IEEE 754 Standard for Floating Point Binary Arithmetic

This computer science video describes the IEEE 754 standard for floating point binary. The layouts of single precision, double precision and quadruple precision floating point binary numbers are described, including the sign bit, the biased exponent and the mantissa. Examples of how to convert between denary (base 10) and single precision floating point binary numbers are shown, including how to round the mantissa correctly. There are also some example conversions that you can try yourself, along with their solutions. The way the exponent bias works and why this particular technique is used in the IEEE 754 standard, rather than two’s complement, is demonstrated. Some reserved exponent values are also mentioned.
0:00 Introduction
0:14 Review of fixed point binary
1:02 Review of floating point binary using two's complement
2:44 The IEEE 754 standard
4:08 Conversion from denary to single precision floating point binary
8:25 The exponent bias
13:42 Example conversions and solutions

[@MuyenKamran1nehal]

This is amazing. You explained in 20 minutes what my professor could not in 90 minutes.

[@neolord50pro77]

How scared and ignorant i was about those floating points, but now i'm feeling enlightened and empowered with knowledge. Thank you!

[@ComputerScienceLessons]

That's music to a teacher's ears :)KD

[@vit6723]

@@ComputerScienceLessons what if the given would be .12?

[@jbkhan1135]

This channel is so incredibly underrated... I wish every compsci student would subscribe to this. I know I could have used it in my uni days. Great work and keep it up!

[@ComputerScienceLessons]

It's very kind of you to say so. Thank you :)KD

[@febiefebriansyah]

@@ComputerScienceLessons 6:44 why the number is chage (19.5.. to 19.5)?
btw, i think the voice volume is too low 🙏

[@raffaelechinotti5244]

@@ComputerScienceLessons i totally agree with him, BECAUSE I AM in university goddamn it can't you be my professor? :l

[@NoName-tj8dm]

By far the best explanation on IEEE 754 and on biased exponent. It's really remarkable that you explained in a lucid manner. Thank you for the series on binary too.

[@ComputerScienceLessons]

Thank you. And you are most welcome :)KD

[@gabrielruszala4432]

This guy's channel name is literally "Computer Science." Epic.

[@jannesvanquaillie9151]

I really started to hate floating-point numbers after trying to do math with it.
But you helped me appreciate it.
This video is SO SUPER good.
Few people are so good at what they do, know what they are talking about AND can even explain it; in your case, you are really good at all of them.
Truly, you are probably one of the reasons that my passion for these kinds of things is so upfront.

[@ComputerScienceLessons]

Thanks for the lovely comment :)KD

[@ganeshharikrishnan]

I've followed this channel for about 2 years now. It's genuinely incredible how he has the capacity to simply the most complex procedures into a single set of steps

[@ComputerScienceLessons]

Thank you :)KD

[@bobdemp8691]

Really good. 20 years ago this went over my head and always meant to revisit it. Very clear explanation.

[@ComputerScienceLessons]

Thank you. You are most welcome :)KD

[@annilator3000]

that's the most comprehensive video I could find on this topic, great job

[@hayden.A0]

Thanks again. Floating point numbers were always very confusing to me until I found your channel

[@ComputerScienceLessons]

Happy to help :)KD

[@poserGirl863]

this is such a good video! I never had the idea of using what you call the "modified gadget" for the place values of 2s complement numbers. I came here thinking I knew my 2s complement and I'm leaving having learned not only the standard for floating point numbers, but an incredibly valuable tool for working with 2s complement. Thank you!

[@LoudnessWar]

I signed in to UA-cam after an eternity solely to post this - your explanation of the Two's Compliment sign bit at 9:41 is absolutely brilliant and intuitive. Well done; it's so simple: the MSB becomes its normal value except negative, and the conversion to decimal "Just Works." I wish I had seen that when I was first learning binary formats as a kid - it's SO much easier to think of it that way than the whole "count backwards from all 1's" way I had conceived of it before.

[@ComputerScienceLessons]

Thank you for taking the time to post such a lovely comment. It makes it all worthwhile. :)KD

[@clarencedomond5837]

This channel is a goldmine, you are saving lives sir please do not stop making videos!

[@ComputerScienceLessons]

You are very kind. More on the way :)KD

[@segafrompk]

This makes it really clear why simple math operations with floating point numbers sometimes give weird X.99999999 results. Great video!

