Hey there! Ready to crack the code on IEEE 754 floating-point numbers? In this video, I break down how to convert decimals to binary and back for 32-bit and 64-bit floats with clear, easy-to-follow steps. From scientific notation to biasing exponents, I’ve got you covered with examples (like 45.84375!). Perfect for computer science students, programmers, or anyone curious about how computers store fractions. Pause, rewind, and learn at your pace! Subscribe for more, check my other videos, and drop a comment with your thoughts!
Introduction to IEEE 754 00:00:00
Prerequisites for Understanding 00:01:03
Steps to Convert Decimal to IEEE 754 00:02:31
Bias and Exponent Explanation 00:03:17
32-bit Float Layout 00:04:09
Sign Bit and Fraction Bits 00:05:44
Example: Decimal to 32-bit Float 00:07:56
Converting 45.84375 to Binary 00:08:34
Scientific Notation for Binary 00:09:05
Biasing the Exponent 00:10:26
Placing Bits in Layout 00:11:17
Final 32-bit Float Representation 00:13:32
Example: IEEE 754 to Decimal 00:14:28
Extracting Sign, Exponent, Fraction 00:15:36
Unbiasing Exponent and Reconstructing 00:16:23
Converting Binary to Decimal 00:18:53
Introduction to 64-bit Floats 00:19:47
64-bit Float Layout and Bias 00:20:05
Example: 64-bit Float to Decimal 00:21:09
Processing 64-bit Float Bits 00:22:05
Reconstructing 64-bit Scientific Notation 00:23:26
Final Decimal Conversion for 64-bit 00:25:00
Conclusion and Recap 00:25:54
Call to Subscribe and Outro 00:26:36
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Hey there! Let’s talk about IEEE 754 floating point numbers stored inside of your machine.
What the heck am I even talking about? Well, uh, there’s a standard called IEEE 754 which governs
how floating point numbers are stored inside of your computer on x8664 machines. So, you know,
of watching this video you probably already know how to convert binary back and forth you know
between binary and decimal and hopefully you understand how to do binary with fractions
if you don’t i have other videos i’ll talk about that in a second and you also understand
that in the machine there is like a slightly different format that’s my doggy slightly
different format for storing signed integers in the machine okay so floating point numbers have
an even different uh format that we have to follow and the hardware is designed that way
hardware is designed that way so before we can really benefit you’re all gonna
see how bad I spell okay before we can benefit from this video your prerequisite
knowledge should be that for starters you should be able to convert back and
forth between binary and decimal including numbers with fractions so if
if you don’t know how to do that yet this video is is not for you just yet you
should go check out my other videos I explain in other videos exactly how to
do that go to the other videos come back and then check this one out after you
know how to convert back and forth between binary and decimal with
fractions the second thing you should know how to do before watching this
video is convert binary numbers with fractions to scientific notation and
and back again. Meaning you should understand scientific notation and you should also understand
how to take a binary number with a fraction and convert it to scientific notation and then
convert it or deconvert it back to like regular form. Again, I have another video for this,
so if you don’t know how to do that yet, you should probably pause this video and then go
Let’s see. For converting from a regular number to IEEE floating point number in the machine using just binary alone is the following.
So I’m going to say basic steps. Step one, convert the decimal number to binary if needed.
And I say if needed because I don’t know, maybe you started with a binary number so you don’t actually need to convert it.
it but I’m just going to assume that you have a decimal number so you’re going to convert it to
binary first then convert the binary number to scientific notation I don’t know if convert is
the right word here because we’re not actually changing the value of the number we’re just
changing the representation so convert it to a scientific notation form maybe would be better
step three we’ll add a bias to the exponents of the scientific notation form what do I mean by
notation form. What do I mean by that? You know, if you have like a binary number and it’s in
scientific notation format, it’s going to look like this. I liked when I’m doing IEEE 754,
I like to keep the right side of the number in decimal. I’m sorry if this confuses you, but
it’s not really necessary to convert this to binary when we’re just trying to convert back
and forth between IEEE. So I always say two to the something power rather than one zero to the
something power because I want to stay in decimal. So I’ll say, you know, two to the seventh power,
right? What this means is that seven is the exponent and I have to bias the exponent before
I put it into the machine. So maybe it’s worth understanding or maybe copy pasting right now
that the bias for 32-bit floating point numbers, I should probably, I’ll paste the whole thing here.
