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0.9 (repeating) is equal to 1
Name: julie a corder and hadley j high
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995
Question:
How is it that 0.9 (repeating) can be equal to the value of 1? That is the
result you get if you set it up as an equation:
9.9(repeating) = 10x
0.9(repeating) = 1x
subtracting gives you 9 = 9x, or 1=x
Please explain!
Replies:
Well, why not? This seems odd because we think the real numbers are repre-
sented by decimals in some exact fashion, when in fact that is not true at
all (this is the most obvious manifestation). Figuring out exactly what
real numbers were took an awfully long time for mathematicians of previous
centuries. One way to think about this particular problem is, 0.9 = 9/10,
0.99 = 99/100 etc., and so what is the number you might expect to get as you
continue that sequence? Mathematically, the limiting value cannot be
anything other than 1, because there is no real number smaller than 1 which
does not eventually also become smaller than one of these fractions
99 . . . 9/100 . . . 0
asmith
Infinity is a difficult concept. You are asking: why is 0.999 . . . with
an infinite number of 9's, equal to the integer 1? Well, it is just a
different way of representing the same number. It even makes sense logical-
ly. Remember how you learned how to count using apples? Let us go back to
that concept. Say you have 9 apples and you give one to a friend. Now what
fraction of the total number of apples have you given away? Right, 1/9.
How do we represent that in decimals? Right, 0.11111 (repeating). Try it
by hand. You will quickly see that the 1's go on forever. Now, how many
apples does your friend have? She has:
(total number of apples) times (her fraction of the total number) =
= (9) times (0.11111 (repeating)) = .99999 (repeating) = 1 apple.
richard a gerber
While I do not disagree with the 2 previous responses I wish to address your
question in a little different light. You give two equations, each with one
unknown. You then combine them (subtraction) like you would if you were
solving a system of two equations with two unknowns: Let me re-write your
equations in a slightly different form --
9.9 (repeating) = 10 X
0.9 (repeating) = 1 Y
subtracting gives 9.0 = 10X - 1Y. This is the result you should have
gotten. You have no a-priori knowledge that the variables in the two
equations are the same. Again, 0.9 (repeating) IS essentially 1.0, but the
two equations you gave do not necessarily have the same value of X just
because you named the variables the same. Algebra instructors will some-
times pose questions of this type to their students to drive home the need
to be careful in choosing names for their variables. Hope this helps!
While my statement in the previous response is true -- that you have to be
careful about naming variables when setting up a system of equations -- it
should have been obvious even to THIS careless scientist that the two
equations given are related by multiplication of both sides by a factor of
0.1. Therefore the two variables ARE identical and the subtraction CAN be
performed as shown.
gregory r bradburn
In fact, 9/9 = 0.9999999999999 repeating IS correct. The really interesting
thing is that this says 0.9999999 repeating equals 1.0000000 repeating.
chaffer
try adding 1/9 to 8/9:
0.11111 . . . + 0.88888 . . . = 0.999999 . . ., (=) 1.0
hawley
This is one of the reasons why the real numbers were not so well understood
by mathematicians until surprisingly recently - the decimal representation
of the real numbers is actually not a 1-to-1 representation - any number
with a repeating 0 at the end of its decimal representation has a second
decimal representation with a repeating 9 at the end.
asmith
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