Now let's look at the number 110.11_{2} using the same relationship between digit, base, and position number. Recall the formula for computing the value of a digit:
DIGIT * BASE^{ POSITION #} |
Since the binary system is a base-2 number system, we can make this formula more specific:
DIGIT * 2^{ POSITION #} |
As we did with the previous decimal example, we can analyze the digits of our binary number 110.11_{2} in a table.
Fours | Twos | Ones | Halves | Fourths | |
2^{2} | 2^{1} | 2^{0} | 2^{-1} | 2^{-2} | |
1 | 1 | 0 | . | 1 | 1 |
Notice that the position values in our table are now powers of two instead of ten. Using our formula, we see that the first digit of our binary number represents the value 1 * 2^{2} or 4 and the second digit represents the value 1 * 2^{1} or 2. Continuing this pattern, we can express the number 110.11_{2} as follows:
1*2^{2 } = 1*4 = 4. 1*2^{1 } = 1*2 = 2. 0*2^{0 } = 0*1 = 0. 1*2^{-1} = 1*.5 = 0.5 1*2^{-2} = 1*.25 = + 0.25 ^{ } 6.75 |
To find the decimal value of a binary number, we simply calculate the value of each binary digit and then sum these values. The animation below demonstrates how to convert the binary number 11101.01_{2} to its decimal value. Click on the "Start Tutorial" button to view the animation.