Okay. So here’s like the format for a 32-bit floating point number. We’re good about this
number. We’re good about this just for a second. The bias for 32-bit floating
point number in IEEE 754 is 127. So that means I pretty much have to just take
the number 7 and bias it by adding 127 to it. After I’ve done that I can convert
the biased exponent to unsigned binary whole number. To an unsigned binary whole
number. So step four here is just regular decimal to binary conversion. So I would take, you know,
seven plus 127. The result of that, which I think is like 134, maybe we’ll just be converted into
binary and unsigned binary integer, not two’s compliment or anything like that. Then step five
is respect the layout. So I just, I just copy pasted the layout up above and the layout is this.
you can see that here let me see if I just maybe paste this down at the bottom
you can see that we actually have 32 bits here if I count them for you real
fast we’ve got one for the sign and then we’ve got I can never remember this I
think it’s eight for the exponent one two three four five six seven eight got
eight bits for the exponent that means I’ve already used nine of 32 bits so
that means I should have like 23 bits here so one two three four five six seven
11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, just to double check.
So we know what the sine bit is.
Actually, we don’t yet know in this video.
So in this video, the sine bit, it’s always going to be a zero if the number is positive
and a one if the number is negative.
So that’s the sine bit.
The biased exponent bits go there.
And then these Fs, those are just the fractional bits.
just the, you know, like the mantissa or the mantissa part of the fraction. So imagine if we
have a number like this, all I’m saying is we would copy paste those numbers right here.
Notice how I’m ignoring that first one dot something, because when you put binary in
scientific notation, the first number is always going to be a one. So it can be assumed. There’s
no point in storing it inside the machine. And then if you don’t have enough digits to
you will literally just pad with zeros on the right side because padding with zeros on the right side of a fraction
doesn’t actually change the value. One, two, three, four.
If I put the zeros on the left side of the fraction, it would make the fractional part smaller and smaller.
So that would be bad. I don’t want to change the value.
And this is the opposite of how you pad a whole number.
If I put numbers on the left side of a whole number part, I’m not changing the value.
But if I put numbers on the right side of a whole number part,
which in this case would be the biased exponent, then I would be changing the value.
So we wouldn’t want to do that.
changing the value so we wouldn’t want to do that. Anyway I’m going to remove this and just stick
with step five so let’s see 5a is place the sine bits step 5b is and I’m going to do an example
for you don’t worry place the exponent bits and then 5c is place the fraction bits everything
the first one dot from scientific notation form.
Again, because the first one is implied, it will never be a zero if you’ve done it correctly.
If you end up with a binary number that looks like this in scientific notation, then you’ve
done something wrong because the first number is supposed to be a one.
Okay, so we have those basic steps.
And now let’s do an example.
1, 2, 3 example. Let’s convert decimal to IEEE 754 32-bit float format.
AKA single precision floating point number. Double precision floating point number is 64-bit.
We’ll do an example for that at the very end of this, but I’m just going to stick with 32 and do
number 45.84375, just double checking, and then we’ll convert to binary. Again, this knowledge
is contained in a different video. So if you don’t know how to do this yet, you’ll need to go see my
other video. And so you can see it’s, you know, 45.84375 in binary is just going to be this.
I’ve padded it with two zeros to the left just because, I don’t know, I have a habit of wanting
wanting everything to look like it’s inside of 8-bit chunks.
You can see there’s 8 bits here.
But really, I can just remove those first two numbers.
It doesn’t matter.
Convert the number to binary.
Then, I have to convert the binary number to scientific notation.
I’m going to put maybe pseudo-scientific notation
because I’m not going to convert the right part
where it’s the exponent into binary.
I’m going to keep it in decimal.
