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DOUG LLYOYD: So hexadecimal numbers,
as if we needed another base number

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scheme right?

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Well, most Western cultures,
as you probably are familiar,

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use the decimal system-- base
10, to represent numeric data.

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We have the digits 0,
1, 2, 3, 5, 6, 7,8,9.

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And if we need to represent
values higher than nine,

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we can combine those digits
using the notion of place value.

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So for 10, we have a 1
digit followed by a 0 digit

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and we intuitively understand
that what we're doing

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there is we're multiplying
the first 1 by 10,

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and then adding 0 for a total of 10.

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Computers do something pretty
similar, as you're probably familiar,

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with the binary system-- base 2.

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The difference there being
that there are only 2 digits

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to work with-- 0 and 1.

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And so our place values,
instead of being one,

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ten, hundred, thousand, as they
would be in the decimal system,

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are one, two, four, eight, and so on.

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Here's the thing though,
those 0's and 1's, especially

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if we're being computer scientists
and we're doing a lot of programming

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or working with computers, were going
to be seeing a lot of binary numbers.

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And those 0's and 1's in large chains
can be very difficult to parse.

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We can't just look at a string of
0's and 1's and necessarily know

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exactly what it is.

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But it's still useful to be able
express data in the same way

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that a computer does.

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We have this notion of the
hexadecimal system, which is

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base 16, instead of base 10 or base 2.

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Which means that we have 16 digits
to work with instead of 10 or 2.

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And it's a much more
concise way to express

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binary information on a computer system,
it's much more human understandable.

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So we have the digits
0 through 9, and then

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we also have these extra six digits-- a,
b, c, d, e, and f, which represent 10,

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our notion of 10, 11, 12,
13, 14 and 15, in decimal.

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Sometimes, by the way, you'll also
see these a through f's as capital A

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through F, which is the
way I tend to do it.

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It's just my preferred
style, but either is fine,

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they both represent pretty
much the same thing.

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>> So why is hexadecimal cool?

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Why do we need to use this
other additional base?

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We already have 2 and
10, why do we need 16?

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Well 16 is a power of 2, and so
each hexadecimal digit, 0 through f,

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corresponds to a unique
ordering, or unique arrangement

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of 4 binary digits, 4 bits.

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And so in that sense, we can express
very long, complex, binary numbers

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in hexadecimal in a
much more concise way,

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without losing information or having to
do particularly cumbersome conversions

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on those numbers.

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>> So, as I just said,
each hexadecimal digit

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corresponds to a unique
arrangement of 4 binary digits.

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So the binary string 0000
corresponds to hexadecimal digit 0.

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0110 corresponds to hexadecimal digit 6.

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And 1111 corresponds
to hexadecimal digit f.

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If you're looking at
this chart, particularly

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if you're looking at the
left side of the chart,

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you can already see there's a
bit of an ambiguity problem here.

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Decimal 0 is pretty much
indistinguishable from hexadecimal 0,

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other than the fact that it's under
a column that says hexadecimal.

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>> But we probably won't always
have that column there.

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Generally when we are expressing
numbers into hexadecimal notation

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to clearly distinguish
them from decimal notation,

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we usually prefix them
with the prefix 0x.

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0x means nothing in reality,
it's just a clue to us as humans

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that what we're about to see,
or about to start parsing,

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is a hexadecimal number.

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Obviously for the higher digits a, b,
c, d, and f, which correspond to 10-15

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it's pretty unambiguous that's
that's a hexadecimal number.

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And in fact, any hexadecimal
number that has letters in it,

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is probably pretty obvious
as a hexadecimal number.

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But, still, for the
sake of clarity, it's

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always a good idea to
prefix every time you

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refer to a digit as a hexadecimal
number by prefixing a 0x.

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>> So, binary, as we
said, has place values.

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There's the ones place, a twos place,
a fours place, and an eights place.

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And decimal also has place values, the
ones, tens, hundreds, and thousands

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that we all may recall
from grade school.

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And hexadecimal is no
exception here, really.

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It also has place values but instead
of being powers of 2 or powers of 10,

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they're powers of 16.

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>> So we see a number like this we
pretty clearly know it's 397, right?

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Well if we see a number like this,
we know this isn't 397 anymore.