[@ComputerScienceLessons]

Thank you :)KD

[@swedishguyonyoutube4684]

4:08 is an incredibly well thought out, easy to follow, pedagogical explanation of conversion! Thanks a bunch!

[@ComputerScienceLessons]

Thank you. Your comment is greatly appreciated. :)KD

[@florisvanderhout2675]

just WOW, you are incredible. I just started with my first year of computer science and this video made everything sooooo clear. I subscribed. So keep up the good work.

[@ComputerScienceLessons]

Thank you for the lovely comment. Big smile here. :)KD

[@kellerw4800]

Thanks so much for this video, excellently paced for taking notes and really absorbing and understanding the information, with lots of great explanations as to WHY steps are taken. TY

[@ComputerScienceLessons]

You are most welcome :)KD

[@roshanafernando2443]

Well explained. Worth watching the video. Keep on doing this. All the best!

[@Sciller4]

Every year, I go on CS competitions (and usually do quite well, I was second in Croatia once), but I could never remember how to use this standard or why things were how they were. Thank you for that!

[@ComputerScienceLessons]

You're most welcome. Good luck with the next competition :)KD

[@MIbrahimKh]

Really made me "floating" with joy, thanks teacher!

[@ComputerScienceLessons]

You are very welcome. You made my day. :)KD

[@Rahul-fq9kf]

Very thankful to you for explaining it so well.

[@kannadiga3644]

excellent session, crisp and complete coverage..

[@ComputerScienceLessons]

Thank you. :)KD

[@estefaniac8260]

Thank you so much for your detailed explanation. It is excellent and exactly what I was looking for to understand this topic.

[@ComputerScienceLessons]

You are most welcome :)KD

[@parsiabolouki7685]

Amazingly great explained! Thank you

[@xinguangmingwo2343]

Your video taught me IEEE 754. Thank you.

[@ComputerScienceLessons]

You're welcome :)KD

[@varma8669]

This was a very helpful video. Came in clutch right before the due date. Thank you!

[@ComputerScienceLessons]

You're very welcome :)KD

[@pinch-of-salt]

More IEEE standard videos!!! Thank you so much. VonC explained it great on stackoverflow but I still referred to this video!

[@ComputerScienceLessons]

Working on it :)KD

[@ieatnoodls]

Explanation clear as the sky, thank you

[@ComputerScienceLessons]

You're welcome :)KD

[@raghunathprabhakar2166]

Thank you so much for explaining this in such a concise manner

[@ComputerScienceLessons]

You're very welcome :)KD

[@JamesBrodski]

Wow, what an amazing video. Thank you so much.

[@LuchsYT]

TYSM! This finally made me understand the IEEE 754 standard for floating point numbers!

[@ComputerScienceLessons]

You are welcome indeed. Delighted to help :)KD

[@caramucha]

Mate, this is brilliant! I am a medical professional trying to grasp the basics of computer anatomy and physiology if you will, and your videos really make everything much more organic! Thanks for this amazing content! I'm looking forward to watching the other series in your channel.

[@ComputerScienceLessons]

You are most welcome. Thanks for taking the time to comment :)KD

[@nitinrawat4522]

Nicely and neatly explained! Got all my answers from the single video. Keep up the good work! XD

[@ComputerScienceLessons]

Delighted to be of service :)KD

[@user-gy8co1gk8o]

Very good explanation, thanks!

[@notgiven3971]

I would just like to say thank you for this video it was very helpful for my upper div assembly lab assignment on this topic. Thank you for the video!

[@ComputerScienceLessons]

You ae most welcome :)KD

[@RawFatGod]

You explain it so clearly

[@user-fg6ng7ej6w]

very clear explanation thanks !

[@swun3367]

your way of saying Mantissa is just so perfect. has to be said.

[@ComputerScienceLessons]

MANTISSA MANTISSA MANTISSA

[@Thomas-ht9cn]

Thank you, sir! This video really helped me grasp the concept.

[@ComputerScienceLessons]

Delighted to help :)KD

[@victorlucki8586]

Loved it! Thank you, as always!

[@ComputerScienceLessons]

You're most welcome. Thanks for the comment :)KD

[@alikhaled844]

The exponent bias part was masterful..

