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Hey friends, welcome to the YouTube channel ALL ABOUT ELECTRONICS.
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So in this video, we will learn about the floating point numbers.
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And we will see that how very large numbers like the mass of the planets or the Avogadro's
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number and similarly very small numbers like the mass of the atom or the Planck's constant
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is stored in the computers.
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So, during the video, we will also see the difference between the fixed point number
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and the floating point numbers.
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And with this comparison, we will understand the importance of the floating point numbers
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in the digital systems.
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So first, let us understand what is the fixed point numbers.
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So in our day-to-day life, we are all dealing with the integers as well as the real numbers.
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Now when these numbers are represented in the fixed point representation, then the position
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of the radix point or the decimal point remains the fixed.
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So all the integers are the example of the fixed point numbers.
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So for the integers, there is no fractional part or in other words, the fractional part
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is equal to zero.
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So by default, the position of the decimal point is at the end of the least significant
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digit.
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Hence, there is no fractional part, so typically, we do not represent this decimal point.
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But we can say that there is a decimal point on the right-hand side of this least significant
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digit.
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And this position of this decimal point will also remain fixed.
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So similarly, for the real numbers, the position of the decimal point is just before the fractional
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part.
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For example, this 11.75 is the real number, where 11 is the integer part and 75 just after
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this decimal point represents the fractional part.
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So when these real numbers are represented in the fixed point representation, then the
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position of this decimal point remains the fixed.
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Now in any digital system, these numbers are stored in a binary format using the certain
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number of bits.
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Let's say in a one digital system, these numbers are stored in a 10-bit format.
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Now the issue with the fixed point representation is that with the given number of bits, the
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range of the numbers that we can represent is very less.
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So if we take the case of the integers, and specifically, an unsigned integer, then in
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the 10-bit format, we can represent any number between 0 and 1023.
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On the other hand, for the signed integers, the MSB is reserved for the signed bit.
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So using the 10 bits, we can represent any number between -512 to +511.
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That means using the 10 bits, the range of the numbers that we can represent is very
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limited.
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So here, basically the range refers to the difference between the smallest and the largest
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number.
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So of course, by increasing the number of bits, we can increase this range.
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But still, if we want to represent the very large numbers, like 10^24 or 10^25, for example
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the mass of the earth, then we need more than 80 bits.
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And the issue of the range becomes even more prominent with the real numbers.
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So when we are dealing with the real numbers, then we always come across this decimal point
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or in general this radix point.
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So the digits on the left of this decimal point represents the integer part and the
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digits on the right represents the fractional part.
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So to store such numbers in a binary format in the computers, some bits are reserved for
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the integer part and the some bits are reserved for the fractional part.
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So let's say, once again, these real numbers are stored in a 10-bit format.
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And out of the 10-bit, the 6 bits are reserved for the integer part and the 4 bits are reserved
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for the fractional part.
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Now when we store these numbers in a binary format, then there is no provision for storing
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this binary point explicitly.
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But here, we have different sections for the integer as well as the fractional part.
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And accordingly, each bit will have its place value.
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So here, just after the 2^0s place, we will assume that there is a binary point.
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So out of the 10 bits, if we reserve 6 bits for the integer part, then for the unsigned
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numbers, we can represent any number between 0 to 63.
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And for the fractional part, the maximum number that we can represent is equal to 0.9375.
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And the minimum number will be equal to 0.0625.
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That means in the 10-bit format, if we want to represent any real number, then the minimum
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non-zero number that we can represent is equal to 0.0625, while if we see the maximum
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number, then that is equal to 63.9375.
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That means in general, in this fixed-point representation, the location of the radix
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point is fixed.
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And once we decide it, then it will not change.
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So in a 10-bit fixed-point representation, once we freeze this specific format, like
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the 6 bits for the integer and the 4 bits for the fraction, then we cannot represent
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any number smaller than this 0.0625.
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For example, if we want to represent this 22.0125 or this 35.0025, then we cannot represent
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it in this 10-bit fixed-point representation.
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So if we want to represent such smaller numbers, then we need to assign more bits for this
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fractional part, like the 5-bit or the 6 bits for the fractional part.
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So, of course, by doing so, certainly we can increase the precision.
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But now, our range will get compromised.
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For example, now we have only 4 bits for the integer part.
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And now, in these 4 bits, we can represent any number between 0 to 15.
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That means in this fixed-point representation, once the location of this radix point is fixed,
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then our range and the precision will also get fixed.
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But in the floating-point representation, it is possible to change the location of
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this radix point or the binary point dynamically.
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For example, for the given number of bits, let's say a 10-bit, if we want more range,
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then we can shift this binary point towards the right.
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Or for example, for some application, if we require more precision, then it is possible
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to shift the radix point towards the left.
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That means using the floating-point representation, it is possible to represent the very large
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numbers like the distance between the planets or the mass of the planets and the very small
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number like the mass of the atom using these floating-point numbers.
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So this floating-point representation provides both good range as well as the precision.
