Floating Point Numbers - Computerphile

Why can't floating point do money? It's a brilliant solution for speed of calculations in the computer, but how and why does moving the decimal point (well, in this case binary or radix point) help and how does it get currency so wrong?
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This video was filmed and edited by Sean Riley.
Computerphile is a sister project to Brady Haran's Numberphile. See the full list of Brady's video projects at: bit.ly/bradychannels

[@MorreskiBear]

Explains weird results I got in BASIC programs 29.99999998 years ago!

[@Daealis]

I got 0.9999... Problems and floating point is one.

[@timothemalahieude5076]

I think this actually used to be a flaw in some banking systems, because programmers initially used floating-point numbers to store accounts money. And then someone took advantage of it by making a lot of transactions from 2 of his accounts, where each transaction made him win a tiny bit of money due to rounding. So in the long term he was able to "create" as much money as he wanted.

[@InsaneMetalSoldier]

Everything is easier to understand if it's explained in british accent

[@MMMowman23]

This is why Minecraft 1.7.3 alpha has block errors after 2 million blocks away from spawn. Pretty cool.

[@oatstralia]

Another issue I've run across with floating point rounding errors is the fact that - for the same reasons outlined in the video - a comparative statement like "0.1 + 0.2 == 0.3" comes out false, and it can be super annoying to pick out where the error is coming from, especially if it's buried in some if statement. For things like currencies, I usually just deal with ints that represent the number of cents (e.g. $2.50 is represented as 250), and divide by 100 in the end... saves me the hair tearing.

[@WarpRulez]

Small correction: You don't need a 64-bit computer to use 64-bit floating point numbers in hardware. For example on Intel processors 64-bit floating point has been supported since at least the 8087, which was a math co-processor for the 16-bit 8086 CPU. (The math co-processor has been integrated into the main CPU since the 80486, which was a 32-bit processor.)

[@Creaform003]

Minecraft stores entities as floating point's, when the world was infinite you could teleport something like 30,000km in any direction and see objects start to move and stutter about including the player. Once you hit 32 bit int overflow the world would become Swiss Cheese, and at the 64 bit int overflow, the world would collapse and crash.

[@themodernshoe2466]

Can you PLEASE do more with Tom Scott? He's awesome!

[@antivanti]

nought (nɔːt) (Mathematics) the digit 0; zero: used esp in counting or numbering For those who were wondering. I guess it has the advantage of being half the number of syllables as "zero".

[@IceMetalPunk]

My calculus teacher in high school loved that word, mainly because when he'd have a subscript of 0, he'd use the variable y just to say, "Well, y-nought?"

[@eisikater1584]

Do you remember the (in)famous Apple IIe 2 squared error? 2+2 yielded four, as it should have been, but 2^2 yielded somothing like 3.9999998, and floating point arithmetic was difficult on 8 bit computers anyway. I once even used a character array to emulate fifteen digits behind the comma, not anything I'd do nowadays, but it worked then.

[@kagi95]

This is great, I've finally understood what it means (and why) when people say "loses precision" when referring to floating points.

[@AtheniCuber]

american here, just hearing 'Not Not Not Not Not Not Not Not Not Not Not seven'

[@novarren]

I've put up English closed captions for this, since I haven't seen anyone else do CCs for this video, which is weird. I think it still needs authorised before it shows up actually on the channel, though.

[@wolverine9632]

I remember the first time I experienced this. I was writing a Pac-Man clone, and I set Pac-Man's speed to be 0.2, where 1.0 would be the distance from one dot to another. Everything worked fine until I started coding the wall collisions, where Pac-Man keeps going straight ahead until hitting a wall, causing him to stop. The code checked to see if Pac-Man's location was a whole integer value, like 3.0, and if it was it would figure out if a wall had been hit. When I tested it, though, Pac-Man went straight through the walls. If I changed the speed to 0.25, though, it worked exactly as expected. I was baffled for a few moments, and then it hit me. Computers don't store decimal values the way you might first expect.

[@chaquator]

So is this why I got like 5.0000003 when I input 25 into my self-written square-root function?

[@raumaankidwai]

chaquator yep

[@junofall]

starting out programming and some bullshit turned out as 4.000000005 instead of 4 so i'm here now xD

[@MatthewChaplain]

I love the way that all of the scrap paper for these series is clearly paper for a tractor-fed printer that's probably been in a box since the 80s.