Just to clarify again,
convert a binary number to scientific notation and you were actually going to publish it somewhere
or give it to someone, then you should convert the right side. It’s just that I’m not going to
because I don’t really need to convert the right side. I’m not interested in having the whole
number in scientific notation in binary. I’m just interested in what the fractional bits are and
then what the exponent is going to do. Like how many positions to the left or right is it actually
going to move the decimal point? And that’s just going to be the number five, five times to the
five you know five times to the left or to the right so I’m gonna go pseudo
scientific notation you pretty much you know as I covered in the different video
all we’re gonna be doing is just moving the decimal point over one two three
four five until there’s a one in the leading spot and then account for it by
saying times two to the fifth power to say the number is a lot bigger than it
looks so that’s that and let’s see if there are any trailing zeros that’s
Let’s see if there are any trailing zeros, that’s okay on the right side of the fraction.
They don’t actually matter.
You can also just delete them if you want.
Then we need to bias the exponents.
So how do we bias the exponent?
The exponent is just the number five.
That’s see, I was telling you like I don’t really care to convert it to binary because
I’m just going to bias it first and then after that I’ll buy it or I’ll convert it to binary.
So biasing it is just basically adding the bias.
just to make sure that it’s on this page and it’s easy to see.
So I’ll do that.
So in the 32-bit float layout, the bias is 127.
So I’m going to add 127 to 5.
The final number is going to be 132.
So that’s the biased exponent now.
Then I convert the biased exponent to binary.
And that’s going to be this.
10000100.
Then set up the layout so you don’t get confused.
This step is crucial.
Everyone always gets confused by the layout because like we’re human beings.
I can’t count 32 zeros that are all like in the same area and like, you know, and not
get confused every single time.
I mean, I guess I won’t get confused half the time, but that’s not good enough.
So step one is place the layout for yourself so it’s easier for you to place the bits.
place the bits and I wrote this at the top 8 bits and 23 bits just as a reminder double check
yourself 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 that’s 23 bits 1 2 3 4 1 2 3 4 that’s
8 bits we don’t need to remember that the sign bit is one bit for now it’s not too hard so now
let’s place it so the first thing to note is that this number was actually positive positive is
the negative sign there, but if it makes you feel better, we’ll just put a positive sign there just to clarify and be specific.
So because the number is positive, the sign bit is going to be zero. If it were negative, it would be one.
Then we will put the biased exponent bits. So we already converted up there. So I’m just going to copy paste it here.
We’re pretty lucky that the bits are just as big as the number of biased exponent bits that are available.
But suppose for the sake of argument that we had,
the sake of argument that we had just like you know this small number here for the biased exponent
what would you do with the remaining spaces you would just pad with zeros to the left
because remember in whole numbers in binary and decimal you pad to the left you’re not actually
changing the value you definitely would not want to pad to the right but you got to put
a bit in every position it’s got to be something so i’m just going to pad with zeros
then we take the fractional part from the scientifically notated form so that’s just
form so that’s just this right here notice how I’m copy pasting every bit
except for the one that starts the number I’m not going to do the one dot
part because that’s always assumed so we don’t need to store it it wouldn’t make
sense I’ll copy paste the fractional part and I’ll just stick it down there
and now we have a bunch of leftover bits that we need to fill out with the
fractional part you pad to the right not to the left so I’m just going to put
zeros to the right because that won’t change the value of the fractional part
Now we’ve got it. I’m just going to double check my work here.
Oh gosh, it’s too hard to read and then a bunch of zeros. Okay.
Then remove these spaces because now that you’ve done your layout,
you actually want to bear in mind that the machine doesn’t have spaces.
The machine is just, you know, a bunch of ones and zeros everywhere.
So I’m just going to carefully remove the two spaces.
And this is it.
for floating point representation of the number of the decimal number positive 45.84375
seems like a lot but it’s not too bad once you kind of get used to the format
and the bias and you already know the other steps of you know converting decimal to binary and back
again so let’s do another number let’s do it backwards so i’m going to open up a new tab here
example, IEEE 754 32-bit float 2 decimal.
