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[CS61C FA20] Lecture 02.3 - Number Representation: Overflow, Sign and Magnitude, One's Complement — Transcript

by CS 61C Departmental · 2,232 words · 368 segments · language en · Watch on YouTube

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0:00and welcome back now let's talk about
0:03number representations
0:06what can we do with representations of
0:07numbers well we can
0:09do everything you can do you understand
0:11in decimal you can add them subtract the
0:13multiply divide compare them
0:15for example and by the way this is
0:16representation in binary
0:18how could we represent
0:22how could we do some addition here's 10
0:251 0 1
0:250 is 10. so 10 10 is 10 remember that
0:28memorize that guy
0:29and let's add that together to seven
0:30seven is one one one well let's do it
0:33together let's actually do it as we did
0:35you're back in first grade zero plus one
0:38one by the way the maximum value in any
0:41of those slots is a one in binary by the
0:43way this is these are all base two
0:45okay this is this is the base two and
0:46base two zero plus one one one plus one
0:49ready it's two which means zero
0:53carry the one because it's kind of the
0:54one bubbled over two
0:56and binary is 0. 1 plus 0 plus 1 is 2
0:590 plus 1 carries 1 plus 1 there again 2
1:030 and the 1 carries to here and you get
1:061 0 0 1. so
1:0910 plus 7 is 17. and this ladies and
1:13gentlemen is 17
1:14because that's 16 plus a 1.
1:18now if i only had
1:21four bits to do this i wouldn't have had
1:23enough room to store that number we're
1:24going to see that a little later but
1:26that's an important piece of that so
1:27anyway the point is i just did a simple
1:28binary addition very very easy
1:31subtraction just as you did before and
1:34how do you tell if a number is bigger if
1:35x
1:36some number of binary numbers bigger
1:37than the other one think about that
1:40is it the total length what if they're
1:42the same length let's think about how
1:43you test
1:44if one number is bigger than the other
1:46with this representation
1:50remember binary bit patterns are simply
1:52representations of numbers
1:54numbers are a theoretical concept the
1:56value pi
1:57theoretically theoretical concept okay
2:00the
2:01numerals are the representations of that
2:04so
2:04123 is a nice example is
2:08a number it's in my head one two three
2:10the way we
2:11write it down so 123 this kind of
2:14concept of 123 things
2:16that could be tally marks that can be
2:18anything that's the number the concept
2:19is a number
2:20the representation is a numeral
2:24digits are always digits of numerals you
2:26don't have digits of numbers if you ever
2:28say that's a digit of a number that's
2:29wrong
2:30it's a digital or numeral numerals are
2:32the written representation of that
2:33theoretical concept of a number so the
2:35idea of abstraction
2:36okay concept representation of that of
2:39that beautiful idea
2:41numbers really have an infinite number
2:42of digits we know that numerals
2:45yeah i just did it that's funny right
2:46see how i did that numerals really have
2:48an infinite number of digits
2:50we always normally the beginning
2:53always is uh zeros we'll talk about how
2:55if the beginning is ones what that means
2:57and other things except for a few of the
2:58right most digits so mostly the
3:00actions in the lower end up because it's
3:01infinite so it'll be lots of lots of
3:03zeros
3:04the leading we just don't normally show
3:05the leading digits there
3:08if the result cannot be stored in the
3:11number of bits we have to allocate that
3:13we call that overflow so here's an
3:16unsigned remember that what we've seen
3:18so far is just always positive numbers
3:20non-negative numbers i'll say so if i go
3:22to zero
3:24and i'm counting i'm gonna use five bits
3:26so five bits i can think
3:27five bits is zero through 31. so
3:30the smallest number is zero the biggest
3:33number is all ones as my 31
3:35if i add one more to that it's going to
3:37wrap it and only have five bits
3:39it's going to wrap around to zero we
3:42call this
3:43overflow okay you haven't heard that
3:45before
3:46if i am subtracting numbers and i'm at
3:49two
3:50here my two i'm at one never zero and i
3:53subtract one more
3:54it'll actually wrap around this way too
3:56it'll go back to all ones
3:57that is not underflow that is
4:01also overflow people often call it
4:03negative overflow
4:04it's not underflow we'll talk about
4:06underflow is in about three lectures
4:09so that's if it's too big okay