[@plutoniumseller]

...that's why the point is floating. *sudden clarity moment*

[@miss1epicness]

Tom Scott's videos strike me (a young CS student) as my favorite. They're really authentic with the "and this is the part where you, a young programmer, start to tear your hair out"--please keep them up, I'm really enjoying it!

[@etmax1]

Well I've been working with computers for some 35 years and while very early compilers used to do 32 bit FP (floating point) around 20 years ago some people got together and settled on standards for floating point on computers and soon after 80 bit FP became a standard even though the computers were only 16 or 32bit at the time. Basically the machine's register size (32bit etc. has nothing to do with the number size usable as a sub-program deals with it. Sure it won't be as time efficient, but that's not what was suggested here.

[@mtveltri]

I have a computer science degree from a top school, and yet nothing was ever explained nearly as well as this. I love this KZbin channel. Absolutely brilliant explaination. Thank you!!

[@BritchesBumble57]

and this is why you should never fuck with floating point numbers

[@JSRFFD2]

OK, so why does the cheap pocket calculator show the correct answer? I thought they used floating point numbers too. Do they have electronics that do computation in decimal?

[@joemaffei]

An even better example of floating point error is trying to add 1/5 and 4/5 in binary, which is similar to the example in the video about adding thirds: 1/5 = 0.0011~ 4/5 = 0.1100~ 1/5 + 4/5 = 0.1111~

[@FhtagnCthulhu]

Please, anyone reading this, don't use float for finances. Its a mistake I see people do all the time, please just don't.

[@paterfamiliasgeminusiv4623]

I appreciate the fact that you get into the topic right at the first second. You don't see this in the world very often.

[@superdau]

You should have mentioned: never use a check for equality with floats. The chances are very high you will never match it. On the other hand I guess every programmer should make that mistake once. E. g. decrease a whole number by 0,1 until you reach zero. Be prepared for a never ending loop.

[@MrShysterme]

Many compilers throw a warning when you try to compare floats. I am not a computer scientist, but instead learned to code for research purposes and fun. There are cases where you have to compare floating point numbers. When I do so, my trick is to come up with some acceptable tolerance around them or to convert to large integers by multiplying or dividing by scalars, rounding, taking ceiling or floor, etc. Basically, I massage into an integer that I can control somewhat or use less than or greater than but watch what I'm up to. I've had many programs not function properly because I did not take into account the behavior of floating point. Is there other techniques that are better that I'm missing? I'm mostly self taught and come up with workarounds on my own.

[@tobortine]

Entirely agree plus there's no reason (that I know of) to use a float for a loop counter.

[@ThisNameIsBanned]

Yea most compilers i used actual see the problem and pop a warning, in Eclipse you can even go as far as using CheckStyle to auto-correct your operations with float (so it will highlight the possible problems right away). Float is great for speed, but you have to do some hacks to get precision again.

[@mountainhobo]

Ran into this 30 years ago, was working in finance and trying my hand at simple applications running fairly complex transactions over many years horizon. Was dumbfounded then. Took me while to learn what was going on. Thanks for the trip down the memory lane.

[@tobortine]

An excellent explanation and a pleasure to watch.

[@EebstertheGreat]

I prefer 3 bit floats. They have 1 sign bit, 1 exponent bit, and 1 significand bit. They can encode ±0, ±1, ±∞, and NaN, which is all you really need.

[@EZCarnivore]

5 years as a programmer, and I finally understand floating point numbers =P

[@EZCarnivore]

I have a college education, they just never explained it in a way that I understood it!

[@WillLorenzoCooper]

+エックザック True it's like you have one shot to learn in a lesson then it's gone. While on here, you can relearn on many videos if we don't quite understand.

[@Jenny_Digital]

Well done on explaining it so well! For a long time I thought it a sod and left it. I finally had to do it in PIC assembly and then after I'd struggled to victory I get told straightforwardly over a single video many years later. Keep up the excellent stuff!

[@Tulanir1]

If you type 0.1 + 0.2 into the python 3.4.1 shell you get 0.30000000000000004 But if you type 1/3 + 1/3 + 1/3 you get 1.0

[@mc85eu]

Exceptionally clear and engaging. You're obvioiusly clever, but are also able to break things down and be interesting. I learned a lot - and not just about floating point. Thank you.

[@majinpe]

Tom Scott's talk is so interesting and easy to understand I love this guy

[@airmanfair]

Aren't floating point errors what they used in Office Space to ripoff Initech?

