Discrete Ideas » repeating http://www.discreteideas.com Thoughts count Thu, 21 Apr 2011 05:24:16 +0000 en hourly 1 http://wordpress.org/?v=3.1.3 Fractured Memories http://www.discreteideas.com/2010/07/fraction-memorization/ http://www.discreteideas.com/2010/07/fraction-memorization/#comments Sun, 04 Jul 2010 04:12:12 +0000 The Count http://www.discreteideas.com/?p=306 Some fractions are a part of everyday life; dimes, quarters, nickels, hours, minutes, seconds, etc. These are relatively easy to manage mainly because we deal with them so often. Everyone just “knows” that 1/2 is 0.5, and 1/4 is .25, and 1/10 is 0.1; we’ve had it ingrained in us through massive amounts [...]]]>

Some fractions are a part of everyday life; dimes, quarters, nickels, hours, minutes, seconds, etc. These are relatively easy to manage mainly because we deal with them so often. Everyone just “knows” that 1/2 is 0.5, and 1/4 is .25, and 1/10 is 0.1; we’ve had it ingrained in us through massive amounts of repetition. I go one step further; I can usually estimate the decimal equivalent of just about any fraction that comes up in my life. Super useful? Maybe not, but it has good show-off value, and I think it’s fun!


Joe commented below and suggested using 0.[3] to denote a 0. followed by an endlessly repeating 3. I thought it was a good idea, so I changed the text below to use it. Hope it’s not confusing.

Learning the first 12 fractions can make it super-easy to do division in your head and produce answers down to the 10ths or even 1000ths quickly and easily. Let’s take a look:

Denominator Values Tips
1/1 1 This is the easy one, put here for completeness. It could be beneficial to remember that any non-zero number N over N = N/N = 1.
1/2 1/2 = 0.5 Sure, it’s simple, but it’s useful when trying to compute 1/20th, etc.
1/3 1/3 = 0.[333]
2/3 = 0.[666] 		That’s 0.[333], where the 3s never stop, also called 0.3 repeating. And yes, 0.9 repeating does equal 1!
1/4 1/4 = 0.25
2/4 = 0.5
3/4 = 0.75 		Here’s the first one where just memorizing keeps you from having to reduce 2/4 = 1/2.
1/5 1/5 = 0.2
2/5 = 0.4
3/5 = 0.6
4/5 = 0.8 		N/5 = 0.(2*N). Note you’re really multiplying N by 2, then dividing by 10 (which just moves the decimal): 3/5 = (3*2)/10 = 0.6!
1/6 1/6 = 0.1[6]
2/6 = 0.[3]
3/6 = 0.5
4/6 = 0.[6]
5/6 = 0.8[3] 		Ok, this one’s not so simple. It helps to realize that 0.[3] / 2 = 0.1[6], and go from there. Having 3/6 = 0.5 in the middle can help too, since 5/6 = 3/6 + 2/6 = 0.5 + 0.[3] = 0.8[3], see?
1/7 1/7 = 0.[142857]
2/7 = 0.[285714]
3/7 = 0.[428571]
4/7 = 0.[571428]
5/7 = 0.[714285]
6/7 = 0.[857142] 		This is by far my favorite fraction. Note that in all cases, all six digits repeat, so 1/7 = 0.142857142857… Also note that the same six digits appear in the same order for all 6 fractions, you just start with a different digit. I use the fact that 14 is half 28 is half (just about) 57 to help remember the digits, too. This is the impressive one, guys. Someone asks, “what’s 1/7th of 100?” and you say “14.2857″ instantly. Nice.
1/8 1/8 = 0.125
2/8 = 0.25
3/8 = 0.375
4/8 = 0.5
5/8 = 0.625
6/8 = 0.75
7/8 = 0.875 		Seems like a lot to know, but most are easily computable from knowing 1/8 and reducing the rest. 5/8 = 4/8 + 1/8 = 0.5 + 0.125 = 0.625
1/9 1/9 = 0.[1]
2/9 = 0.[2]
…
7/9 = 0.[7]
8/9 = 0.[8] 		Just take the numerator and repeat it over and over. And again, 9/9 = 0.[9] = 1. Also of note, any number N (up to 99) over 99 0.[N] too, but use both digits, so 5/99 = 0.[05], 63/99 = 0.[63], etc. This continues for 999, 9999, etc.
1/10 1/10 = 0.1
2/10 = 0.2
…
8/10 = 0.8
9/10 = 0.9 		These are pretty self-evident. You’re dividing by 10, so just slide the decimal place.
1/11 1/11 = 0.[09]
2/11 = 0.[18]
3/11 = 0.[27]
4/11 = 0.[36]
5/11 = 0.[45]
6/11 = 0.[54]
7/11 = 0.[63]
8/11 = 0.[72]
9/11 = 0.[81]
10/11 = 0.[90] 		See what’s happening? N/11 = 0.[N*9] repeating, with both digits repeating (Note, 1*9 = 09 in this case). This becomes obvious when you think that 11/11 must equal 0.[9], so dividing that by 11 must divide each of those 99s in the decimal by 11 as well: 0.[9] / 11 = 0.[09].
1/12 1/12 = 0.08[3]
2/12 = 0.1[6]
3/12 = 0.25
4/12 = 0.[3]
5/12 = 0.41[6]
6/12 = 0.5
7/12 = 0.58[3]
8/12 = 0.[6]
9/12 = 0.75
10/12 = 0.8[3]
11/12 = 0.91[6] 		I must admit, I don’t really have these memorized. I know that 1/12 = 0.08[3] and work from there. 7/12 = 6/12 + 1/12 = 0.5 + 0.08[3] = 0.58[3], etc. Since half the values for N reduce to smaller fractions, this is where I leave off memorizing.