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This is the hexadecimal
number three-nine-seven.

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It's not 397, it means
something different,

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because we're using powers of 16 as all
of our place values instead of powers

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of 10.

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In fact, the place values here would
be the ones place, the sixteens place,

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and the two-hundred-fifty-sixes place,
which correspond to our idea of a ones

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place, tens place, and a hundreds
place, if the number was 397.

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But since it's 0x 397, we have
a ones place, sixteens place,

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and a two-hundred-fifty-sixes place.

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Or, a 16 to the 0 place, which is 1.

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A 16 to the first power place, 16.

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A 16 squared place, 256, and
so on, and so on, and so on.

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So this number is really 3 times 16
squared, plus 9 times 16, plus 7.

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I didn't do the math here, but it's not
397, it's much, much larger than that.

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>> Similarly, we could have 0x adc,
well that's a times 16 squared.

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Or if we translate that to our notion
of decimal numbers, that's 10 times

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16 squared, plus d times
16, or plus 13 times 16.

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And don't worry if you haven't memorized
that d is 13, or anything like that,

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there's not too many
of these letter digits

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and it'll become
intuitive pretty quickly.

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So again this is 10 times 16 squared,
plus 13 times 16, plus 12 times 1.

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So 0x adc.

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>> So, as I said, every
group of 4 binary digits

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corresponds to a single
hexadecimal digit,

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and so it's actually really
easy to change back and forth

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between hex and binary.

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If you have this long string of
binary digits, all you need to do

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is start grouping them right
to left as groups of 4.

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And then you can consolidate
them into hexadecimal numbers,

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severely limiting the number of
digits you have to process mentally.

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Instead of 32 0's and 1's,
as we'll see in a second,

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you might be able to get it down
to just 8 hexadecimal digits, a lot

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more concise.

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>> The charts a few slides back will
help you to figure out this mapping,

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although, again you'll
memorize it pretty quickly.

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We'll go through an example right now.

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So if we have a number like this,
this really large binary number,

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or what appears to be
a large binary number.

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And the reason I say that, it's
just so-- it's a behemoth, right?

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There's so many 0's and 1's there.

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But we probably don't
really have a sense of what

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the magnitude of this number really is.

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We don't have any idea what it
would correspond to a decimal.

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And in fact we won't even see what it
corresponds to in decimal right now.

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We might be able to
express this in a way that

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would give us some more information
about just how big this number is.

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>> So let's go to that conversion process.

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The first thing we need
to do is we want to group

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these digits out into groups
of 4, starting from the right

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and working to the left.

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There happen to be 32 digits
here, which means we have

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a nice clean break of 8 groups of 4.

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Remember that each group of
4 here, uniquely corresponds

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to a hexadecimal digit.

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So we'll start again building our
number from the right, and working left.

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Well what's 1101?

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Well we do the math out in our head,
we have 1 in the eights place, a 1

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in the fours place, a 0 in the twos
place, and a 1 in the ones place.

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That's 8 plus 4 plus 1,
which we would know as 13.

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But we probably wouldn't write 13 out,
because we're working with hexadecimal.

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We need to convert it to the hexadecimal
equivalent of 13, which is d.

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>> 0011, well that's a 0 in the
eights place, a 0 in fours place,

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a 1 in the twos place,
and a 1 in the ones place.

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That's 3.

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I mean keep doing this
again, we have here 9.

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And then 11, but that's b, recall.

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2, 10-- or a-- 6, and 4.

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And so that very large string
of 0's and 1's of the top

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is more concisely expressed
in hexadecimal as 0x 46a2b93d.

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>> Well, OK, we've learned a new
cool skill, what's the point?

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We might not use this all the
time, as we're going to soon see,

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we use hexadecimal quite
a lot as programmers.

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Not necessarily for the
purpose of doing math with it,

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but because a lot of times
memory addresses in our system

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are represented in hexadecimal.

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It's a really concise way to express
otherwise cumbersome, binary numbers.

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And so, again, you may
not-- you're probably

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not going to do any math
with it, you are not

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going to be multiplying
hexadecimal numbers together,

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or doing anything weird like that.

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But it is a useful skill to have
so you can express and understand

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memory addresses, and other
ways of using data in C.

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>> I'm Doug Lloyd, this is CS50.

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