4:13think how do you represent negative
4:15numbers though
4:17so so far all we've seen is unsigned
4:19numbers and by the way if you've ever
4:21seen
4:21in c this is called an unsigned integer
4:24or
4:25u for unsigned int this end is going to
4:28be a number 8
4:2916 32 which tells you actually the width
4:31of it because
4:32this is ambiguous how many bits is an
4:35unsigned int it's ambiguous
4:36so we started in c99 which they've
4:39introduced in subsequent versions of c
4:41now up to c18
4:43we call them you int with a number there
4:46underscore t that's the type for that so
4:49you should use that
4:49it's called int types.h we'll get that
4:51when we talk about c later
4:53so far we know unsigned numbers starts
4:55at zero and goes up
4:57well how do we do a negative number
4:59there are numbers on the number line
5:00there are numbers over here
5:02uh be nice be able to represent those at
5:04some point um of course if you owed
5:06money maybe no
5:07don't represent those at all that'd be
5:08good not to represent those because i
5:10owe money so the way one way to think
5:12about it is let's just borrow some let's
5:14actually borrow some space from it so i
5:16got
5:17five bits let's borrow a bit
5:20let me tell you my story so there's a
5:23story i need to tell you here just to
5:24put a context to this
5:27it was a rich king kingdom and he had
5:30three wise people three wise thinkers
5:34and he said to the wise thinkers i want
5:35you to teach me all there is to know
5:37about
5:38economics and they went away and
5:40scurried away and they came back a year
5:41later
5:41you know five tomes they had written
5:43this is all we understand of economics
5:45we've searched the world
5:46this is the world's knowledge of
5:47economics king says no time no time
5:50make it shorter go back and make it
5:51shorter they came back we created this
5:54single book no time sends them away
5:57another year comes back
5:58we have this pulp fiction novel to
6:00explain economics no no time to they
6:02come back
6:02next year we created this comic book
6:04graphic novel
6:0510 pages nope no time left them we
6:07cringle a single sheet of paper
6:09executive summary no time no time comes
6:12back
6:13so they get grayer and older and the
6:15beards and the hair
6:17they come back he says sire king we have
6:21come up with the theory of economics
6:24in four words and how it became english
6:28says yes go ahead i have time for that
6:31ain't no free lunch
6:34what it means is you can't get something
6:35for nothing if you want to be able to
6:37now represent negative numbers
6:38you got to give something you got to
6:40lose some numbers that you used to
6:41represent now you won't be able to
6:43represent them anymore so
6:44if we borrow a bit now we can't go as
6:46high and positive
6:48but now we can do negatives so here's
6:49the idea the leftmost bit is going to be
6:52my sign
6:53okay so you look at that leftmost bit
6:55that's positive negative zero is p by
6:57the way it'd be really nice if we in all
6:59these
6:59in all these representations all zeros
7:02were zero
7:02okay so all zero is this that way
7:06the when the first when the first bit is
7:08zero nothing changes
7:10now the biggest number i can represent
7:11is 15.
7:13b4 was 31 before but this
7:17allows me to represent the negative
7:18space this is now
7:20uh negative zero negative
7:25zero negative
7:28one etc so now i'm going to show you
7:32this way of thinking about this
7:34this is a
7:38binary odometer back in your olden days
7:41of
7:41driving an old car that had an analog
7:44odometer the mileage reading
7:47as you go from all zeros you get brand
7:48new car all zeroes and you start to
7:50drive it it's all zeros let's say it's
7:51binary then all zero's in one then all
7:53zero's one zero then all zeros one zero
7:55zero et cetera
7:56by the way i want you to be able to
7:57think about counting
7:59in binary see this spinal odometer
8:03your hand can be this binary odometer
8:05watch this
8:06here we go okay all zeros
8:09repeat after me try this at home okay
8:11hand this is zeros
8:12this is my as you see it this is the uh
8:15one spot
8:17two spot four spot eight spot sixteenth
8:19spot let's count
8:21zero one two
8:24three four i have to block it see
8:27otherwise i'm on
8:28youtube five okay that's five
8:32six seven is harder
8:36eight nine okay let's see eight
8:40eight this is harder because nine okay
8:42go it
8:43ten eleven twelve
8:47twelve thirteen fourteen
8:51fourteen 15 cool right
8:5516 17 yos 17 dude
8:59okay et cetera et cetera et cetera 31.
