has a weight of 2 (\[{{2}^{1}}\]), the third a weight of 4 (\[{{2}^{2}}\]), the fourth a weight of 8 (\[{{2}^{3}}\]) and so on
Therefore, the decimal equivalent of 10011 is
\[\begin{align}
  & =1\times {{2}^{4}}+0\times {{2}^{3}}+0\times {{2}^{2}}+1\times {{2}^{1}}+1\times {{2}^{0}} \\
 & =1\times 16+0\times 8+0\times 4+1\times 2+1\times 1 \\
 & =16+0+0+2+1 \\
 & =19 \\
\end{align}\]
Hence, the correct option is A.

Note:
When we convert numbers from binary to decimal, or decimal to binary, subscripts are used to avoid errors. Converting binary to decimal (base-2 to base-10) or decimal to binary numbers (base10 to base-2) can be done in a number of different ways as shown above. When converting decimal numbers to binary numbers it is important to remember which is the least significant bit (LSB), and which is the most significant bit (MSB).