Binary to Decimal Converter
Enter a binary number.
(101011)_{2} = (43)_{10}
SOLUTION
We multiply each binary digit by its place value and add the products.
(101011)_{2} = (1 × 2^{5}) + (0 × 2^{4}) + (1 × 2^{3}) + (0 × 2^{2}) + (1 × 2^{1}) + (1 × 2^{0})
= (1 × 32) + (0 × 16) + (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1)
= 32 + 8 + 2 + 1
= (43)_{10}
OTHER INFORMATION
Binary and decimal numbering systems are commonly used in mathematics, computer science and electrical engineering.
Binary Numbering System (Base 2): In the binary system, numbers are represented using one of two digits: 0 and 1. Each digit is called a bit. For a whole number, the right-most bit has a place value of 2^{0} = 1, the next bit has a place value of 2^{1} = 2, then 2^{2} = 4, 2^{3} = 8, and so on, doubling with each position to the left. For a binary number with a radix point, the place value of the left-most digit in the fractional part is 2^{–1}, the next bit on the right has a place value 2^{–2}, and so on.
Decimal Numbering System (Base 10): In the decimal system, numbers are represented using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each digit has a place value of 10 raised to a power depending on its position in the number.
Each binary number has a unique representation in decimal system. Binary to decimal (bin to dec) conversion is the process of converting a binary number into its equivalent decimal representation. To convert a binary number to decimal we multiply each digit by its place value and add the products.
Each place value in a binary number can be represented using an exponential number with a base of 2. The exponent of the ones digit is zero, and it increases by 1 for each digit moved to the left. In other words,
Each digit of the binary number above is marked with its place value. The place value of the ones digit is 1, the next digit's place value is 2, which is twice that of 1, and the subsequent digit's place value is 4, which is twice that of 2, and so on.
The place value of the ones digit is 1, and as we move to the right, the place value halves, resulting in place values in fraction form such as ..., ..., ..., and so on.
Find the decimal representation of the binary number 1011_{2}.
The place values, from left to right, are 8, 4, 2, and 1, respectively. To find the decimal equivalent, we multiply each place value by the corresponding digit and then add the products together.
1011_{2} = 1 · 8 + 0 · 4 + 1 · 2 + 1 · 1
= 8 + 2 + 1
= 11
So, the decimal representation of 1011_{2} is 11_{10}.
Find the decimal equivalent of the binary number 110.11_{2}.
The place values of 110.11_{2} are shown below.
Multiply those values with the corresponding digits and add the products.
... ... ...
...
...
...
Find the decimal equivalent of the largest 8-digit binary number.
The highest possible value of a binary digit is 1. Therefore, the largest 8-digit binary number is 11111111_{2}. Its decimal equivalent is:
11111111_{2} = 1 · 2^{7} + 1 · 2^{6} + 1 · 2^{5} + 1 · 2^{4} + 1 · 2^{3} + 1 · 2^{2} + 1 · 2^{1} + 1 · 2^{0}
= 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1
= 255_{10}
Alternatively, we can find the decimal equivalent of the smallest 9-digit binary number and subtract 1 to reach the result faster. The smallest 9-digit binary number is 100000000_{2}.
100000000_{2} = 2^{8}
= 256_{10}
That number is 1 more than the largest 8-digit binary number.
11111111_{2} = 256 - 1
= 255_{10}
If 1a001_{2} = 2b_{10}, find the sum a + b.
Let's find the decimal equivalent of the binary number 1a001_{2}.
1a001_{2} = 16 + 8 · a + 1
= 17 + 8a
There are only two possible values for a: 0 and 1. If a were 0, the decimal equivalent of this number would be 17_{10}. However, since 2b_{10} cannot equal 17_{10}, a cannot be 0. Therefore, a = 1. When we calculate the decimal equivalent for this value:
17 + 8 * 1 = 25_{10}
Consequently, b = 5. The sum of the numbers a = 1 and b = 5 is 6.
The table below provides the decimal equivalents of the binary numbers up to 100.