There you have them, the first 12 fractions for easy memorization. Amaze your friends! Astound your kids! Become even more of a know-it-all than you already are! I joke, but I guess you’d be surprised how often I use these, I know I am.

The Count

P.S. Please excuse my use of * to denote repeating decimals, I’d be happy to hear of a better symbol, since my font doesn’t allow lines across the top of text

]]> http://www.discreteideas.com/2010/07/fraction-memorization/feed/ 5 Mom was right http://www.discreteideas.com/2010/05/mom-was-right-repeating-9/ http://www.discreteideas.com/2010/05/mom-was-right-repeating-9/#comments Sun, 09 May 2010 14:00:44 +0000 The Count http://www.discreteideas.com/?p=279 Ah, Mother’s Day. Admittedly, Math isn’t the first thing you think about when scrambling for that 1-800-Flowers phone number (I always forget that one) or that last box of chocolate from the drugstore. However, one of the memories that sticks in my mind most about my Mom is arguing with her about Math [...]]]>

Ah, Mother’s Day. Admittedly, Math isn’t the first thing you think about when scrambling for that 1-800-Flowers phone number (I always forget that one) or that last box of chocolate from the drugstore. However, one of the memories that sticks in my mind most about my Mom is arguing with her about Math during dinner about .9 repeating and 1. Before you think we’re crazy, please keep in mind she’s a Math teacher, and I have some small interest in the subject myself. So here’s to you, Mom. I finally realized you were right some time ago, but don’t think I ever said so.


I didn’t really have any good arguments for my side of the case. What can I say? I was 8 or 9 and saying things like “1 minus 0.9 repeating is 0.0 repeating with a 1 at the end” made perfect sense to me. I also have to say I didn’t listen that much to Mom’s arguments, so I’m not sure she used these following proofs, but I’m pretty sure they should convince just about anyone.

First, there’s simple addition. If you agree that 1/3 = 0.3333(repeating), then:

Start with 1/3rd 1 / 3 = 0.33333333…
Add another 1/3rd 2 / 3 = 0.66666666…
Add a final 1/3rd 3 / 3 = 1 = 0.99999999…


Ok, what about the algebraic solution?

Start with: X = 0.99999999…
Multiply both sides by 10: 10X = 9.99999999…
Subtract the first row from the second: 9X = 9
Now divide both sides by 9: X = 1


See how we started with X = 0.999999… and ended with X = 1? That means they’re the same!

Finally, there is the infinite geometric series, where each term is a set ratio of the previous term. In the case of 0.99999…, we can say that this is the sum of 0.9 + 0.09 + 0.009… This gives us an initial term of 0.9, and a ratio between terms of 1/10.

Geometric series sum is this formula, where A is the initial term, and R is the ratio between terms: S = A(1-(R^N)) / (1-R)
Since we want N to be infinite (the 9′s do go on forever) and |R| < 1, then R^N becomes 0: S = A(1 – 0) / (1 – R)
The initial term in the series A is 0.9, and the ratio between terms R is 0.1 S = 0.9(1) / (1 – 0.1)
Algebra time, see how the series sum becomes 1? S = 0.9 / 0.9 = 1


I admit, that last one wouldn’t have made much sense to me at 9 years old. It just goes to show how many different ways you can prove that 0.9(repeating) is the same as 1. I hope my Mom reads this and realizes that it’s finally sunken into my brain that she was right… on at least this one occasion. If you ever lose an argument with your Mom, make sure you let her know about it, too. Happy Mother’s Day to you all.

The Count

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