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Name: julie a corder  and  hadley j high
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Date: Around 1995

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!

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


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.


try adding 1/9 to 8/9:

0.11111 . . . + 0.88888 . . . = 0.999999 . . ., (=) 1.0


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.


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