Subtitles section Play video Print subtitles >> DAVID J MALAN: All right, so this is CS50. And this is clearly a Friday. And this is the end of week one. So you may recall that we left off, last time, with a cliffhanger of sorts. Whereby we expose this lie, that no matter what you've been taught growing up, 1 divided by 10 is not, in fact, 0.1 or 0.100 or 0.10000. In fact, if we compile this program, as we did yesterday, with, make imprecision, and then do dot, slash, imprecision, we learned that this is what's 1 divided by 10 actually is. So that's not really quite the case. But this does hint at some fundamental limitation of computers. And indeed, among the things we're going to do today is take a look at why this has happened, what implications this has, how humanity has failed to grasp this in some very serious situations-- the result of the which has been quite tragic and expensive-- and also take a look at how we can actually defend against these kinds of limitations. So intuitively, perhaps, why is 1 divided by 10, according to my computer here, not just 1/10, 0.10? Yeah. What do you think? >> AUDIENCE MEMBER: The radix is different? >> DAVID J MALAN: The what is? Oh, the radix is different? So not quite. It's actually more fundamental to the hardware. Other thoughts? Yeah. AUDIENCE MEMBER: They represent numbers differently? DAVID J MALAN: So they-- Right. They represent numbers differently. Inaccurately, apparently. So that is-- Well, differently from what? Or from whom? From us? AUDIENCE MEMBER: Yeah. They don't use the decimal system to [INAUDIBLE]. DAVID J MALAN: OK. So in a sense, they don't use the decimal system. Underneath the hood, everything is, indeed, binary, and that's related, in fact. But it's even a simpler explanation than that. Yeah. >> AUDIENCE MEMBER: They only have so many bits. They can only store up to a certain extent, like with the decimals. >> DAVID J MALAN: Yeah. This is what's really getting at it. It turns out that computers typically, will always only use a finite amount of information to represent something. Right? After all, we only have a fixed amount of hard disk space. We only have a fixed amount of RAM, or computer memory. And if you only have a fixed amount of something, surely you can't actually count up to infinity or any number you want. You, kind of, have to pick and choose what range of values you're going to support. >> And so for instance, a week or two ago, when we talked about ASCII, and we talked about 8 bits, or a byte, so to speak, the biggest number we could represent with 8 bits was what? 255. And we can represent 256 total values but not if we spend one of them on 0-- Well, but if we spend one of them on 0, then the biggest number is, of course, 255. >> So it turns out, that this is the case too, in this context. We started talking, last time, about floating-point numbers, which are different from integers, in that they have a decimal point and, hopefully, some numbers after that, but they are too. A computer is only typically going to use 32 bits, maybe 64 bits to represent a floating point value. So even though we've grown up learning mathematics and knowing that you can absolutely have an infinite number of numbers after the decimal point, not if you have finite memory. You, kind of, have to round, or pick and choose which numbers you're going to represent. And so you can think of, in a sense, this being the computer's closest approximation to the value 1/10 that it can get, with just 32 or so bits. And it's not just 1/10. >> For instance, if I change this to 1/3, which we think is even simpler. So 1 divided by 3. Let me save the file. Let me go ahead and recompile it. And let me re-run it. Here too, apparently, 1/3 is not 0.3 and then an infinite number of 3's thereafter. You have this imprecision at the end. So we humans are correct, and what you learned is, in fact, correct, but we're bumping up against some limitations. >> And what I thought we'd do today, is begin by looking at, frankly, the tragic consequences of this sometimes, when mankind does not quite implement for this reality and these limitations. And we'll see a series of vignettes from the History Channel that takes a look at how things have gone wrong. It's about 8 minutes long, and we'll come back after this and take a look at exactly what else can go wrong. If we could dim the lights-- >> [VIDEO PLAYBACK] >> -Computers, we've all come to accept the often frustrating problems that go with them. Bugs, viruses, and software glitches are small prices to pay for the convenience. But in high-tech and high-speed military and space program applications, the smallest problem can be magnified into disaster. On June 4, 1996, scientists prepared to launch an unmanned Ariane 5 rocket. It was carrying scientific satellites designed to establish precisely how the Earth's magnetic field interacts with solar winds. The rocket was built for the European Space Agency and lifted off from its facility on the coast of French Guiana. >> -And about 30 seconds into the flight, they first noticed something was going wrong. That the nozzles were swiveling in a way they really shouldn't. Around 40 seconds into the flight, clearly the vehicle was in trouble, and that's when they made the decision to destroy it. A Range Safety Officer with tremendous guts pressed the button, blew up the rocket before it could become a hazard to public safety. >> -This was the maiden voyage of the Ariane 5, and its destruction took place because of a flaw embedded in the rocket's software. -The problem on the Ariane was that there was a number that required 64 bits to express, and he wanted to convert it to a 16 bit number. They assumed that the number was never going to be very big, that most of those digits in the 64 bit number were 0's. They were wrong. >> -The inability of one software program to accept the kind of number generated by another was at the root of the failure. Software development had become a very costly part of new technology. The Ariane 4 rocket had been very successful, so much of the software