9:02isn't that neat
9:02so what you just saw in this binary
9:03diameter you can do with your hand and i
9:05encourage you to memorize how to do that
9:06zero one two three that's pretty cool do
9:08that as fast as you can see
9:10see how we can do it and maybe you're
9:11more dexterous than i am i can't get the
9:13eights up there
9:14so well so
9:17notice that the odometer is good it
9:18starts at the beginning and goes up
9:20continuous
9:20nice as i as the binary goes up the
9:23number gets bigger
9:23kind of makes sense however watch what
9:26happens to the binary odometer with sine
9:28magnitude
9:30okay that's good oh wait now as odometer
9:33is going up i'm counting
9:34and going away from zero the wrong way
9:36i'm going positive positive positive and
9:38then i go negative
9:40that's hard we don't like that so much
9:42passive we went always went the same way
9:43always went bigger so
9:45think about that as one of the problems
9:47with sine magnitude
9:48so another problem so that was one
9:51problem is this
9:52when a weird direction the second is to
9:55build
9:55circuitry around this is actually hard
9:57because you got to deal with the fact
9:58that it's positive or negative and what
9:59if there one's a positive one's negative
10:01hard okay annoying you got two zeros
10:04that's annoying uh does that mean for
10:07programming is it equal to one of those
10:08you have to check for two patterns of a
10:09zero now that's
10:10that's annoying i'd mention the binary
10:12odometer does this weird thing
10:13um so actually we don't usually use sine
10:15magnitude except for in rare cases often
10:17in signal processing
10:19let's try it again how about if we just
10:21flip the bits let's just try it's
10:23another
10:23attempt these are all people sitting in
10:25a you know with a white board thinking
10:27let's try flipping all the bits when
10:29it's negative so here let's try it again
10:31so seven is two zeros and three ones
10:34okay what would negative seven be flip
10:38the bits
10:39now one one zero zero zero there's
10:42minus seven i get it so in a way it
10:45looks
10:46like it actually looks like that leading
10:48zero look at this got the same kind of
10:49nice thing is
10:50positive numbers have leading zeros
10:52here's the leading zeros
10:54negatives have leading ones but it's not
10:55psi magnitude
10:58what's minus zero zero zero well all
11:01ones so i still have this problem that
11:03there's two zeros we've we see that
11:05already um
11:07how many positive numbers okay how many
11:09ask yourself how many positive numbers
11:10and how many negative numbers you might
11:11have
11:12in n bits okay
11:15so but the problem you see how the
11:17problem is let's let's watch let's
11:19look at the odometer just to see ready
11:20here's the odometer you go up
11:22and then if you notice
11:26all ones is at the far side on the left
11:29so actually as the dominant goes up
11:30after zero zero zero one one one one and
11:32it wraps to the next number
11:33it's kind of like back in the day of you
11:35know you have all zero and all nines it
11:37wraps to
11:37one in all zeros in decimal
11:40we odometer is now fixed but this is
11:43still overlap so you're thinking well
11:44couldn't we do mostly this but just take
11:47this bottom here's what you want to do
11:48right
11:49take this bottom guy and just shift it
11:51left
11:52then you wouldn't have the overlap and
11:54it would all work out and in fact
11:56that's what we do that's the so the
11:58shortcoming
12:00arithmetic is complicated still got the
12:02two zeros although we got the nice
12:03odometer to do the right thing okay so
12:05in historically it was used for a while
12:07and then eventually abandoned because
12:09this next one is better
12:12we'll teach you what the next one is on
12:13the next lecture
12:15we'll see you there

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