[@user-om2ev8wz6g]

Thank you Kevin Drumm for this amazing video

[@ComputerScienceLessons]

My pleasure :)KD

[@rpop7911]

Very well explained, thank you so much!

[@ComputerScienceLessons]

You're very welcome :)KD

[@petar7867]

Video is very easy to understand, thank you very much!

[@ComputerScienceLessons]

You are very welcome :)KD

[@spacemule1]

High quality! Thanks brother

[@MCMeru]

Incredibly well explained!!!

[@ComputerScienceLessons]

Thank you. You made my day :)KD

[@izotovadaria7151]

Thank you so much! This is so helpful for my classes!

[@ComputerScienceLessons]

You're very welcome :)KD

[@YnjeviYhhei]

Very well explained thank you

[@elchemico100]

Really helpful tutorial! Thank you very much!

[@ComputerScienceLessons]

You're very welcome :)KD

[@alphanumeric2702]

Brilliant presentation!

[@ComputerScienceLessons]

Thank you :)KD

[@ErosMarques]

thanks, this was really helpful!

[@roman_mf]

Thank you so much, an explanation well done!

[@ComputerScienceLessons]

You are very welcome. :)KD

[@realtissaye]

DUDE THIS VIDEO IS SO AWESOME

[@ComputerScienceLessons]

You are most welcome. Thanks :)KD

[@fromthebeattothesaber1419]

Thanks a lot for this great explanation
I am from Germany and i am currently playing with the x32 osc software which uses these floation points to control the faders of each single channel
There is no open source javascript implementation of this calculation on the internet
And also no good explanation
With the help of this video I've managed to create my own conversion function
Thanks a lot

[@tadikasxd9546]

Amazing video, Thank you so much

[@ComputerScienceLessons]

You're very welcome :)KD

[@ahmadhosseinzadeh3141]

Thanks for the splendid video!
Will you record a video on rounding up/down the floating point formats and the associated errors?

[@axelboy9080]

it was wonderful..
you taught me Sth..
I LOVE YOU !

[@ComputerScienceLessons]

Thank you. Love U 2
:)KD

[@randyzakyasuchrady1488]

Thanks for the explanation!

[@ComputerScienceLessons]

You are very welcome :)KD

[@from2Dto3Dto4D]

thanks, very helpful and concise

[@ComputerScienceLessons]

You're welcome :)KD

[@slava_in3757]

Amazing explanation to say the least!

[@ComputerScienceLessons]

Thank you :)KD

[@ghattassaliba4910]

I love your work and explanation, you deserve to be the president 💪💪

[@ComputerScienceLessons]

Thank you so much. Not sure I would want the job of president though! :)KD

[@stopthecap8810]

This is such a phenomenal video. The pace at which the animations/charts line up with your words greatly help with understanding this. To be honest, your channel is like a wikipedia for basic data representation, and should be something every teacher recommends. The only question is what exactly do you mean by 'the value of the exponent specifies how far to the left or to the right the binary should float to obtain the value in the register?' That part wasn't clear. Do you mean how much it should float to determine the mantissa? Also, what part of the floating point concept tells us where the 'decimal' starts? Is it the exponent?

[@larrywu8977]

Really Really helpful, thank you!

[@ComputerScienceLessons]

Delighted to help :)KD

[@koraltayeb8681]

FANTASTIC!! THANK YOU

[@ComputerScienceLessons]

You're most welcome :)KD

[@bakaphu1992]

at 06:46 the header changes from example 19.5375 to 19.25 randomly, even though the example is still done with the first number.

[@TurgutKalfaoglu]

very good thank you!

[@ComputerScienceLessons]

You're very welcome :)KD

[@ofd3769]

u are the best one thanks a lot

[@ComputerScienceLessons]

You're very kind. Thank you :)KD

[@fkarls9282]

A really good explanation and a nice visualisation, great video. Can you also make a video about the Quine-McCluskey method ?

[@ComputerScienceLessons]

I'll take a look at it. Sounds like an interesting programming project. :)KD

[@singmantkpss]

Thx god i found this gem

[@ComputerScienceLessons]

Thank you :)KD

[@Antagon666]

This is one of the nicer takes

[@ComputerScienceLessons]

Thank you :)KD

[@bakaphu1992]

thank you! I think at 14:49 it's rather how many places you have to get to the right instead of left right? step 3

[@marouaabdi611]

Thank you for your brilliant explanation sir , but I have a question . in the last two examples when we converted the number 123 to binary , why one started with 01 and the other with 11 ?

