Math, asked by gamingtoxic791, 8 hours ago

The dimonisolon of a fraction is more then
twoice the norainator if the sum of the
fraction f te reciproca i 2 16
bi 21

Answers

Answered by sb3sharma

Answer:

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

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

Other Binary Fraction Examples

0.11 = (1×2-1) + (1×2-2) = 0.5 + 0.25 = 0.7510

11.001 = (1×21) + (1×20) + (1×2-3) = 2 + 1 + 0.125 = 3.12510

1011.111 = (1×23) + (1×21) + (1×20) (1×2-1) + (1×2-2) + (1×2-3)

= 8 + 2 + 1 + 0.5 + 0.25 + 0.125 = 11.87510

So to find the binary equivalent of the decimal integer: 11810

118 (divide by 2) = 59 plus remainder 0 (LSB)

59 (divide by 2) = 29 plus remainder 1 (↑)

29 (divide by 2) = 14 plus remainder 1 (↑)

14 (divide by 2) = 7 plus remainder 0 (↑)

7 (divide by 2) = 3 plus remainder 1 (↑)

3 (divide by 2) = 1 plus remainder 1 (↑)

1 (divide by 2) = 0 plus remainder 1 (MSB)

Then the binary equivalent of 11810 is therefore: 11101102 ← (LSB)

So to find the binary fraction equivalent of the decimal fraction: 0.812510

0.8125 (multiply by 2) = 1.625 = 0.625 carry 1 (MSB)

0.625 (multiply by 2) = 1.25 = 0.25 carry 1 (↓)

0.25 (multiply by 2) = 0.50 = 0.5 carry 0 (↓)

0.5 (multiply by 2) = 1.00 = 0.0 carry 1 (LSB)

Thus the binary equivalent of 0.812510 is therefore: 0.11012 ← (LSB)

We can double check this answer using the procedure above to convert a binary fraction into a decimal number equivalent: 0.1101 = 0.5 + 0.25 + 0.0625 = 0.812510

Binary Fraction Example No2

Find the binary fraction equivalent of the following decimal number: 54.6875

First we convert the integer 54 to a binary number in the normal way using successive division from above.

54 (divide by 2) = 27 remainder 0 (LSB)

27 (divide by 2) = 13 remainder 1 (↑)

13 (divide by 2) = 6 remainder 1 (↑)

6 (divide by 2) = 3 remainder 0 (↑)

3 (divide by 2) = 1 remainder 1 (↑)

1 (divide by 2) = 0 remainder 1 (MSB)

Thus the binary equivalent of 5410 is therefore: 1101102

Next we convert the decimal fraction 0.6875 to a binary fraction using successive multiplication.

0.6875 (multiply by 2) = 1.375 = 0.375 carry 1 (MSB)