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Hexadecimal number system | Applying mathematical reasoning | Pre-Algebra | Khan Academy - YouTube
Channel: Khan Academy
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- [Voiceover] We're all
familiar with the base 10
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number system, were
often called the decimal
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number system, where we have 10 digits
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Zero, one, two, three, four,
five, six, seven, eight, nine.
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Now, we started to see that we can
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have alternate number system.
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We can have a base two number system,
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or it's the binary number system,
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where instead of 10 digits
you only have two digits.
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Each place, instead of being a power
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of ten is going to be a power of two.
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Now you can imagine that
we can keep extending this.
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We can extend to base three,
four, five, six, seven,
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eight, nine, or we could even go above 10.
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What I want to show you
in this video is a fairly,
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heavily used number system
that is larger than,
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or that has more digits than
base 10, and that base is 16.
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Base 16, often called the hexadecimal.
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Hexadecimal number system.
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As you can imagine, instead of only
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having 10 digits, it is going to have 16.
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What are those digits going to be?
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As we'll see, instead of
the place is being powers
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of two or powers of ten,
there will be powers of 16.
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Let's see, we can reuse
the existing 10 digits
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from the decimal number system.
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We can reuse zero, one, two, three,
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four, five, six, seven, eight, nine,
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but then we're going to need
to have six more digits.
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The convention is to use
the first six letters.
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A, B, C, D, E, and F.
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You might say this is crazy.
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These are letters, not numbers,
but remember these are just
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arbitrary squiggles of
ink on a piece of paper.
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These are just arbitrary symbols
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that we're grown to associate with things.
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You're grown to associate
this symbol right over here
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with eight thing, with
the word eight which you
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associate with when you
see that many objects.
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If you're thinking in hexadecimal,
this isn't the letter A
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that makes you want to
say "ah", or the letter B
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that makes you want to say "bababababa".
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This is, literally, this represents
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if you had 10 things laying around.
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You would say, "I have
A things over there."
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If you have 11, you'd say,
"I have B things over there."
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12, C things.
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13, instead of saying,
"I have 13 things there",
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"I have D things there."
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Instead of saying, "I have 14",
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you could say, "I can
have E things there."
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Instead of saying, "I have 15",
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you could say, "I have F things there."
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Now, how does that help?
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Well, let's see if we can represent
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the same number 231, or 231 in decimal.
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If we can represent that
same number in hexadecimal.
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What I'll do is I'll give
you what the number is,
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and then I'll show you how we convert it.
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I'll show you the place value,
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and I'll show you how we convert it.
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231 in hexadecimal.
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231 in hexadecimal is the number E seven.
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E seven.
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Once again, you're
like, "This looks crazy.
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"This is like I'm playing
like battleship or something."
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What's E seven?
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This is a number and I would say yes.
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This is a number.
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Now remember, base 16.
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What are these place values represent?
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This first place represents
16 to the zero power
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or still represents the ones place.
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This is the ones place.
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This is seven ones.
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Now, what is this place here represents?
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Well, in base 10, that
was 10 to the first power.
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In base two, that was
two to the first power.
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On base 16, this is going to be,
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I'll leave those there,
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in base 16, this is going
to be 16 to the first power.
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This is literally, well let me write out
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the word, this is literally sixteens.
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This is E sixteens plus seven ones.
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Let me write that down.
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This is E sixteens plus seven ones.
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That's what this number represents.
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Now, if we want to start rewriting this
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or re-conceptualizing it in our decimal
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number system, what is E sixteens?
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Well, the E if we think
in decimal, E is 14.
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E is 14.
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This is really, we can
really think of this
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if you want to think them decimals.
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This is 14 sixteens.
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It's 14 sixteens.
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Well, that's just the
same thing as 14 times 16.
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14 times 16 is equal to 224.
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Maybe I should do that in same color.
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This thing right over
here is going to be 224.
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14 sixteens, 14 times 16
is 224 plus seven ones.
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Well, 224 plus 7 is
going to be give you 231.
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Hopefully, you can appreciate it.
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You can represent the same quantity
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in any of these different number systems.
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In any number that you
can represent in decimal,
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you can also represent
that number in binary,
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or in hexadecimal, or in
base three, or in base 60,
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or in base 31, whatever you want to do.
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You might have noticed the pattern.
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The more symbols that we have,
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so in base 16, you have 16 symbols,
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the less place values we need
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to represent the same quantity.
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One way to think about it is each
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of the places are
containing more information.
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This is one of 16 characters.
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While this over here is
only one of two characters.
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This is one of ten characters.
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The more symbols that
you have, the more digits
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that you could put in each
place, the less places
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that you need to represent
a given quantity.
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Another way to think about it
is when you have a high base,
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like base 16, as you take
powers of 16, the next
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place right over here would be 16 squared,
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which, of course, is two
hundred and, wait a minute, 256.
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You're clearly going to be able
to represent bigger numbers
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faster, I guess you could
say, or with less digits.
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It's just an interesting thing to observe.
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But hopefully, you're
going to kick out of,
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as much of a kick out of base 16
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as I do, and it's actually useful.
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This actually is used if
you look at most web pages.
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If you look at the actual code for there,
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or I guess you could say the
formatting line, the HTML
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for the webpage, when they specify colors,
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they tend to specify in hexadecimal.
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That's because they're specifying
the colors, the intensity
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of the red, the green, or the
blue, between zero and 255.
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Two digits of hexadecimal
are perfect for that,
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because if you think
about it, what is F F?
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What would this be if you
rewrite it in the decimal number
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system, and I encourage you
after this video is done,
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I encourage you to do that to
figure that out on your own.
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If you really want to do something fun,
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let me give you another one.
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Try to figure out what A F three is.
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Again, this isn't very specialized.
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I just wanted to give you another
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interesting thing to work on.
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