[@Czeckie]

Congratulations on Tom Scott! He seems like a great teacher, I enjoyed this video very much - it doesn't matter that i knew the stuff already

[@rodcunille4800]

Before anything, great explanation!!! That is why supermarket's checkout's systems expresses the currencies in hundreds and not in decimals, at the end, your bill will be divided by 100 and you'll get your exact amount to pay

[@MenacingBanjo]

Wow! I had wondered for years why Microsoft Excel would always end up with weird figures that would just show up way out in the 15th-or-so decimal places of the returned values from SUM formulas. Now I know. Thank you!

[@Zeuskabob1]

Hah, this reminds me of a programming exercise that I had to undertake in Algorithms 2. The teacher wanted us to calculate a continuous moving average for a set of values. Since the data requirement was so minimal, I decided to store the last n digits in an array, and cycle through them when new numbers appeared. When needed, the moving average was calculated by adding the numbers together and dividing by n. My program would fail the automated test, because it failed to include the almost 3% error that the professor had gotten by updating a floating point average value for every step of the calculation. I had to teach about 5 other students about the fact that their program was too accurate, and needed to be downgraded.

[@KoyasuNoBara]

Thank you! I came across this problem in some homework recently. I stored a decimal number plus an incrementing number in the float variable that the teacher provided in his premade node, and started getting this problem. The variable was supposed to hold user input anyway*, and since the problem didn't show up when I did it that way, I didn't bother worrying about it. It did confuse the hell out of me, though, so I'm glad to find an explanation. *I got tired of doing the user input part while trying to test my Linked List, because it involved about six different questions to answer per node.

[@imanabu5862]

I was only one minute into the video and you already answered my questions! I am no a specialist and literally have no idea what floating points are/ after hours of searching this is the first video that makes sense to me !! thanks

[@thecassman]

Fantastic explanation! Not heard it explained anywhere this clearly before and it's a great way to explain it. Keep these videos going Brady / Sean! As somebody who programmed for a payroll firm for 5 years, and now program for a credit risk firm - i totally appreciate the benefits of using decimal types!

[@krishnasivakumar2479]

I'm loving this channel. It's heaven.

[@cbbuntz]

Floating point is quite nice for audio. It means that very quiet passages have roughly as much accuracy as loud passages. It can also be annoying for audio too. If you want to add DC offset for some types of processing, the accuracy plummets (as I pointed out in another comment here). example: If you have a quiet passage and you do something like x = in + 1; out = x - 1; out will have much lower accuracy than in.

[@midinerd]

Because we do not perceive loudness in a linear form, the amount of positions offered to 'louder' portions is significantly larger than those allocated to the 'quiet' portions, and thus it is easiest to tell the bit-depth of the recording by listening to the quietest passages, loudest.

[@Moonteeth62]

And yet they tend to use absolute values that scale to the DAC values. Like 8,12,16,24 bit PCM. In the end ALL codecs have to feed an digital to analog converter that is N number of bits. Is there an audio format that uses floating point values? I don't know of one, but that doesn't mean anything.

[@cbbuntz]

Moonteeth62 Most DAWs process audio with floating point. That's why Nuendo sounded better ProTools years. It's also much easier to be "sloppy" with floating point. You don't have to worry about clipping at all; With 32 bit float, you essentially have a scalable 24 bit PCM. However, I don't know of any modern floating point ADCs or DACs. There were some in early digital years to squeeze the most out of 12 bits and that sort of thing. Converters on early AMS and EMT gear from the late 70's early 80's were that way. Our senses are generally logarithmic loudness, brightness, skin pressure. Interestingly they're have been studies that indicate that tribes without civilized culture contact often have a logarithmic perception of numbers, as in they would "count" (so to speak) in exponents. 1 2 4 8 16 etc It makes sense to me though. If you have a million dollars, you don't care about 10 bucks. If you've got 40 bucks, you care about 10 bucks.

[@StephenDahlke]

Eagerly awaiting the follow-up video that talks about the actual storage and operation of these numbers. My Numerical Analysis course spent several lectures on it, and wrapping your head around storing a number like 123.456 in IEEE 64-bit floating point is about as much fun as dealing with time zones. :)

[@RitchieFlick]

Have been programming for a while now (self-taught, 5 years in school, 1 semester in university) and I wasn't even aware of this. I LIKE :)

[@mikelipsey8837]

This guy is a very good teacher.

