We realize that decimal (or denary) numbers utilize the base ten (base-10) numbering system where each digit in a decimal number is permitted to take among ten possible values in the number of 0 to 9. So moving from to left along a decimal number, each digit could have a value ten times higher than the digit to its immediate right.
Source: online text to binary conversion
But and also each digit being ten times bigger compared to the previous number as we move from right-to-left, each digit may also be ten occasions smaller than its neighbouring number as we move along in the contrary direction from left-to-right.
However, after we reach zero (0) and the decimal stage, we need not just stop, but can continue moving from left-to-right along the digits producing what exactly are generally called Fractional Numbers.
AN AVERAGE Fractional Number
Within this decimal (or denary) number example, the digit immediately to the proper of the decimal point (#5) will probably be worth one tenth (1/10 or 0.1) of the digit immediately left of the decimal point (#4) which as a multiplication value of 1 (1).
Thus as we undertake the quantity from left-to-right, each subsequent digit will be one tenth the worthiness of the digit immediately to its left position, and so forth.
Then your decimal numbering system uses the idea of positional or relative weighting values creating a positional notation, where each digit represents a different weighted value based on the position occupied either side of the decimal point.
Thus mathematically in the typical denary numbering system, these values are generally written as: 40, 31, 22, 13 for every position left of the decimal point inside our example above. Likewise, for the fractional numbers to right of the decimal point, the weight of the quantity becomes more negative giving: 5-1, 6-2, 7-3 etc.
So we are able to see that every digit in the typical decimal system indicates the magnitude or weight of this digit within the quantity. Then your value of any decimal number will become add up to the sum of its digits multiplied by their respective weights, so for our example above: N = 1234.56710 in the weighted decimal format this will be equal too:
1000 + 200 + 30 + 4 + 0.5 + 0.06 + 0.007 = 1234.56710
or it may be written to reflect the weighting of every denary digit:
(1×1000) + (2×100) + (3×10) + (4×1) + (5×0.1) + (6×0.01) + (7×0.001) = 1234.56710
or even in polynomial type as:
(1×103) + (2×102) + (3×101) + (4×100) + (5×10-1) + (6×10-2) + (7×10-3) = 1234.56710
We can also utilize this notion of positional notation where each digit represents a different weighted value dependant on the positioning it occupies in the binary numbering system. The difference this time around is usually that the binary number system (or just binary numbers) is usually a positional system, where in fact the different weighted positions of the digits are to the energy of 2 (base-2) rather than 10.
The binary numbering system is a base-2 numbering system which contains just two digits, a “0” or a “1”. Thus each digit of a binary number may take the “0” or the “1” value with the positioning of the 0 or 1 indicating its value or weighting. But we are able to likewise have binary weighting for values of significantly less than 1 producing what exactly are called unsigned fractional binary numbers.
Similar to decimal fractions, binary numbers may also be represented as unsigned fractional numbers by placing the binary digits to the proper of the decimal stage or in this instance, binary point. Thus all of the fractional digits to the proper of the binary stage have particular weightings which are negative powers of two, creating a binary fraction. Quite simply, the powers of 2 are negative.
So for the fractional binary numbers to the proper of the binary stage, the weight of every digit becomes more negative giving: 2-1, 2-2, 2-3, 2-4, and so forth as shown.
Binary Fractions Example
Let’s suppose we have the following binary number of: 1101.01112, what will be its decimal number equivalent.
1101.0111 = (1×23) + (1×22) + (0×21) + (1×20) + (0×2-1) + (1×2-2) + (1×2-3) + (1×2-4)
= 8 + 4 + 0 + 1 + 0 + 1/4 + 1/8 + 1/16
= 8 + 4 + 0 + 1 + 0 + 0.25 + 0.125 + 0.0625 = 13.437510
Hence the decimal equivalent number of 1101.01112 is given as: 13.437510