1
3
LN N
1
1
o
1
Therefore, (231)0
(11100111),
Conversion of Binary Numbers to Decimal Numbers
We use multiplication method to convert binary numbers into decimal numbers.
Example 1: Convert the binary number 1101 into a decimal number.
Solution: To convert a binary number into decimal number,
1. Starting from the right most digit write all the digits in different rows of a table, as
shown below.
2. Multiply each digit by base 2 with its power based on the position (0, 1, 2, 3, ...) of
the digit.
3. Add the products obtained.
Digit
Power of base
Calculation
Product
1
1 x 2°
1 x1
1
0
0 x 21
0X2
0
1
1 x 22
1 x4
4
23
Answers
Explanation:
Binary Fractions use the same weighting principle as decimal numbers except that each binary digit uses the base-2 numbering system
We know that decimal (or denary) numbers use the base ten (base-10) numbering system where each digit in a decimal number is allowed to take one of ten possible values in the range of 0 to 9. So moving from right to left along a decimal number, each digit will have a value ten times greater than the digit to its immediate right.
But as well as each digit being ten times bigger than the previous number as we move from right-to-left, each digit can also be ten times smaller than its neighbouring number as we move along in the opposite direction from left-to-right.
However, once we reach zero (0) and the decimal point, we do not need to just stop, but can continue moving from left-to-right along the digits producing what are generally called Fractional Numbers.
A Typical Fractional Number
a fractional number
Here in this decimal (or denary) number example, the digit immediately to the right of the decimal point (number 5) is worth one tenth (1/10 or 0.1) of the digit immediately to the left of the decimal point (number 4) which as a multiplication value of one (1).
Thus as we move through the number from left-to-right, each subsequent digit will be one tenth the value of the digit immediately to its left position, and so on.
Then the decimal numbering system uses the concept of positional or relative weighting values producing a positional notation, where each digit represents a different weighted value depending on the position occupied either side of the decimal point.
Thus mathematically in the standard denary numbering system, these values are commonly written as: 40, 31, 22, 13 for each position to the left of the decimal point in our example above. Likewise, for the fractional numbers to right of the decimal point, the weight of the number becomes more negative giving: 5-1, 6-2, 7-3 etc.
So we can see that each digit in the standard decimal system indicates the magnitude or weight of that digit within the number. Then the value of any decimal number will be equal 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 could be written to reflect the weighting of each 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 form as:
(1×103) + (2×102) + (3×101) + (4×100) + (5×10-1) + (6×10-2) + (7×10-3) = 1234.56710
We can also use this idea of positional notation where each digit represents a different weighted value depending upon the position it occupies in the binary numbering system. The difference this time is that the binary number system (or simply binary numbers) is a positional system, where the different weighted positions of the digits are to the power of 2 (base-2) instead of 10.
Binary Fractions
The binary numbering system is a base-2 numbering system which contains only two digits, a “0” or a “1”. Thus each digit of a binary number can take the “0” or the “1” value with the position of the 0 or 1 indicating its value or weighting. But we can also have binary weighting for values of less than 1 producing what are called unsigned fractional binary numbers.
Similar to decimal fractions, binary numbers can also be represented as unsigned fractional numbers by placing the binary digits to the right of the decimal point or in this case, binary point. Thus all the fractional digits to the right of the binary point have respective weightings which are negative powers of two, creating a binary fraction. In other words, the powers of 2 are negative.
So for the fractional binary numbers to the right of the binary point, the weight of each digit becomes more negative giving: 2-1, 2-2, 2-3, 2-4, and so on as shown.
Binary Fractions
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