Okay, so suppose we started with the following number.
Start with this number.
Oh my gosh, right?
Remember the layout.
So in the layout we have, use the layout.
Maybe I should copy paste the layout again one more time just to make sure that it’s
easy to see.
it’s easy to see. So I’m going to paste it up at the top and well, no, maybe I want this because
this is like a little bit better. I want that version. Okay. So 32 bits, a bias of 127. So
we’re going to use the layout. I’m just going to copy paste the original number and then add
spaces because I know, well, I guess I can copy paste the layout itself right above that number.
I know I need a space right there that’s the sine bit and then I need a space
after that and notice how the numbers all line up so there’s nothing missing
just to double-check yourself okay so I have the layout I know right away that
the number is going to be positive because the sine bit is zero sign is
positive okay so then I need to pull the biased exponent will be biased
the exponent bits only so I’m just going to copy paste those bits and then I’ll
pull out the fractional part first pull out the fractional part I’m going to
copy paste all those bits and maybe I’ll just say zero point those bits because
that’s what I originally grabbed and I can delete all the zeros on the right
side because again that won’t change the value of the fractional part so I’m
going to delete make sure you’re only deleting zeros though delete that now I
I need to unbias the biased exponents.
First, actually, I should, well, let’s convert the biased exponent to decimal.
A little easier.
I guess you don’t have to do that, but it’s a little easier for me.
Pull the biased exponent.
So this number is going to be 132, which I think is what we had last night.
Let me just double check that.
132.
Yeah.
132, yeah so the biased exponent that we pulled out is 132 then we unbiased the
biased exponent so just subtract the bias again in 32-bit format here the bias
is 127 so I’m going to subtract 127 instead of adding 127 so now I know the
real bias or sorry I know the real exponent is 5 then I can recreate the
I’ll put sudo here just because it’s not, you know, I’m not, I’m not converting binary for,
for the entire thing. So that means I can take the fractional part here and I can say it’s going to be
one dot the fractional part. Maybe I should remove that zero here to make it less.
I don’t know. Cause if I put a one there, it kind of feels like I’m saying the number is
that it’s one dot something.
I don’t like that I’m putting a zero there.
I’m just gonna erase it
because it’s really gonna end up being a one.
And if I say pull the fractional part
and I put a one there,
then it doesn’t it kind of sound like
I’m pulling the one from somewhere?
No, the one is implied.
Anyway, so now I can recreate the number
in pseudo scientific notation.
I can say times two to the something power.
We know that the exponent is five.
Again, it’s pseudo scientific notation
because if five we’re gonna do the complete,
complete, you know, true scientific notation. The right side should also be in binary, but I’m not
going to do it. I’m a little bit lazy because we don’t need that IEEE. So it’s going to be two to
the fifth power. Then if I undo the scientific notation, I’m really just going to take this
scientifically notated version and I’m going to say, well, I’m going to move the decimal point to
the right five times because that’s what the part on the right says. So I’m going to go one, two,
1, 2, 3, 4, 5, stick it there, remove the original decimal point, and then I can remove
the exponent part. So then let me just double check my work. 1, 0, 1, 1, 0, 1, 1, 1, 0, 1. Okay.
So we’ve got that now. Then all we have to do is convert this number from binary to decimal.
The final number here is just, you know, it’s going to be the same number that we worked with
number that we worked with previously so I’m just going to copy paste it there is
you know of course this video is not about converting back and forth from
binary to decimal with fractions check out my other video if you want to learn
how to do that but once we do that you know this number becomes that number
and and now we know how to convert back and forth between a decimal number and
for 32-bit floats in the machine.
No problem.
A lot of steps, but no problem once you understand them.
Okay, now let’s do an example
with 64-bit floating point numbers.
Honestly, this is not gonna be more difficult.
In 64-bit floats, the idea and the format and the steps,
it’s all gonna be the exact same thing.
It’s just that we have more bits
for the biased exponent and the fraction,
and then the bias itself will be a little bit higher.
