Directions: Convert the following binary to decimal.
1. $11011_2=$
2. $101101_2=$
3. $111011_2=$
4. $10101_2=$
5. $1010101_2=$
Answer (1)
Convert each binary number to decimal by multiplying each digit by the corresponding power of 2 and summing the results.
1101 1 2 = ( 1 × 2 4 ) + ( 1 × 2 3 ) + ( 0 × 2 2 ) + ( 1 × 2 1 ) + ( 1 × 2 0 ) = 27
10110 1 2 = ( 1 × 2 5 ) + ( 0 × 2 4 ) + ( 1 × 2 3 ) + ( 1 × 2 2 ) + ( 0 × 2 1 ) + ( 1 × 2 0 ) = 45
11101 1 2 = ( 1 × 2 5 ) + ( 1 × 2 4 ) + ( 1 × 2 3 ) + ( 0 × 2 2 ) + ( 1 × 2 1 ) + ( 1 × 2 0 ) = 59
1010 1 2 = ( 1 × 2 4 ) + ( 0 × 2 3 ) + ( 1 × 2 2 ) + ( 0 × 2 1 ) + ( 1 × 2 0 ) = 21
101010 1 2 = ( 1 × 2 6 ) + ( 0 × 2 5 ) + ( 1 × 2 4 ) + ( 0 × 2 3 ) + ( 1 × 2 2 ) + ( 0 × 2 1 ) + ( 1 × 2 0 ) = 85
The decimal equivalents are: 27 , 45 , 59 , 21 , 85 .
Explanation
Problem Analysis and Conversion Method We need to convert the given binary numbers to their decimal equivalents. The binary numbers are: 1101 1 2 , 10110 1 2 , 11101 1 2 , 1010 1 2 , 101010 1 2 . To convert a binary number to decimal, we multiply each digit by 2 n , where n is the position of the digit from the right, starting from 0, and then sum the results.
Converting 11011 (binary) to Decimal
1101 1 2 = ( 1 × 2 4 ) + ( 1 × 2 3 ) + ( 0 × 2 2 ) + ( 1 × 2 1 ) + ( 1 × 2 0 ) = ( 1 × 16 ) + ( 1 × 8 ) + ( 0 × 4 ) + ( 1 × 2 ) + ( 1 × 1 ) = 16 + 8 + 0 + 2 + 1 = 27
Converting 101101 (binary) to Decimal
10110 1 2 = ( 1 × 2 5 ) + ( 0 × 2 4 ) + ( 1 × 2 3 ) + ( 1 × 2 2 ) + ( 0 × 2 1 ) + ( 1 × 2 0 ) = ( 1 × 32 ) + ( 0 × 16 ) + ( 1 × 8 ) + ( 1 × 4 ) + ( 0 × 2 ) + ( 1 × 1 ) = 32 + 0 + 8 + 4 + 0 + 1 = 45
Converting 111011 (binary) to Decimal
11101 1 2 = ( 1 × 2 5 ) + ( 1 × 2 4 ) + ( 1 × 2 3 ) + ( 0 × 2 2 ) + ( 1 × 2 1 ) + ( 1 × 2 0 ) = ( 1 × 32 ) + ( 1 × 16 ) + ( 1 × 8 ) + ( 0 × 4 ) + ( 1 × 2 ) + ( 1 × 1 ) = 32 + 16 + 8 + 0 + 2 + 1 = 59
Converting 10101 (binary) to Decimal
1010 1 2 = ( 1 × 2 4 ) + ( 0 × 2 3 ) + ( 1 × 2 2 ) + ( 0 × 2 1 ) + ( 1 × 2 0 ) = ( 1 × 16 ) + ( 0 × 8 ) + ( 1 × 4 ) + ( 0 × 2 ) + ( 1 × 1 ) = 16 + 0 + 4 + 0 + 1 = 21
Converting 1010101 (binary) to Decimal
101010 1 2 = ( 1 × 2 6 ) + ( 0 × 2 5 ) + ( 1 × 2 4 ) + ( 0 × 2 3 ) + ( 1 × 2 2 ) + ( 0 × 2 1 ) + ( 1 × 2 0 ) = ( 1 × 64 ) + ( 0 × 32 ) + ( 1 × 16 ) + ( 0 × 8 ) + ( 1 × 4 ) + ( 0 × 2 ) + ( 1 × 1 ) = 64 + 0 + 16 + 0 + 4 + 0 + 1 = 85
Final Answer Therefore, the decimal equivalents of the given binary numbers are: 1101 1 2 = 27 , 10110 1 2 = 45 , 11101 1 2 = 59 , 1010 1 2 = 21 , 101010 1 2 = 85 .
Examples
Binary numbers are used extensively in computer science and digital electronics. For example, when you save a file on your computer, the data is stored as a sequence of binary digits (bits). Understanding how to convert between binary and decimal is crucial for understanding how computers process and store information. Another application is in digital communication, where binary signals are used to transmit data over networks.