created for it was also used on the Ariane 5. >> -The basic problem was that the Ariane 5 was faster, accelerated faster. And the software hadn't accounted for that. >> -The destruction of the rocket was a huge financial disaster, all due to a minute software error. But this wasn't the first time data conversion problems had plagued modern rocket technology. >> -In 1991, with the start of the First Gulf War, the Patriot missile experienced a similar kind of number conversion problem. As a result, 28 people, 28 American soldiers were killed and about 100 others wounded, when the Patriot, which was supposed to protect against incoming scuds, failed to fire a missile. >> -When Iraq invaded Kuwait and America launched Desert Storm in early 1991, Patriot missile batteries were deployed to protect Saudi Arabia and Israel from Iraqi Scud missile attacks. The Patriot is a US, medium-range surface-to-air system manufactured by the Raytheon company. >> -The size of the Patriot interceptor, itself, is about, roughly 20 feet long. And it weighs about 2000 pounds. And it carries a warhead of about-- I think it's roughly 150 pounds. And the warhead, itself, is a high explosive, which has fragments around it. The casing of the warhead is designed to act like buckshot. >> -The missiles are carried, 4 per container, and are transported by a semi-trailer. >> -The Patriot anti missile system goes back at least 20 years now. It was originally designed as an air defense missile to shoot down enemy airplanes. In the First Gulf War, when that war came along, the Army wanted to use it to shoot down scuds, not airplanes. The Iraqi Force was not so much of a problem, but the Army was worried about scuds. And so they tried to upgrade the Patriot. >> -Intercepting an enemy missile traveling at mach five was going to be challenging enough. But when the Patriot was rushed into service, the Army was not aware of an Iraqi modification that made their scuds nearly impossible to hit. >> -What happened, is the scuds that were coming in, were unstable, they were wobbling. The reason for this, was the Iraqis, in order to get 600 kilometers out of a 300 kilometer range missile, took weight out of the front warhead and made the warhead lighter. So now, the Patriot's trying to come at the scud, and most of the time, the overwhelming majority of the time, it would just fly by the scud. >> -Once the Patriot system operators realized the Patriot missed its target, they detonated the Patriot's warhead, to avoid possible casualties if it was allowed to fall to the ground. >> -That was what most people saw, those big fireballs in the sky, and misunderstood as intercepts of Scud warheads. Although in the night skies, Patriots appeared to be successfully destroying scuds, at Dhahran, there could be no mistake about its performance. There, the Patriot's radar system lost track of an incoming Scud, and never launched, due to a software flaw. It was the Israelis who first discovered that the longer the system was on, the greater the time discrepancy became, due to a clock embedded in the system's computer. >> -About 2 weeks before the tragedy in Dhahran, the Israelis reported to the Defense Department, that the system was losing time, that after about 8 hours of running, they noticed that the system was becoming noticeably less accurate. The Defense Department responded by telling all of the Patriot batteries to not leave the systems on for a long time. They never said what a long time was. 8 hours? 10 hours? 1000 hours? Nobody knew. -The Patriot battery stationed at the barracks at Dhahran, and its flawed internal clock, had been on over 100 hours on the night of February 25. >> -It tracked time to an accuracy of about 1/10 of a second. Now 1/10 of a second is an interesting number because it can't be expressed in binary, exactly. Which means, it can't be expressed, exactly, in any modern digital computer. It's hard to believe, but use this as an example. Let's take the number, 1/3. 1/3 cannot be expressed in decimal, exactly. 1/3 is 0.333 going on for infinity. There's no way to do that, with absolute accuracy, in decimal. That's exactly the same kind of problem that happened in the Patriot. The longer the system ran, the worse the time error became. >> -After 100 hours of operation, the error in time was only about 1/3 of a second. But in terms of targeting a missile traveling at mach 5, it resulted in a tracking error of over 600 meters. It would be a fatal error for the soldiers at Dhahran. >> -What happened, is a Scud launch was detected by early-warning satellites, and they new that the Scud was coming in their general direction. They didn't know where it was coming. >> -It was now up to the radar component of the Patriot system, defending Dhahran, to locate and keep track of the incoming enemy missile. >> -The radar was very smart. It would actually track the position of the Scud and then predict where it probably would be, the next time, the radar sent a pulse out. That was called the range gate. >> -Then, once the Patriot decides enough time has passed to go back and check the next location for this detected object, it goes back. So when it went back to the wrong place, it then sees no object and it decides that there was no object, it was a false detection, and drops the track. The incoming Scud disappeared from the radar screen, and seconds later, it slammed into the barracks. The Scud killed 28 and was the last one fired during the First Gulf War. Tragically, the updated software arrived at Dhahran the following day. The software flaw had been fixed, closing one chapter in the troubled history of the Patriot missile. >> [END PLAYBACK] >> DAVID J MALAN: So we'll take a look at some similar limitations in just a bit. But first, let's transition to a few FYI's. So one-- this weekend, there will be super sections, which are meant to supplant regularly scheduled sections, which will start up a week hence. Take a look at the CS50's website for more information on those. They will also be filmed and streamed live for those unable to attend.