So let me copy paste the format for you to see.
at for you to see. So in IEEE 64-bit floats, IEEE 64-bit float, well IEEE 754, 64-bit float,
you know we refer to these floats as doubles right double precision floating point numbers.
Notice how instead of 8 bits for the biased exponent we have 11 so we can represent a much
notice how for the uh the fraction part instead of having 23 bits we have 52 bits so we can also
represent much more precisely that’s really the only difference let’s uh oh sorry the other
difference is also that the bias is now equal to 1023 instead of 127 so what is that like
10 times larger of a number that we can represent or something like that so mostly we get more
If you don’t like it, sorry, it’s the way it is.
It’s in the hardware.
So let’s do an example for a 64-bit float for a double.
64-bit example.
Let’s say we started with the following number.
Inside of the machine.
Like we looked inside of the machine and we grabbed these bits.
And I’ll get rid of the spaces.
So we have like a giant, big, huge number.
We have like a giant big huge number.
We’re trying to figure out, you know, what is this number in decimal.
So I’m going to add spaces to respect the layout.
I’m going to just copy paste the layout right here.
I’m going to say respect the layout.
I’m going to paste it and then that helps me line everything up.
So I’m going to say the sine bit is the first bit and then I’ll put a space there so I know.
All right, the sine bit is a one.
That means it’s going to be a negative number.
The biased exponent bits are there.
The fractional bits are there.
bits are there so well first thing I can just do is say that the sine is going to be negative
because the sine bit is a one
and then I’ll pull the biased exponent bits which is going to be this number right here
should pull the fraction bits next. Fraction bits are just here I’m gonna
copy paste those here and then I can remove the zeros. Be very careful you’re
not removing any ones. So there was a lot of wasted bits there. That’s okay. I’ll
convert the biased exponent to decimal. So that’s going to be…
I think and I did not do that in my head.
I have an answer sheet.
Don’t worry.
I’m not that cool.
Then we’ll unbiased or de-bias unbiased the biased exponents by just subtracting the bias.
Remember in this 64 bit format, the bias is 1023.
So that means I got to take 1031 minus 1023 and that will be equal to 8.
Okay.
Okay, now we kind of have everything we need to sort of reconstruct the scientific notation part, right?
We have the fraction bits, that’s the most important part.
Oh, and then also the sine bits.
So I can say negative something.
It’s obviously going to be one dot something because even though the number is negative,
it will still always start with a one.
That rule doesn’t change.
Negative one dot something and I’ll just paste the, whoops, paste the fraction bits.
I think I did paste the right thing.
right thing so we paste the fraction bits and then we have to use the regular exponent the
unbiased exponent the non-biased exponent two times times two to the oops times two to the eighth power
now this is our number in scientific notation um
convert to scientific notation i’ll put p for pseudo scientific notation
and then now we have to convert to just regular format okay so I’m going to
copy paste this right here and if it’s 2 to the eighth power that means the real
number is going to be bigger than it looks in scientific notation even though
if you are very familiar with scientific notation you kind of already know that
it looks big but you know if we just if we’re only talking about the bits on the
1, 2, 3, 4, 5, 6, 7, 8.
Put the decimal point there.
Remove the original decimal point.
Now we no longer need 2 to the 8th power.
And again, just as a reminder, if it was 2 to the negative 8th power,
you would be moving the decimal point in the other direction.
So now we’ve got this number 111, 010101, 0101, 0101, 0101.
Then a 0.01.
Okay, so then convert to decimal.
doesn’t talk about how to convert back and forth from binary to decimal find my other videos but
basically uh it’s going to end up being six uh negative four six nine point two five
okay we now have done a quick practice starting with a giant number of bits and slowly converting
them to the real uh floating point number in decimal that uh that they represent i’m not going
going to do another number in the reverse direction because we already did that for 32 bits
and again with 64 bits the only real difference is just you have more exponent and fraction bits
and the bias is a higher number and that’s it so like all the other steps are just the same
I just wanted to show you one time with a 64-bit number in one direction but
I think at this point we’re good thank you so much for watching this video I hope you learned
had a little bit of fun. I’ll see you in the next video.
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