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So now, let's see how to represent these floating-point numbers.
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So the representation of this floating-point number is very similar to how we are representing
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the decimal numbers in the scientific notation.
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So in the scientific notation, the radix point or the decimal point is set in such
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a way that we have only one significant digit before the decimal point.
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So for the integers, by default, the radix point or this decimal point is set to the
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right-hand side of this least significant digit.
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So here, to represent this number in the scientific notation, the decimal point is shifted to
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the left-hand side by the five decimal places.
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And that is why, here the exponent is equal to 5.
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So as you can see over here, we have only one significant digit before the decimal point.
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But if the same number is represented like this, then that is not the scientific notation.
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Because if you see over here, then the digit before the decimal point is 0.
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But in the scientific notation, it has to be non-zero.
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Similarly, if you take this number, then in the scientific notation, this is how it can
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be represented.
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So here, for the scientific notation, the decimal point is shifted to the right by three
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decimal places.
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And that is why over here, in the exponential term, we have this 10 to the power minus 3.
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So in the scientific notation, we have total two components.
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That is the significand and the exponent.
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So here, in the second representation, if you see the significand, then that is equal
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to 4.345.
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And similarly, the exponent is equal to minus 3.
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And here of course, since we are representing the decimal numbers, so the base of the exponent
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is equal to 10.
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So here in the scientific notation, we are normalizing the numbers so that we have only
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one significant digit before the decimal point.
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And because of this normalization, it is possible to represent all the numbers in a uniform
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fashion.
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For example, if we take the case of this number, then the same number can also be represented
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like this.
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And of course, the value of the number will still remain the same.
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But as you can see, all these representations are different.
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So that is why it is good to have a uniform representation for each number.
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So in general, we can say that in a scientific notation, this is how the decimal number is
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represented, where this D represents the decimal digit.
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So similarly, this floating point representation is very similar.
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And here, this B represents the binary digits.
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So if you see this floating point representation, then it consists of the three parts.
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That is sign, fraction, and the exponent part.
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And here, the base of the exponent is equal to 2.
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So in this representation also, first binary numbers are normalized in this format.
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So in a scientific notation, we have seen that we must have only one significant digit
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before the decimal point.
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Now in the case of the binary, we have only two digits, that is 1 and 0.
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And therefore in the binary, the only possible significant digit is equal to 1.
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That means in this floating point representation, this significant digit just before the binary
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point will always remain 1.
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So we can say that, this is the general representation for the floating point number.
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So now let's see, how to normalize any binary number and how to represent it in the floating
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point representation.
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So let's say, this is our binary number.
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And we want to represent this number in the normalized form.
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So for that, we need to shift this binary point in such a way that just before the binary
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point, the significant digit is equal to 1.
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That means here, we need to shift the binary point to the left by 2 bits.
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And that is why over here, this exponent is equal to 2.
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That means whenever we shift the radix point to the left by a 1-bit position, then the
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exponent will increase by 1.
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So here, since the radix point is shifted to the left side by 2 bits, so the exponent
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will increase by 2.
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So similar to the left-hand side, when the radix point is shifted to the right by a 1-bit
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position, then the exponent will decrease by 1.
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For example, if we have this number and to represent this number in a normalized form,
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we need to shift the binary point to the right side by 2 bits.
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And that is why here the exponent will decrease by 2.
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Or in other words, here this exponent is equal to minus 2.
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So these two representations are in the normalized form.
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So in this way, we can normalize any binary number and we can represent it in the floating
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point form.
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So now, let's see how this floating point number is actually stored in the memory.
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So while storing, the 1-bit is reserved for the sign bit.
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That means while this number is stored, then the MSB will represent the sign bit.
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So if this bit is 0, then it means that the number is positive.
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And whenever this bit is equal to 1, then it indicates that the number is negative.
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So after the sign bit, the few bits are reserved for storing the exponent value.
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And then the remaining bits are reserved for storing this fractional part.
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So now if you see this significand, then here the integer part of this significand will
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always remain 1.
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And therefore, this 1 is not stored and instead of that only the fractional part is stored.
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So this fractional part is also referred as the mantissa or the significand.
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That means while storing this floating point number, we have total 3 parts.
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That is sign, exponent and the mantissa.
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Now to store this floating point number, a certain standard has been defined.
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Like how many bits will be reserved for the exponent as well as the mantissa part.
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And similarly, how to store this mantissa as well as the exponent part.
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Because this exponent part if you see, then it can be positive or the negative.
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That means we need to decide how to store this exponent part.
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So to store such numbers, a common standard has been defined.
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And one such commonly used standard is the IEEE 754.
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So in the next video, we will see the format of this IEEE standard and we will understand
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that as per this standard, how the floating numbers are stored.
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But I hope in this video, you understood the difference between the fixed point numbers
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and the floating point numbers.
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And using this floating point representation, how it is possible to represent the very large
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numbers or the very small numbers with good precision.
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So if you have any question or suggestion, then do let me know here in the comment section
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below.
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