Floating Point Numbers | Fixed Point Number vs Floating Point Numbers

00:13:39
https://www.youtube.com/watch?v=zVM8NKXsboA

Summary

TLDRThe video introduces floating-point numbers, explaining how computers store large and small numbers such as planetary masses and atomic weights. It contrasts fixed-point representation, where the decimal point is fixed, with floating-point representation that allows dynamic placement of the decimal point. This flexibility helps to represent a wider range of numbers with precision. Scientific notation is used as an analogy for understanding how floating-point numbers work with components like the sign, significand, and exponent. The video also hints at the IEEE 754 standard, which governs the storage format for floating-point numbers and promises further exploration in a subsequent video. Overall, it emphasizes the importance and utility of floating-point numbers in computing.

Takeaways

  • 🔢 Understand the difference between fixed and floating-point numbers.
  • 🌐 Floating-point allows representation of very large and small numbers.
  • ✍️ Similar to scientific notation with a sign, fraction, and exponent.
  • 📝 Fixed-point has a limited range, affecting precision.
  • 💻 Floating-point numbers consist of the sign, exponent, and mantissa.
  • 📊 Provides a better range and precision than fixed-point.
  • 🔍 Normalization involves adjusting the position of the binary point.
  • 📈 Floating-point supports large number ranges, enhancing computing capabilities.
  • 🔍 In scientific notation, important for understanding number representation.
  • 📘 IEEE 754 is a standard for floating-point number storage.

Timeline

  • 00:00:00 - 00:05:00

    The video introduces floating point numbers, exploring how they allow computers to store extremely large numbers, like the mass of planets, and very small numbers, such as the mass of atoms. It begins by differentiating between fixed point and floating point numbers, emphasizing the importance of floating point numbers in digital systems. Fixed point numbers have a constant decimal point position; for integers, the decimal is at the end of the number. Digital systems typically store numbers in binary format, limited by the number of bits, restricting the range of numbers that can be represented, particularly with fixed point integers and real numbers. The video explains how binary storage divides bits between integer and fractional parts, and how fixed bit allocation can limit number representation and precision. In contrast, floating point allows dynamic radix point shifts, enhancing range and precision, enabling representation of both vast and minute values, using a structure akin to scientific notation.

  • 00:05:00 - 00:13:39

    The video continues by explaining how floating point numbers are represented, similar to scientific notation with components like significand and exponent. It shows how binary numbers are normalized in floating point representation to have a significant digit of 1 before the binary point. The storing of floating point numbers in memory is also covered, where part of the bits are allocated to represent the sign, exponent, and mantissa. The video touches on the IEEE 754 standard for floating point storage, noting its role in standardizing how these numbers are saved, though details are to be explored in a subsequent video. The explanation emphasizes floating point representation's ability to store large and small numbers with high precision, surpassing fixed point limitations by allowing flexible bit distribution for different number types.

Mind Map

Video Q&A

  • What is a fixed-point number?

    In fixed-point representation, the position of the decimal point remains fixed, typically allowing straightforward representation of integers and some real numbers.

  • What are the limitations of fixed-point representation?

    Fixed-point representation has a limited range and precision, making it unsuitable for very large or very small numbers unless a large number of bits are used.

  • How does floating-point representation help in computing?

    Floating-point representation allows the decimal point to be moved, enabling computers to represent very large or very small numbers with good precision.

  • What is scientific notation?

    Scientific notation is a way of expressing numbers where one significant digit is placed before the decimal point, accompanied by an exponent of 10.

  • How are floating-point numbers similar to scientific notation?

    Floating-point numbers use a similar concept to scientific notation, normalizing the number so that a significant digit appears before the point, accompanied by an exponent.

  • How are floating-point numbers stored in memory?

    Floating-point numbers are stored with a sign bit, a few bits for the exponent, and the remaining bits for the mantissa (fractional part).

  • What is the IEEE 754 standard?

    The IEEE 754 standard defines a common way to represent floating-point numbers, specifying the number of bits used for the exponent and mantissa.

  • What is normalization in floating-point representation?

    Normalization is adjusting the floating-point number to ensure a significant digit appears before the point, maintaining a uniform format.