[@Frasenius9]

Yay for more of Tom Scott!

[@nickgrossman2385]

this is a great video, I never really understood floating point numbers until now. good job!

[@swal593868]

Thank you. Just what I needed, trying to learn Java and was getting confused when it was appropriate to use the 'float' keyword. Love your definition of a float "standard notation for Base 2.

[@veloxsouth]

I would have loved to see more about the reserved cases of the floating point standard such as NaN, as well as some more on the topic of normalization. Hope there's more to this in some "extra bits"

[@Kd8OUR]

You guys provide a far better lesson than anything I got in school.

[@jmbrjmbr2397]

Thanks to this channel for these videos, they are so valuable for me.

[@Wilc0]

Tom, I really hope there are a lot more videos coming from you, I /love/ the way you talk! Maybe an idea for a future topic, the differences and down/upsides to all the different coding languages?

[@0pyrophosphate0]

This actually cleared up FP logic for me a lot. Thanks for that.

[@CoxTH]

To come back to the 3d game example: In older versions of Minecraft there is a bug caused by floating point rounding errors. Because the game world is procedurally generated those errors cause errors is the world generation. They only become significant when one of the coordinates is at about +/- 12.5 million, but there the world generation changes significantly.

[@frollard]

This primer now makes it all make sense. I've written a few programs where floats were required and they always pooped the bed somehow - to solve it I was doing type conversions to to integer math, then converting back...such a pain!

[@RMoribayashi]

My old HP-25c and 15c had Engineering notation. It was Scientific notation but the exponent always a multiple of 3 with up to three digits left of the decimal point. Getting your answers formatted 823.45x10⁻⁶ instead of 8.2345x10⁻⁴ made working with the Metric system effortless, absolutely brilliant.

[@RachelMant]

Would be fantastic to see a video done on Fixed Point, which is the other way to solve the Floating Point accuracy issues for some small length of numbers, especially as you can store the decimal component as an integer and do some clever maths computing the overflow quantity back into the integer component. This is actually how programs like Excel solve the problem when you click the Currency button.

[@marko.rankovic]

In Python, doing the 0.1 + 0.2 gave me 0.3000000... Then the last digit was 4.. This lesson helped me understand the situation :)

[@at-excel]

Thanks a lot for your explanation! Floating Point Numbers are a very big problem in spreadsheet calculations most People are not aware of. If you do a lookup or if-function the normal user expects, that his numbers are correct and not smaller or bigger. I allways use to round a value to cut the error off.

[@SerpentStare]

Tom is very passionate about his numbers being correct. Some of his explanations of these errors almost sound as though the numbers have personally offended him. I find it rather charming.

[@jgcooper]

it took me a long while, but i just noticed that the "/" was added at the end. awesome!

[@salmanbukhari2008]

This is beauty. Well presented and many thanks.

[@TechLaboratories]

Thank you for what's probably the best explanation of floating point arithmetic that now exists! It's easy from this starting point to extrapolate into understanding floating point operations (FLOPs), and half (16bit), single (32 bit), and double (64bit) precision floating point numbers! Thanks Tom!

[@AlexVineyard]

Thank you Sir Dude, I finally understand!

[@deantheobald1393]

an important piece of information i needed to know, will definitely look here first for any other needs i have on this subject... thanks!

[@TheyZebra]

Awesome as usual! Thanks for the knowledge. Subscribed.

[@cfhay]

What I missed in the video is, the explanation how can the calculator show 0.1 + 0.2 as 0.3 still, when it uses floating point numbers. The calculator stores more precision than it shows on the screen, and the leftover digits are used for rounding, showing most of the stable calculations without error. This is important, because programmers needs to be aware of these errors, how to handle them, what error is tolerable, and what precision they need to keep up that requirement.

[@nlgatewood]

I run into this problem all the time at work...I deal a lot with currency. great explanation

[@BooBaddyBig]

Actually, in graphics it's also a problem, even being off by a fraction of a pixel can give a visible dust of gaps between surfaces. Careful rounding usually helps with floating point issues.

[@murphy54000]

Uhm. Pixels can't move, and there are less than 3000 tall/long on most screens. I fail to see how it could make a difference.

[@BooBaddyBig]

For example in 3D graphics you also have rotation, translation and perspective transformations and they can individually or in combination put the edges of surfaces right either side of the centre of the pixel, so the point used for the pixel isn't touching either surface.