[@fay7725]

I'm recommending this video to my college friends. Thank you so much
Edit : also, you got a new subscriber

[@rosiRose676]

Thank You !!!!

[@ComputerScienceLessons]

You are most welcome :)KD

[@maxhmetonxrono]

Thank you for the amazing video you are very good Teacher!!! I want to ask why we div with 2^-118 at 20:27
???

[@lilultime6555]

So good

[@ComputerScienceLessons]

Thank you :)KD

[@oviya.n1317]

excellent

[@ComputerScienceLessons]

Thank you :)KD

[@jessebelleman8070]

You are a legend!

[@ComputerScienceLessons]

You are too kind :)KD

[@brascoarts2634]

If you think about it, there's always gonna be 1 at the end💥

[@Lights.Camera.ActionYT]

Please represent 0.1 + 0.2 in IEEE 754 format and also how to equate exponents if we have -4 and -3

[@kendalwilliams5128]

Thanks!

[@ComputerScienceLessons]

You're welcome :)KD

[@yiming9569]

Could you explain why the normalization of the mantissa is "1.1 * 2^-4" instead of "0.11 * 2^-3" as you explained that normalization for positive numbers should be in the form of 0.1xxxxxx * 2^x in the video of "Binary 6 Normalized Floating Point Binary Fractions"?

[@pataruutu1]

i, and many others i suppose, would appreciate if you could do a video on guard, round and sticky bits.

[@ComputerScienceLessons]

OK - I'll see what I can do :)KD

[@mdrezaulkarim47]

wow..its awesome

[@ComputerScienceLessons]

Thank you :)KD

[@shahebmiah2492]

I haven't spent so long on a topic like this. Looks so simple but so confusing.

[@ComputerScienceLessons]

It may be worth going back to the start of my playlist on binary and working your way through theses. This video is the last in a series.

[@shahebmiah2492]

@@ComputerScienceLessons from this video i put everything together with what i learned and i understand it now. Thank you

[@xianhui5302]

thank you T.T

[@ComputerScienceLessons]

You're very welcome :)KD

[@mattstyles4283]

Great video! Why is the leftmost bit of the mantissa always 1? And how is this different to the 0.1 / 1.0 rule of normalised floating point binary?

[@mattstyles4283]

Is it something to do with the fact that IEEE format separates the sign bit?

[@abdalrhmanbadran138]

thanxxxxxxx youuuu relay greaaaaaat

[@ComputerScienceLessons]

Thank you very much indeed :)KD

[@chidon7465]

Ah now it works!

[@ComputerScienceLessons]

:)

[@bradthompson1209]

I would have liked to see you convert -3.303139... E-36 back to a 32 bit binary number. I am struggling with how to deal with the base 10 exponent. It's not obvious to me yet.

[@shahebmiah2492]

Why is there a remainder 1 after when no division took place? could i please know

[@rohitpant6473]

Thanks :)

[@ComputerScienceLessons]

YW :)KD

[@tungtruong9232]

very helpful, but why in your first example, you want to convert the value 19.25 to binary but in step 2 you also use 19.59375, can you explain me, very thanks, you lesson is very valuable

[@saiankitsahoo3944]

BRILLIANT . PERIOD
SAY NO MORE

[@TurgutKalfaoglu]

Small error: While explaning converting from 19.59375, the title of the slide switches to converting to 19.25. the bottom of the slide seems to continue on the 19.59375 however.

[@ComputerScienceLessons]

Ahhh! Thanks :)KD

[@keonscorner516]

That’s the IEEE 754 floating point double precision 32-bit format
can you explain IEEE 754 floating point double precision 64-bit format

[@Karim-nq1be]

Your video is great. One of the most boring stuff I had to learn in my life.

[@ComputerScienceLessons]

:)KD

[@floopiiisss9295]

I am confused on how he got 4 at 7:26 can anyone please explain?

[@ermanerbak]

VERY GOOD JOB. But question changing on 6.45 :))

[@96erardo]

I would love to read the original IEEE document on the subject, but it seems that is only accessible paying and i dont have the money right now :(

[@ComputerScienceLessons]

Yes - it's a shame they don't publish it for free. :)KD