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made this actual software, decided that the maximum number of coins
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the user can accumulate is, indeed, 4 billion,
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because they chose in their code to use not a long long, apparently,
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but just an integer, an unsigned integer, only a positive integer, whose
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max value is roughly that.
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Well, here's another funny one.
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So in the game Civilization, which some of you might be familiar, with
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it turns out that years ago there was a bug in this game whereby
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if you played the role of Gandhi in the game,
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instead of him being very pacifist, instead was incredibly, incredibly
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aggressive, in some circumstances.
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In particular, the way that Civilization works is that if you, the player,
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adopt democracy, your aggressiveness score gets
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decremented by two, so minus minus, and then minus minus.
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>> So you subtract 2 from your actual iterating.
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Unfortunately, if your iterating is initially 1, and you subtract 2 from it
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after adopting democracy as Gandhi here might
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have done, because he was very passive-- 1 on the scale of aggressiveness.
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But if he adopts democracy, then he goes from 1 to negative 1.
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>> Unfortunately, they were using unsigned numbers,
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which means they treated even negative numbers as though they were positive.
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And it turns out that the positive equivalent of negative 1,
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in typical computer programs, is 255.
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So if Gandhi adopts democracy, and therefore has
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his aggressiveness score decreased, it actually rolls around to 255
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and makes him the most aggressive character in the game.
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So you can Google up on this.
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And it was, indeed, an accidental programming bug,
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but that's entered quite the lore ever since.
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>> That's all fun and cute.
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More frightening is when actual real world devices, and not games,
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have these same bugs.
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In fact, just a year ago an article came out about the Boeing 787 Dreamliner.
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>> And the article at first glance reads a little arcane.
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But it said this, a software vulnerability in Boeing's
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new 787 Dreamliner jet has the potential to cause pilots
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to lose control of the aircraft, possibly
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in mid-flight, the FAA officials warned airlines recently.
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It was the determination that a model 787
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airplane that has been powered continuously for 248 days
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can lose all alternating current, AC, electrical power due to the generator
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control units, GCUs, simultaneously going into fail safe mode.
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It's kind of losing me.
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But the memo stated, OK, now I got that, the condition was caused by a software
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counter internal to the generator control
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units that will overflow after 248 days of continuous power.
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We are issuing this notice to prevent loss
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of all AC electrical power, which could result
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in loss of control of the airplane.
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>> So, literally, there is some integer, or some equivalent data type,
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being used in software in an actual airplane
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that if you keep your airplane on long enough, which apparently
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can be the case if you're just running them constantly and never unplugging
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your airplane, it seems, or letting its batteries die,
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will eventually count up, and up, and up, and up, and up, and up.
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>> And, by nature, a finite amount of memory
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will overflow, rolling back to zero or some negative value,
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a side effect of which is the frighteningly real reality
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that the plane might need to be rebooted, effectively,
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or might fall, worse, as it flies.
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So these kinds of issues are still with us,
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even-- this was a 2015 article, all the more frightening
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when you don't necessarily understand, appreciate, or anticipate
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those kinds of errors.
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>> So it turns out there's one other bad thing about data representation.
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It turns out that even floats are kind of flawed, because floats, too,
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I proposed are 32 bits, or maybe 64 if you use a double.
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But that's still finite.
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>> And the catch is that if you can put an infinite number of numbers
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after the decimal point, there is no way you
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can represent all the possible numbers that we were taught
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in grade school can exist in the world.
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A computer, essentially, has to choose a subset of those numbers
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to represent accurately.
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>> Now, the computer can round maybe a little bit,
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and can allow you to roughly store any number you might possibly want.
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But just intuitively, if you have a finite number of bits,
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you can only permute them in so many finite ways.
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So you can't possibly use a finite number
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of permutation of bits, patterns of zeros and ones,
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to represent an infinite number of numbers,
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which suggests that computers might very well be lying to us sometimes.
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>> In fact, let's do this.
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Let me go back into CS50 IDE.
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Let me go ahead and create a little program
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called Imprecision, to show that computers are, indeed, imprecise.
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>> And let me go ahead and start with some of that code from before,
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and now just do the following.
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Let me go ahead and do printf, percent f, backslash n, 1 divided by 10.
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In other words, let's dive in deeper to 1/10, like 1 and divided by 10.
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Surely, a computer can represent 1/10.
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>> So let's go ahead and make imprecision.
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Let's see.
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Format specifies type double.
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But the argument has type int.
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What's going on?
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>> Oh, interesting, so it's a lesson learned from before.
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I'm saying, hey, computer show me a float with percent f.
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But I'm giving it 2 ints.
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So it turns out, I can fix this in a couple of ways.
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