[@murphy54000]

BooBaddyBig Well I don't make graphics, so I'm going to say that it still doesn't really make sense, but I trust what you say regardless.

[@Patman128]

New languages (like Clojure) are starting to include fractions as a data type. If the numerator and denominator are arbitrary precision (integers that scale as necessary) then you can represent and do operations on any rational number with no error.

[@DrMcCoy]

Sure; just keep in mind that this introduces both memory and computational overhead. :)

[@cosmemiralles1295]

Finally, I start to understand what the floating point is!!!

[@coolboyrocx]

I already understood this concept but I thought it was a really good explanation regardless. Some tricky things in here to get your head around initially so hopefully people find it useful!

[@Goodwithwood69]

Bloody hell! A video I understood right to the end! I'm usually lost two thirds of the way through on Braidy's videos,

[@oO_ox_O]

Could do a continuation where NaN and all the other fun stuff from IEEE 754 are explained and while at it maybe also talk about the different rounding modes and encodings of negative numbers.

[@marcuswilliams3455]

Very excellent explaining of Floating Point Numbers. Now it be great to explain about Decimal floating-point (DFP) aka IEEE 754-2008. As per financial transactions, people often balk at the one language which utilize BCD numbers for representing currency.

[@TheWeepingCorpse]

Loved learning how to program the FPU in assembler years back, I now use SIMD in my game engine.

[@dordijebojovic5582]

Ahhh, finally undersood this.THANK YOU!

[@MusicByNumbersUK]

thanks! Happy to see this as it was one of the things I was hoping to see! :) as an addition, it would be great to get an explanation on floating point from the perspective of the mantissa and exponent part and how these are implemented in binary but that might be taking it too far right? ;)

[@Adiy.]

Excellent video, Really Helped me!

[@ThePhiSci]

I watch Computerphime's videos for this guy.

[@ivokroon1099]

He says that the last didgit is not a problem, but once I tried to make a timer which kept adding 0.1 to itself, and when it reached a number (for example 3), it would activate another script. I did never activate because of those floating point numbers. Thanks for the explonation, computerphile!!!

[@ericwu6428]

Wow. I remember this from my early times in programming. Now I am learning programming in school and those programming languages are so smart that they fix these errors for us. It makes me kind of sad to think that in the future these things will be done for us, and an understanding of these sort of things is going to be obsolete.

[@leeannison5048]

This was very helpful, thank you.

[@SirCrest]

Tom Scott is probably my favourite guest.

[@PullerzCoD]

Thats first maths process is gonna save me a lot of anguish

[@TheKazragore]

It's so nice to hear someone say "maths" on KZbin.

[@nO_d3N1AL]

Wow, didn't know there was so much to floating point numbers. Nice video

[@samanthabayley2194]

This finally explains strange results in a program I wrote for first year uni. The program was meant to calculate change in the fewest possible coins but whenever I had 10c left it would always give two 5c coins, for the life of me I couldn't figure out why now I know. It also used to happen with the train ticket machines so at least I'm not alone in making that error.

[@mailsprower1]

Thanks for closing that tag at the end, Brady! My OCD is finally fixed!

[@MrTridac]

The problem is that the tags from the first few videos are still open. I guess the nesting depth is still a few dozen levels deep. And you have to include the opening tags at the end of all these early videos too. INSANE. O.o

[@mailsprower1]

I tried not to think about that... I tried.

[@MrTridac]

mailsprower1 Brady should make a "closing tags only" video or put them in as "cameo" here and there in future videos. :)

[@ronaldlijs]

Well explained, keep them up!

[@LeBonkJordan]

Jan Misali basically said (heavy paraphrasing here) "0.3" in floating point doesn't really represent exactly 0.3; it represents 0.3 _and_ every infinitely-many other real numbers that are close enough to 0.3 to be effectively indistinguishable to the format. Basically, every floating point number is actually a range of infinitely many real numbers.

[@ltcameron1]

Great video. Very many thanks. Makes floating point so much easier to understand.Bonus that it's a young brit! Well done.

[@devingreenfield8047]

you make so much more sense than my numerical methods prof, Thank you!

[@mxkep]

I never thought floating point could be described in such an interesting way!

[@TheSam1902]

What a brilliant video, thank you, I wasn't aware of this problem and I'll think about it next time I deal with things like currency etc ^^

[@tiagoling15]

For the first time in my life I now understand what floating point is. Thank you.