This video is brought to you by Extreme
Finitism dot com why 0.9 recurring
does not equal one how can we get
0.9 recurring well if we
assume one-third equals 0.3
recurring then we can multiply both
sides by 3 to get one equals 0.9
recurring but the validity of this
argument depends on the validity of the
starting assumption can we assume that
one-third equals 0.3
recurring if we divide one by three does
it result in 0.3
recurring? Before we tackle one-third
let's first look at a well-behaved
decimal expansion that is to say one
that ends nicely if we want to see the
decimal expansion of a fraction we need
to perform the division operation and so
when we do the division operation on a
fraction like one eighth we are performing a
process where we get nought point one
then we get a two and then we get five
and at this point the division ends and
then we can put an equal sign in and say
one eighth equals 0.125 now if we try
and do the same with one-third we start
doing the process and it seems to be
working ok but we keep getting more and
more threes and we soon realize this is
going to keep going on and on and on and
so we hit this big problem that the
process does not end we don't get to the
nice end point where we can put the
equal sign in like we could do with one eighth
we can of course use values like pie e
square root of two and one-third in
equations it's only when we try to
convert these values to decimal numbers
that we encounter this endlessness
problem and since the decimal expansion
has no end point we are forced to
introduce a rounding error
we want a numeric answer unlike the
example with one-eighth we do not get a
fixed result here our division process
usually meant by naught point three
recurring meaning one a decimal number
with infinitely many digits notice the
expression infinitely many meaning two
the limit of the series where the nth
term is 3 over 10 to the power n but
this second
meaning effectively says that when we
write 0.3 recurring it's
just another way of writing the fraction
one-third it says nothing about decimal
numbers so for 0.3
recurring to be a decimal we are left
with meaning one to investigate so what
does infinitely many digits mean well as
point three recurring as a series of
terms three tenths three hundredths
never reaches one third although no
finite size will make the sum of the
series equal one third the idea of
infinitely many is that somehow the sum
of the series can equate to one third
one difficulty in understanding the
concept of infinitely many is how can
the value stop at one-third if the
series does not end is the idea of
infinitely many a mistake that
ultimately leads to contradictions or
can we just call the things we don't
understand paradoxes and move on like
0.3 three recurring. 0.9
recurring is also supposed to contain
infinitely many digits and so if we
could prove mathematically that
0.9 recurring doors equate to one
then this would suggest that infinitely
many does have a real meaning so here is
the famous 10x minus X proof we start
off with x equals 0.9
recurring then we multiply both sides by
10 to get 10 x equals nine point nine
recurring then we take away what we
started with and on the left hand side
we get 10x minus X is 9x on the right
hand side we get nine point nine
recurring minus 0.9 recurring
equals nine we divide throughout by nine
and we end up with x equals 1
and since we started off with x equals
0.9 recurring we appear to
have proved that 0.9recurring equals
one the issue here is the subtraction
operation how do we determine if the
endless part of X exactly matches the
endless part of 10x now it appears
obvious that the trailing parts cancel
out but this is what we call an informal
proof in order to be absolutely sure we
need to do this with mathematical rigor
in a rigorous approach we would show
that the first n terms cancel each other
out then all subsequent terms can be
deemed to cancel out and we will have a
rigorous proof that 0.9
recurring equals 1 and so let's do a
more rigorous approach to this proof we
start off by expressing 0.9
recurring as a series up to the nth term
we're assuming that there are an endless
number of terms after the end of term
infinitely many of you prefer but we
only need to consider up to the nth term
because all of these subsequent terms
can be said to have canceled out if we
can show that all terms up to the nth
term cancel out and so we start with x
equals our series up to the nth term on
the line above we write 10 x equals 10
times the series up to the nth term next
we do the subtraction just like in the
informal proof here a lot of the middle
terms cancel out and we are left with 9
x equals nine minus nine over 10 to the
power n and when we divide throughout by
nine we get x equals one minus one over
10 to the power n now we could argue
that this extra term does tend towards 0
as n increases what no value of n can
make it equal to zero effectively we
have shown that 0.9 recurring
does not equal one so where does this
leave us our rigorous proof gave this
result
in the proof we showed that the endless
terms do not all cancel out and we
showed that for any value of n, 0.9
recurring cannot equate to one note
that if we divide throughout by three we
see that for any value of n naught point
three recurring cannot equate to
one-third which means we were wrong to
assume 0.3 recurring equals
one-third at the start of this video but
if 0.9 recurring is not a
decimal with infinitely many digits then
what exactly is it and how are these
objects defined without using words like
infinitely many and how can these
objects be used in mathematics these
questions will be answered in the next
video about why 0.9 recurring
does not equal one this video was
brought to you by Extreme Finitism
dot com in association with Reveal
Computer Software