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  • 00:00:06
    Hey friends, welcome to the YouTube channel ALL ABOUT ELECTRONICS.
  • 00:00:10
    So in this video, we will learn about the floating point numbers.
  • 00:00:14
    And we will see that how very large numbers like the mass of the planets or the Avogadro's
  • 00:00:20
    number and similarly very small numbers like the mass of the atom or the Planck's constant
  • 00:00:26
    is stored in the computers.
  • 00:00:28
    So, during the video, we will also see the difference between the fixed point number
  • 00:00:32
    and the floating point numbers.
  • 00:00:34
    And with this comparison, we will understand the importance of the floating point numbers
  • 00:00:38
    in the digital systems.
  • 00:00:40
    So first, let us understand what is the fixed point numbers.
  • 00:00:45
    So in our day-to-day life, we are all dealing with the integers as well as the real numbers.
  • 00:00:51
    Now when these numbers are represented in the fixed point representation, then the position
  • 00:00:55
    of the radix point or the decimal point remains the fixed.
  • 00:00:59
    So all the integers are the example of the fixed point numbers.
  • 00:01:03
    So for the integers, there is no fractional part or in other words, the fractional part
  • 00:01:08
    is equal to zero.
  • 00:01:10
    So by default, the position of the decimal point is at the end of the least significant
  • 00:01:15
    digit.
  • 00:01:16
    Hence, there is no fractional part, so typically, we do not represent this decimal point.
  • 00:01:22
    But we can say that there is a decimal point on the right-hand side of this least significant
  • 00:01:27
    digit.
  • 00:01:29
    And this position of this decimal point will also remain fixed.
  • 00:01:33
    So similarly, for the real numbers, the position of the decimal point is just before the fractional
  • 00:01:38
    part.
  • 00:01:39
    For example, this 11.75 is the real number, where 11 is the integer part and 75 just after
  • 00:01:48
    this decimal point represents the fractional part.
  • 00:01:51
    So when these real numbers are represented in the fixed point representation, then the
  • 00:01:55
    position of this decimal point remains the fixed.
  • 00:01:58
    Now in any digital system, these numbers are stored in a binary format using the certain
  • 00:02:03
    number of bits.
  • 00:02:04
    Let's say in a one digital system, these numbers are stored in a 10-bit format.
  • 00:02:10
    Now the issue with the fixed point representation is that with the given number of bits, the
  • 00:02:14
    range of the numbers that we can represent is very less.
  • 00:02:18
    So if we take the case of the integers, and specifically, an unsigned integer, then in
  • 00:02:24
    the 10-bit format, we can represent any number between 0 and 1023.
  • 00:02:29
    On the other hand, for the signed integers, the MSB is reserved for the signed bit.
  • 00:02:35
    So using the 10 bits, we can represent any number between -512 to +511.
  • 00:02:41
    That means using the 10 bits, the range of the numbers that we can represent is very
  • 00:02:46
    limited.
  • 00:02:47
    So here, basically the range refers to the difference between the smallest and the largest
  • 00:02:52
    number.
  • 00:02:53
    So of course, by increasing the number of bits, we can increase this range.
  • 00:02:58
    But still, if we want to represent the very large numbers, like 10^24 or 10^25, for example
  • 00:03:05
    the mass of the earth, then we need more than 80 bits.
  • 00:03:10
    And the issue of the range becomes even more prominent with the real numbers.
  • 00:03:14
    So when we are dealing with the real numbers, then we always come across this decimal point
  • 00:03:19
    or in general this radix point.
  • 00:03:22
    So the digits on the left of this decimal point represents the integer part and the
  • 00:03:26
    digits on the right represents the fractional part.
  • 00:03:29
    So to store such numbers in a binary format in the computers, some bits are reserved for
  • 00:03:34
    the integer part and the some bits are reserved for the fractional part.
  • 00:03:39
    So let's say, once again, these real numbers are stored in a 10-bit format.
  • 00:03:44
    And out of the 10-bit, the 6 bits are reserved for the integer part and the 4 bits are reserved
  • 00:03:48
    for the fractional part.
  • 00:03:50
    Now when we store these numbers in a binary format, then there is no provision for storing
  • 00:03:55
    this binary point explicitly.
  • 00:03:57
    But here, we have different sections for the integer as well as the fractional part.
  • 00:04:02
    And accordingly, each bit will have its place value.
  • 00:04:06
    So here, just after the 2^0s place, we will assume that there is a binary point.
  • 00:04:12
    So out of the 10 bits, if we reserve 6 bits for the integer part, then for the unsigned
  • 00:04:17
    numbers, we can represent any number between 0 to 63.
  • 00:04:22
    And for the fractional part, the maximum number that we can represent is equal to 0.9375.
  • 00:04:29
    And the minimum number will be equal to 0.0625.
  • 00:04:34
    That means in the 10-bit format, if we want to represent any real number, then the minimum
  • 00:04:38
