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.

[@gabrielruszala4432]

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

[@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

[@f.e.febriansyahpusatedukas8816]

@@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

[@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

[@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?

[@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

[@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

[@hayden.A0]

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

[@ComputerScienceLessons]

Happy to help :)KD

[@LoudnessWar]

I signed in to KZbin 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

[@annilator3000]

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

[@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

[@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

[@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

[@alikhaled844]

The exponent bias part was masterful..

[@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

[@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!

[@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

[@xinguangmingwo2343]

Your video taught me IEEE 754. Thank you.

[@ComputerScienceLessons]

You're welcome :)KD

[@MIbrahimKh]

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

[@ComputerScienceLessons]

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

[@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

[@ieatnoodls]

Explanation clear as the sky, thank you

[@ComputerScienceLessons]

You're welcome :)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

[@roshanafernando2443]

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

[@kannadiga3644]

excellent session, crisp and complete coverage..

[@ComputerScienceLessons]

Thank you. :)KD

[@Rahul-fq9kf]

Very thankful to you for explaining it so well.

[@swun3367]

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

[@ComputerScienceLessons]

MANTISSA MANTISSA MANTISSA

[@LuchsYT]

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

[@ComputerScienceLessons]

You are welcome indeed. Delighted to help :)KD

[@raghunathprabhakar2166]

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

[@ComputerScienceLessons]

You're very welcome :)KD

[@aprendendoaainvestir]

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

[@Marijanus]

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

[@varma8669]

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

[@ComputerScienceLessons]

You're very welcome :)KD

[@parsiabolouki7685]

Amazingly great explained! Thank you

[@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

[@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!

[@JamesBrodski]

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

[@Thomas-ht9cn]

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

[@ComputerScienceLessons]

Delighted to help :)KD

[@RawFatGod]

You explain it so clearly

[@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

[@user-om2ev8wz6g]

Thank you Kevin Drumm for this amazing video

[@ComputerScienceLessons]

My pleasure :)KD

[@user-fg6ng7ej6w]

very clear explanation thanks !

[@izotovadaria7151]

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

[@ComputerScienceLessons]

You're very welcome :)KD

[@petar7867]

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

[@ComputerScienceLessons]

You are very welcome :)KD

[@rpop7911]

Very well explained, thank you so much!

[@ComputerScienceLessons]

You're very welcome :)KD

[@axelboy9080]

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

[@ComputerScienceLessons]

Thank you. Love U 2 :)KD

[@victorlucki8586]

Loved it! Thank you, as always!

[@ComputerScienceLessons]

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

[@spacemule1]

High quality! Thanks brother

[@MCMeru]

Incredibly well explained!!!

[@ComputerScienceLessons]

Thank you. You made my day :)KD

[@YnjeviYhhei]

Very well explained thank you

[@alphanumeric2702]

Brilliant presentation!

[@ComputerScienceLessons]

Thank you :)KD

[@elchemico100]

Really helpful tutorial! Thank you very much!

[@ComputerScienceLessons]

You're very welcome :)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?

[@Antagon666]

This is one of the nicer takes

[@ComputerScienceLessons]

Thank you :)KD

[@roman_mf]

Thank you so much, an explanation well done!

[@ComputerScienceLessons]

You are very welcome. :)KD

[@slava_in3757]

Amazing explanation to say the least!

[@ComputerScienceLessons]

Thank you :)KD

[@tadikasxd9546]

Amazing video, Thank you so much

[@ComputerScienceLessons]

You're very welcome :)KD

[@from2Dto3Dto4D]

thanks, very helpful and concise

[@ComputerScienceLessons]

You're welcome :)KD

[@realtissaye]

DUDE THIS VIDEO IS SO AWESOME

[@ComputerScienceLessons]

You are most welcome. Thanks :)KD

[@ahmadhosseinzadeh3141]

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

[@ErosMarques]

thanks, this was really helpful!

[@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

[@randyzakyasuchrady1488]

Thanks for the explanation!

[@ComputerScienceLessons]

You are very welcome :)KD

[@singmantkpss]

Thx god i found this gem

[@ComputerScienceLessons]

Thank you :)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.

[@larrywu8977]

Really Really helpful, thank you!

[@ComputerScienceLessons]

Delighted to help :)KD

[@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 ?

[@koraltayeb8681]

FANTASTIC!! THANK YOU

[@ComputerScienceLessons]

You're most welcome :)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

[@fay7725]

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

[@ofd3769]

u are the best one thanks a lot

[@ComputerScienceLessons]

You're very kind. Thank you :)KD

[@TurgutKalfaoglu]

very good thank you!

[@ComputerScienceLessons]

You're very 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"?

[@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 ???

[@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

[@oviya.n1317]

excellent

[@ComputerScienceLessons]

Thank you :)KD

[@lilultime6555]

So good

[@ComputerScienceLessons]

Thank you :)KD

[@rosiRose676]

Thank You !!!!

[@ComputerScienceLessons]

You are most welcome :)KD

[@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

[@brascoarts2634]

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

[@Karim-nq1be]

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

[@ComputerScienceLessons]

:)KD

[@jessebelleman8070]

You are a legend!

[@ComputerScienceLessons]

You are too kind :)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

[@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?

[@kendalwilliams5128]

Thanks!

[@ComputerScienceLessons]

You're welcome :)KD

[@mohamedjimale9258]

i'm confused on the last question of this video, where did you get 10 to the -36 from when you were times it by -3.3031391268106278973921496695945?

[@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.

[@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

[@shahebmiah2492]

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

[@xianhui5302]

thank you T.T

[@ComputerScienceLessons]

You're very welcome :)KD

[@mdrezaulkarim47]

wow..its awesome

[@ComputerScienceLessons]

Thank you :)KD

[@ermanerbak]

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

[@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

[@HexadecimalDump]

Do you know where I can read more about this theory of converting denary number to pure binary, or maybe you could explain it? I understand why repeatedly dividing whole part by 2 works, but I can't understand how repeatedly multiplying fractional part by 2 gives us binary representation. I really want to understand this part and not just memorise it

[@ComputerScienceLessons]

I think converting between binary and denary by successively multiplying or dividing by two is nice 'trick' but I prefer to teach binary from first principles. It's important to realise that each bit has a place value. Perhaps my binary playlist will help. kzbin.info/www/bejne/mXuxnmxpbb2ql9E :)KD

[@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

[@saiankitsahoo3944]

BRILLIANT . PERIOD SAY NO MORE

[@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

[@imraulemmaka511]

What is a fixed-sized register?? Could you link any related resource, please? and in the beginning, you said -" the value of exponent specifies how far to the left or the right the binary point should float in order to obtain the actual value in the register." - Could you please explain "register" here? is the value in register or what!!

[@ComputerScienceLessons]

A register is the smallest unit of storage found in the CPU. In an older machine, a register could hold 32 bits. Modern machines have 64 bit registers (or even 128 bit registers). Registers are always fixed in size. I pointed out that they were fixed in size to emphasise that register size imposes a limitation on the size of the number it can store. I strongly recommend that you take a look at my playlist on binary which explains this in more detail. kzbin.info/aero/PLTd6ceoshprcpen2Jvs_JiuvWvqIAkzea :)KD

[@Lights.Camera.ActionYT]

can i have ieee 754 format for 0.1 + 0.2

[@lucaspaiva8410]

Obrigado amigo você e Jesus são quase a mesma pessoa, salvou muito

[@ComputerScienceLessons]

Obrigado. Você é muito gentil. :)KD