Why aren't repeating decimals (such as 1/3) infinitely large?

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  • #1
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I had a slightly embarrassing conversation with a 10 year old:

Me: Did you know the fraction 1/3 can also be written as 0.3333...
Her: When does it end?
Me: Never, keeps going to infinity.
Her: Then if every 3 makes the number larger, and there are infinite 3s, does that mean it keeps getting bigger until infinity?
Me: *blank stare*

How would you explain this to a 10 year old?
 

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  • #2
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Well, convergence of infinite series is probably a bit much to expect a 10 year old to understand ;) Have you tried saying "try it yourself, and let me know when you get to four" ?
 
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  • #3
mjc123
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Try the old - is it Zeno's Paradox? If you are going from A to B, first go halfway there, then half the remaining distance, and so on. Because each step is half the remaining distance, you never quite get to B (except at infinity). I think that may be easier to visualise.
 
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  • #4
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Her: Then if every 3 makes the number larger, and there are infinite 3s, does that mean it keeps getting bigger until infinity?
##0.\bar{3}=0.333\ldots ## is just a representation of ##\frac{1}{3}## which is finite. It only shows, that ##0.333\ldots ## isn't optimal. For instance, we could chose a basis ##3## instead of ##10##. Then it becomes ##\frac{1}{3}=_3 \,0.1##. There is no need to use decimal numbers (basis ##10##). Our computers use powers of ##2## as basis units, the Babylonians used ##60##. In the end it's probably the number of our fingers which determined the ##10##, and that it is optimal for payments, as you only need coins ##1,2,5## to get all numbers. So you see, ##10## is somewhat deliberately chosen, and some quotients cannot be written finitely in this basis. Another basis - another representation. However, with the better notation ##\frac{1}{3}## everything is fine.
 
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  • #5
Because each time you add a three, you also dilute it by another 10x, so the new 3's get weaker and weaker the farther you go. Only by adding together the infinite
series of terms after the decimal point, does it actually get to exactly 1/3. The value is always slightly under 1/3 until you do the infinite summation. Have the child image a drop of food coloring in a glass, a bowl, a bathtub, a pond, a lake, an ocean. As you proceed, the change in color you've made becomes insignificant.
 
  • #6
Another option is to take 0.333333333333... and re-write this as 0.33333 + 0.0000033333333... , they will agree that it is the same. Now ask if
0.000004 is always bigger than 0.00000333333333333... no matter how many 3's you include. If the answer is yes, the infinite sum cannot get
over 0.333334 Push this idea to the N'th decimal point instead of the 6th, they will see the infinite sum must converge to 1/3.
 
  • #7
As the power of "three" becomes weaker and weaker, the ability of the number getting larger will also be weaker and weaker, until it infinitely approaches##\frac{1}{3}##.
 
  • #8
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I had a slightly embarrassing conversation with a 10 year old:

Me: Did you know the fraction 1/3 can also be written as 0.3333...
Her: When does it end?
Me: Never, keeps going to infinity.
Her: Then if every 3 makes the number larger, and there are infinite 3s, does that mean it keeps getting bigger until infinity?
Me: *blank stare*

How would you explain this to a 10 year old?
"Nope. Depends how small the numbers you're adding: Keep adding 1/2 to a number, then you get 1, 1 1/2, 2, 2 1/2 and it gets bigger and bigger. Now try adding 1/2, then 1/3, then 1/4 and so on. Adding smaller and smaller numbers but that too will continue growing. However, if the numbers you add continue to be a lot smaller, say, adding 1/2 then 1/4, then 1/8, then 1/16, well that won't grow very big, but will get closer and closer to one and never get bigger even if you add an infinite number of them: If you add the right kind of smaller and smaller numbers to a number, even an infinite number of them, the number will only grow so big and that is what happens when you keep adding three's to 0.333... It only gets so big, 1/3 and never bigger. "
 

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