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Okay, in this segment weíre going to talk about how to convert a base-10
to base-2, a number from base-10 to base-2.
So Iím going to show you through example, Iím not going to give you the proof for it.
It can be proven, the method which Iím going to show you, youíll be able to show
by proof why this works to convert a base-10 number to a base-2 number, but
the only thing which Iím going to concentrate on in this segment is the method itself, how to convert a
base-10 number to base-2 number.|So for example, Iím going to take the same number,
11.1875, which I had previously, and this is in base-10.
How do I convert this number into base-2?|What is the
base-2 equivalent?|So this is the question mark right here, radix point,
another question mark, base-2.|If Iíve got to find out what should I fill in
for this question mark, and what should I fill in for this question mark.
So letís go ahead and start with the integer part, which is the part before the
decimal point in this case, or what we call the radix point for any base.
So letís go ahead and see what the equivalent of 11 is.|So Iím going to draw a table here, thatís the best way
to go ahead and convert these numbers in decimal notation.|So Iím going to say,
hey, Iím going to put the quotient here,
and Iím going to put the remainder here, and Iíll tell you what that means.
So Iím going to take this number 11, and what Iím going to do is, Iím going to divide this by 2,
and the reason why Iím dividing by 2 is thatís what base.|So Iím going to take the 11 number,
divide it by 2, what do I get as the quotient?|I get quotient is 5, and what is the remainder?
The remainder is 1.|Now I take the quotient which I just got, which was
5, thatís this number right here, I divide that by 2, and I get 2
as my quotient and again 1 as the remainder.
Now I take the number 2 and I again divide it by 2, so youíre dividing by the
same number here, the quotient is 1, and the remainder is
0, of course.|Now Iím left with a quotient of 1, I get
1 divided by 2.|If I do 1 divided by 2, I get a quotient of 0,
and a remainder of 1.|And thatís where I stop, as soon as
the quotient becomes 0, when Iím looking at the integer part, I stop the whole
process.|Now, what I will have is that this one is a0,
this one is a1, this one is, I call a2, and this one I call a3, so
this is the remainder which I get in the first step, second step, third step, fourth step.
So the equivalent of 11 in the base-2 will be equal to
a3, a2, a1, and a0,
so the equivalent will be 1011
base-2.|So you do need to look at it this way, that if you are not
going to do it this way where you are going to signify a0, a1, a2, and a3,
what youíve got to understand is that the first number which goes here is the last number which you calculated, and
the next number which should go there is the second-to-last number which you calculated, and so on and so forth, so youíre going backwards.
So the equivalent of 11, this should be base-10, thatís wrong, so the
equivalent of the number of 11 in the base-10 is 1, which is this 1, then 0,
which is this 0, then this 1, which is this 1, and then this 1 is right here,
so you go backwards to look at the equivalent number.|Now if youíre going to go ahead and try to find out
what the decimal equivalent, or base-10 equivalent, of this number is, youíre going to exactly get 11
from there, so I will leave that as an exercise for you to do.|Now letís go ahead and look
at the fractional part, which is 0.1875, how do we go about finding
the equivalent of that?|So Iíve got 0.1875,
which is in base-10, and Iíve got to
find out what is it in base-2, so thatís the rest of the
number here.|So in this case, what you do is, rather than dividing by 2,
you multiply by 2.|So let me draw a table here, thatís the best
way to do these problems when you are a beginner.
So Iím going to get 0.1875,
thatís the number which I have, and I will multiply by 2, and I get
0.375.|So thatís the number which I am getting,
and then what Iíve got to look at is the number after
the decimal point, and then Iíve got to also
look at the number before
the decimal point.|So when I multiply 0.1875 by 2 I get
0.375, the number after the decimal point is still 0.375,
the number before the decimal point is just 0.|So thatís how Iíve got to extract it, so
Iíve got to extract the integer part of this number and the fractional part of this number, so the
fractional part of the number is still 0.375, and the number before the decimal point
is 0.|Now I take 0.375 and I multiply
by 2, I get 0.750.|So in this case again here, the
number which is after the decimal point is 0.750, the number before
the decimal point is still 0.|Now I go ahead and
take, again, I get 0.75 now,
multiply by 2, now I get 1.50.
So you might have been wondering, why are those numbers 0 before the decimal point, and thatís the way it turned out to be.
Now 0.75 multiplied by 2 is 1.50, so the number after the
decimal point is 0.500, and the number
before the decimal point is 1.|Now Iím left with
0.5, which is the number after the decimal point, those are the numbers which I have put in here,
so I get 0.5 times 2, I get 1.00.|The number
after the decimal point is 0.00,
and the number before the decimal point is simply 1.|And thatís when I
stop, when I find out that the number which is left after the decimal point
is simply 0, thatís when you stop the whole process, that you donít need to do it any further,
that you have obtained what the equivalent binary number is for that.
And now what Iím going to do is Iím going to call this to be a(-1),
Iím going to call this to be a(-2), Iím going to call this to be a(-3),
and this to be a(-4).|So the equivalent of
0.1875 base-10 is
equal to a(-1), a(-2), a(-3),
a(-4) base-2, which in this case would
be 0, thereís a radix point right here, radix
point here, 0011 base-2.
So in this case, when you are writing down the equivalent of the fractional part, you start from the top, you get
0, 0, 1, 1 ñ 0011,
as opposed to when you were doing the integer part, you started from the bottom, but here you start from the top,
0, 0, 1, 1 ñ 0011.
So if you look at the overall number, which is 11.1875 base-10,
then the whole equivalency in the binary notation
will be whatever is the equivalent of 11, which turned out to be 1011, so
thatís what we found on the previous part of this segment, so we got
1011 as the equivalent of 11, and for 0.1875,
we just calculated is the radix point right here, 0011
base-2.|So as an exercise, what you need is
go ahead and see whether you can go from here to here, and be sure that
you calculate this to be equal to 11.1875 in base-10.
And thatís the end of this segment.
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