    non-zero number that we can represent is equal to 0.0625, while if we see the maximum
  • 00:04:44
    number, then that is equal to 63.9375.
  • 00:04:49
    That means in general, in this fixed-point representation, the location of the radix
  • 00:04:53
    point is fixed.
  • 00:04:55
    And once we decide it, then it will not change.
  • 00:04:58
    So in a 10-bit fixed-point representation, once we freeze this specific format, like
  • 00:05:03
    the 6 bits for the integer and the 4 bits for the fraction, then we cannot represent
  • 00:05:08
    any number smaller than this 0.0625.
  • 00:05:12
    For example, if we want to represent this 22.0125 or this 35.0025, then we cannot represent
  • 00:05:20
    it in this 10-bit fixed-point representation.
  • 00:05:23
    So if we want to represent such smaller numbers, then we need to assign more bits for this
  • 00:05:28
    fractional part, like the 5-bit or the 6 bits for the fractional part.
  • 00:05:33
    So, of course, by doing so, certainly we can increase the precision.
  • 00:05:38
    But now, our range will get compromised.
  • 00:05:41
    For example, now we have only 4 bits for the integer part.
  • 00:05:46
    And now, in these 4 bits, we can represent any number between 0 to 15.
  • 00:05:52
    That means in this fixed-point representation, once the location of this radix point is fixed,
  • 00:05:57
    then our range and the precision will also get fixed.
  • 00:06:01
    But in the floating-point representation, it is possible to change the location of
  • 00:06:05
    this radix point or the binary point dynamically.
  • 00:06:08
    For example, for the given number of bits, let's say a 10-bit, if we want more range,
  • 00:06:13
    then we can shift this binary point towards the right.
  • 00:06:17
    Or for example, for some application, if we require more precision, then it is possible
  • 00:06:23
    to shift the radix point towards the left.
  • 00:06:26
    That means using the floating-point representation, it is possible to represent the very large
  • 00:06:30
    numbers like the distance between the planets or the mass of the planets and the very small
  • 00:06:35
    number like the mass of the atom using these floating-point numbers.
  • 00:06:40
    So this floating-point representation provides both good range as well as the precision.
  • 00:06:46
    So now, let's see how to represent these floating-point numbers.
  • 00:06:51
    So the representation of this floating-point number is very similar to how we are representing
  • 00:06:55
    the decimal numbers in the scientific notation.
  • 00:06:58
    So in the scientific notation, the radix point or the decimal point is set in such
  • 00:07:03
    a way that we have only one significant digit before the decimal point.
  • 00:07:08
    So for the integers, by default, the radix point or this decimal point is set to the
  • 00:07:13
    right-hand side of this least significant digit.
  • 00:07:17
    So here, to represent this number in the scientific notation, the decimal point is shifted to
  • 00:07:22
    the left-hand side by the five decimal places.
  • 00:07:25
    And that is why, here the exponent is equal to 5.
  • 00:07:29
    So as you can see over here, we have only one significant digit before the decimal point.
  • 00:07:34
    But if the same number is represented like this, then that is not the scientific notation.
  • 00:07:40
    Because if you see over here, then the digit before the decimal point is 0.
  • 00:07:45
    But in the scientific notation, it has to be non-zero.
  • 00:07:48
    Similarly, if you take this number, then in the scientific notation, this is how it can
  • 00:07:54
    be represented.
  • 00:07:56
    So here, for the scientific notation, the decimal point is shifted to the right by three
  • 00:08:01
    decimal places.
  • 00:08:03
    And that is why over here, in the exponential term, we have this 10 to the power minus 3.
  • 00:08:08
    So in the scientific notation, we have total two components.
  • 00:08:12
    That is the significand and the exponent.
  • 00:08:15
    So here, in the second representation, if you see the significand, then that is equal
  • 00:08:20
    to 4.345.
  • 00:08:22
    And similarly, the exponent is equal to minus 3.
  • 00:08:26
    And here of course, since we are representing the decimal numbers, so the base of the exponent
  • 00:08:31
    is equal to 10.
  • 00:08:33
    So here in the scientific notation, we are normalizing the numbers so that we have only
  • 00:08:38
    one significant digit before the decimal point.
  • 00:08:41
    And because of this normalization, it is possible to represent all the numbers in a uniform
  • 00:08:46
    fashion.
  • 00:08:47
    For example, if we take the case of this number, then the same number can also be represented
  • 00:08:53
    like this.
  • 00:08:55
    And of course, the value of the number will still remain the same.
  • 00:08:58
    But as you can see, all these representations are different.
  • 00:09:02
    So that is why it is good to have a uniform representation for each number.
  • 00:09:07
    So in general, we can say that in a scientific notation, this is how the decimal number is
  • 00:09:12
    represented, where this D represents the decimal digit.
  • 00:09:16
