When does 1/2 + 2/3 = 3/5?

We all remember how to solve 1/2 + 2/3 in standard arithmetic. You convert each fraction to the least common denominator (6), and find that 3/6 + 4/6 = 7/6.

But at times, it can be useful to use a non-standard arithmetic in which 1/2 + 2/3 = 3/5. One example is in sports.

Baseball arithmetic

If a batter gets 1 hit in 2 at-bats one day, and then 2 hits in 3 at-bats the next day, what is the person’s overall batting average?

The answer is easily found as 3/5. The overall batting average is found by summing up the total hits by the total at-bats.

What we have is a different kind of addition, which can be called baseball arithmetic and follows the rule:

The interesting thing is baseball arithmetic can be useful in simplyfing a variety of formulas.

Update 4-13-12: I was informed this operation is formally known as the mediant.

Advertising click-through ratios

Any time you wish to aggregate subtotals, baseball arithmetic would be useful. For example, suppose your blog ads showed 10 clicks out of 1,000 impressions one day (1 percent click-through), and then 20 clicks out of 1,500 (1.33 percent click-through), you could easily use baseball arithmetic to solve that the overall click-through ratio is 30/2,500, or 1.2 percent.

Hospital surival rates

Similarly, if a hospital loses 1 patient out of 1,000 to a procedure one day, and then 1 patient to 1,200 the next, the overall rate of losing patients is 2/2,200 or 0.9 percent.

Other applications

There are in fact applications to Archimedes law of the lever and the concept of pseudoperspective in computer science.

The explanation comes from Dr. Ron Golman of Rice University, which is where I learned about baseball arithmetic.

Check out these lecture slides (pdf) for the details: Baseball, Classical Mechanics, and Computer Graphics.

(hat tip: Augarithms newsletter)

Dr. Goldman’s ending slide points out that math is not just some set of rules you should accept, but rather the study of the structure of the rules. Always be on the lookout for novel ways to describe the structures and you might find something new and interesting.



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  • http://www.franchise-info.ca michael_webster

    Thanks, Presh.  This was both fun and informative.  It makes a sophisticated reader appreciate the assumptions behind any axiom scheme.  I liked it.

  • jmhajek

    Well, it’s cheating. In the first case, 1/2 means “one half”, in the other, “one for two”. They are completely different things. 

    Note also that 2/4 != 1/2 in the second case. 

  • Sauron

    It’s kind of cheating, but in a good way.  My abstract algebra prof really liked to talk about notation.  He pointed out how our notation is ultimately arbitrary and, for example, there’s no real reason why 2 + 2 isn’t represented as 2 & 2.  He also liked to point out that reusing notation can sometimes lead one to believe that other properties should follow as well when that’s not necessarily true.  For example, while scalar arithmetic holds that xy = yx it is not the case in matrix multiplication that AB = BA even though one might expect it to transfer.

    One of the big things, though, was how a good notation, while potentially leading to the dangers of the second point, will let us communicate much more effectively.  This post is one very good example of a good choice of notation with a small caveat.  It’s not “cheating” so much as it is a definition of the symbol in a given context to allow us to work more efficiently.  We can also just say it’s simply mathematics.

  • john

    This “baseball addition” is a weighted average where the denominator (or sample size) is the weight.

    jmhajek calls it cheating, and I sort of agree. It’s basically a matter of semantics (the definition of “+”) and by creating your own definition of words, pretty much any “statement” can be true. On the other hand, while “+” is overwhelmingly used in the standard arithmetic sense, there are branches of mathematics which commonly use it with variations of the definition. This apparently is one of them.

    Sauron makes some good points. The point that the reuse of notation can lead to false assumptions I find of particular interest here.

    Using standard arithmetic, a/c+b/d=xa/xc+yb/yd, as each term is unchanged, and so the sum is unchanged as well. But for “baseball addition” this is NOT true. It is a weighted average, and while the terms are unchanged, the weights are not. This means the ratios cannot be simplefied seperately.

    For instance, 10 clicks out of 1000 impressions is the same as 1 in 100. 20 in 1500 is one in 75. With “baseball addition” (I’ll use the notation”[b+]“), one might be tempted to say it’s 10/1000[b+]20/1500=1/100[b+]1/75=2/175=1.14%. Or 10/1000[b+]20/1500=1%[b+]1.33%=2.33/2%=1.17%.

    However, if the same factor is used in the simplefication of each term, the weights stay proportional and it works. So deviding all the clicks and impressions by 10: 1/100[b+]2/150=3/250=1.2%.

    a/c[b+]b/d=xa/xc[b+]xb/xd is true.

    If using weighted averaging instead of “baseball arithmetic”, each ratio can be simplefied seperately as long as the weights are proportional to the sample sizes and not the new demominators. 2 to 3 is the same ratio as 1000 to 1500, so the weighted average can be written (2/100+3/75)/5=18/1500=1.2%.

    This can all really just be viewed as making sure the units are consistant.

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