0_{2} = 0_{10} |
1_{2} = 1_{10} |
10_{2} = 2_{10} |
11_{2} = 3_{10} |
100_{2} = 4_{10} |
101_{2} = 5_{10} |
110_{2} = 6_{10} |
111_{2} = 7_{10} |
1000_{2} = 8_{10} |
1001_{2} = 9_{10} |
1010_{2} = 10_{10} |
1011_{2} = 11_{10} |
1100_{2} = 12_{10} |
1101_{2} = 13_{10} |
1110_{2} = 14_{10} |
1111_{2} = 15_{10} |
10000_{2} = 16_{10} |
10001_{2} = 17_{10} |
10010_{2} = 18_{10} |
10011_{2} = 19_{10} |
10100_{2} = 20_{10} |
10101_{2} = 21_{10} |
10110_{2} = 22_{10} |
10111_{2} = 23_{10} |
11000_{2} = 24_{10} |
11001_{2} = 25_{10} |
11010_{2} = 26_{10} |
11011_{2} = 27_{10} |
11100_{2} = 28_{10} |
11101_{2} = 29_{10} |
11110_{2} = 30_{10} |
11111_{2} = 31_{10} |
100000_{2} = 32_{10} |
100001_{2} = 33_{10} |
100010_{2} = 34_{10} |
100011_{2} = 35_{10} |
100100_{2} = 36_{10} |
100101_{2} = 37_{10} |
100110_{2} = 38_{10} |
100111_{2} = 39_{10} |
101000_{2} = 40_{10} |
101001_{2} = 41_{10} |
101010_{2} = 42_{10} |
101011_{2} = 43_{10} |
101100_{2} = 44_{10} |
101101_{2} = 45_{10} |
101110_{2} = 46_{10} |
101111_{2} = 47_{10} |
110000_{2} = 48_{10} |
110001_{2} = 49_{10} |
110010_{2} = 50_{10} |
110011_{2} = 51_{10} |
110100_{2} = 52_{10} |
110101_{2} = 53_{10} |
110110_{2} = 54_{10} |
110111_{2} = 55_{10} |
111000_{2} = 56_{10} |
111001_{2} = 57_{10} |
111010_{2} = 58_{10} |
111011_{2} = 59_{10} |
111100_{2} = 60_{10} |
111101_{2} = 61_{10} |
111110_{2} = 62_{10} |
111111_{2} = 63_{10} |
1000000_{2} = 64_{10} |
1000001_{2} = 65_{10} |
1000010_{2} = 66_{10} |
1000011_{2} = 67_{10} |
1000100_{2} = 68_{10} |
1000101_{2} = 69_{10} |
1000110_{2} = 70_{10} |
1000111_{2} = 71_{10} |
1001000_{2} = 72_{10} |
1001001_{2} = 73_{10} |
1001010_{2} = 74_{10} |
1001011_{2} = 75_{10} |
1001100_{2} = 76_{10} |
1001101_{2} = 77_{10} |
1001110_{2} = 78_{10} |
1001111_{2} = 79_{10} |
1010000_{2} = 80_{10} |
1010001_{2} = 81_{10} |
1010010_{2} = 82_{10} |
1010011_{2} = 83_{10} |
1010100_{2} = 84_{10} |
1010101_{2} = 85_{10} |
1010110_{2} = 86_{10} |
1010111_{2} = 87_{10} |
1011000_{2} = 88_{10} |
1011001_{2} = 89_{10} |
1011010_{2} = 90_{10} |
1011011_{2} = 91_{10} |
1011100_{2} = 92_{10} |
1011101_{2} = 93_{10} |
1011110_{2} = 94_{10} |
1011111_{2} = 95_{10} |
1100000_{2} = 96_{10} |
1100001_{2} = 97_{10} |
1100010_{2} = 98_{10} |
1100011_{2} = 99_{10} |
1100100_{2} = 100_{10} |
Binary to decimal converter,
You can use binary to decimal converter in two ways.
You can enter a binary number into the input box and then click the '
You can click on the DIE ICON next to the input box to generate a random binary number, which will be automatically entered into the calculator. The result and explanations will then appear below the calculator. You can also create your own examples and practice using this feature.
To find the decimal equivalent of another binary number, click on the CLEAR button to clear the input box.
You can copy the generated solution by clicking on the 'Copy Text' link located below the solution panel.
You can also download the solution as an image file with a .jpg extension by clicking on the 'Download Solution' link located at the bottom of the solution panel. You can then share the downloaded image file.
Binary to Decimal Converter