    So similarly, this floating point representation is very similar.
  • 00:09:21
    And here, this B represents the binary digits.
  • 00:09:24
    So if you see this floating point representation, then it consists of the three parts.
  • 00:09:29
    That is sign, fraction, and the exponent part.
  • 00:09:33
    And here, the base of the exponent is equal to 2.
  • 00:09:37
    So in this representation also, first binary numbers are normalized in this format.
  • 00:09:43
    So in a scientific notation, we have seen that we must have only one significant digit
  • 00:09:47
    before the decimal point.
  • 00:09:49
    Now in the case of the binary, we have only two digits, that is 1 and 0.
  • 00:09:55
    And therefore in the binary, the only possible significant digit is equal to 1.
  • 00:10:00
    That means in this floating point representation, this significant digit just before the binary
  • 00:10:04
    point will always remain 1.
  • 00:10:07
    So we can say that, this is the general representation for the floating point number.
  • 00:10:12
    So now let's see, how to normalize any binary number and how to represent it in the floating
  • 00:10:17
    point representation.
  • 00:10:20
    So let's say, this is our binary number.
  • 00:10:23
    And we want to represent this number in the normalized form.
  • 00:10:26
    So for that, we need to shift this binary point in such a way that just before the binary
  • 00:10:31
    point, the significant digit is equal to 1.
  • 00:10:35
    That means here, we need to shift the binary point to the left by 2 bits.
  • 00:10:40
    And that is why over here, this exponent is equal to 2.
  • 00:10:44
    That means whenever we shift the radix point to the left by a 1-bit position, then the
  • 00:10:48
    exponent will increase by 1.
  • 00:10:50
    So here, since the radix point is shifted to the left side by 2 bits, so the exponent
  • 00:10:55
    will increase by 2.
  • 00:10:57
    So similar to the left-hand side, when the radix point is shifted to the right by a 1-bit
  • 00:11:02
    position, then the exponent will decrease by 1.
  • 00:11:05
    For example, if we have this number and to represent this number in a normalized form,
  • 00:11:11
    we need to shift the binary point to the right side by 2 bits.
  • 00:11:15
    And that is why here the exponent will decrease by 2.
  • 00:11:18
    Or in other words, here this exponent is equal to minus 2.
  • 00:11:22
    So these two representations are in the normalized form.
  • 00:11:26
    So in this way, we can normalize any binary number and we can represent it in the floating
  • 00:11:31
    point form.
  • 00:11:33
    So now, let's see how this floating point number is actually stored in the memory.
  • 00:11:38
    So while storing, the 1-bit is reserved for the sign bit.
  • 00:11:42
    That means while this number is stored, then the MSB will represent the sign bit.
  • 00:11:47
    So if this bit is 0, then it means that the number is positive.
  • 00:11:52
    And whenever this bit is equal to 1, then it indicates that the number is negative.
  • 00:11:57
    So after the sign bit, the few bits are reserved for storing the exponent value.
  • 00:12:03
    And then the remaining bits are reserved for storing this fractional part.
  • 00:12:07
    So now if you see this significand, then here the integer part of this significand will
  • 00:12:12
    always remain 1.
  • 00:12:14
    And therefore, this 1 is not stored and instead of that only the fractional part is stored.
  • 00:12:21
    So this fractional part is also referred as the mantissa or the significand.
  • 00:12:26
    That means while storing this floating point number, we have total 3 parts.
  • 00:12:30
    That is sign, exponent and the mantissa.
  • 00:12:33
    Now to store this floating point number, a certain standard has been defined.
  • 00:12:38
    Like how many bits will be reserved for the exponent as well as the mantissa part.
  • 00:12:42
    And similarly, how to store this mantissa as well as the exponent part.
  • 00:12:46
    Because this exponent part if you see, then it can be positive or the negative.
  • 00:12:51
    That means we need to decide how to store this exponent part.
  • 00:12:55
    So to store such numbers, a common standard has been defined.
  • 00:12:59
    And one such commonly used standard is the IEEE 754.
  • 00:13:03
    So in the next video, we will see the format of this IEEE standard and we will understand
  • 00:13:08
    that as per this standard, how the floating numbers are stored.
  • 00:13:13
    But I hope in this video, you understood the difference between the fixed point numbers
  • 00:13:16
    and the floating point numbers.
  • 00:13:19
    And using this floating point representation, how it is possible to represent the very large
  • 00:13:23
    numbers or the very small numbers with good precision.
  • 00:13:27
    So if you have any question or suggestion, then do let me know here in the comment section
  • 00:13:31
    below.
  • 00:13:32
    If you like this video, hit the like button and subscribe to the channel for more such
  • 00:13:36
    videos.
Tags
  • floating-point numbers
  • fixed-point numbers
  • digital systems
  • binary representation
  • scientific notation
  • computer memory
  • IEEE 754
  • normalization
  • precision
  • range