Is there a Development of "Average Algebra"?

[WWGD]

Hi All, wondering if this below has been developed already:
There is a context in which adding fractions by the rule : a/b + c/d = (a+c)/(b+d) : say we are considering

the grade in a course where a lot of exams are administered, and the total is considered over, say 1000. Then, if we get 90/100 in one quiz and 47/50 , the total average so far is given by (90+47)/(100+50) , not the standard way of adding fractions. Just curious if there is some known structure in Mathematics using this as an operation.
Thanks.

 

[fresh_42]

Given this rule we get a/b + a/b = (a+a)/(b+b) = 2a/2b = a/b which isn't possible for a ≠ 0. One has to build a whole new arithmetic and I doubt that this can be done without contradictions.

 

[WWGD]

Given this rule we get a/b + a/b = (a+a)/(b+b) = 2a/2b = a/b which isn't possible for a ≠ 0. One has to build a whole new arithmetic and I doubt that this can be done without contradictions.

But you don't necessarily simplify by common factors here, because you lose information. This property would tell you that the average of two of the same ratio is the ratio. If you have 15/20 in an exam, this is not quite the same as 3/4. So we could define a/b -c/d as (a-b)/(c-d). But yes, this simplification issue needs to be addressed.

Interestingly, I think 0 here would be 0/0 : 0 successes in 0 trials means that the success ratio has not changed. Then 0/0 makes sense in this context, but a/0 for a>0 does not. I don't have the whole thing figured out yet, clearly.

 

[fresh_42]

But you don't necessarily simplify by common factors here, because you lose information. This property would tell you that the average of two of the same ratio is the ratio.

But this is essentially what the ordinary arithmetic mean does: $\frac{\frac{a}{b} + \frac{a}{b}}{2} = \frac{a}{b}$

One certainly has to begin at the start: which elements are considered, which operations with which rules are allowed. I think a neutral notation like a circle instead of the plus sign would help to distinguish between known operations in ℚ and the new one, for otherwise there will be too much confusion. E.g. algebras are defined in every thinkable way. I have to look it up but in genetics they use some funny algebras, multiplication rules, resp. These might come close to your rule.

(I once met some analysts who (seriously) wanted to apply your rule on their index calculations based on market capitalization.)

 

[Mentallic]

Numberphile has actually made a video on this

 

[Mark44]

Hi All, wondering if this below has been developed already:
There is a context in which adding fractions by the rule : a/b + c/d = (a+c)/(b+d) : say we are considering

the grade in a course where a lot of exams are administered, and the total is considered over, say 1000. Then, if we get 90/100 in one quiz and 47/50 , the total average so far is given by (90+47)/(100+50) , not the standard way of adding fractions. Just curious if there is some known structure in Mathematics using this as an operation.
Thanks.

Another context is "baseball arithmetic." A player who has 3 hits for 5 times at bat in one game has a batting average for that game of 3/5 (normally presented as .600). If he gets 1 hit out of 4 "at bats" in the next game (1/4 or .250), his average for the two games is 4/9 (= .444), calculated as $\frac{3 + 1}{5 + 4}$.

 

[pwsnafu]

Just curious if there is some known structure in Mathematics using this as an operation.
Thanks.

It's called the mediant of two rational numbers. But it is not monotonic leading to the Simpson[/PLAIN] paradox: we can find $\frac{a}{b} < \frac{A}{B}$ and $\frac{c}{d} < \frac{C}{D}$ but $\frac{a+c}{b+d} > \frac{A+C}{B+D}$.

 

[WWGD]

Another context is "baseball arithmetic." A player who has 3 hits for 5 times at bat in one game has a batting average for that game of 3/5 (normally presented as .600). If he gets 1 hit out of 4 "at bats" in the next game (1/4 or .250), his average for the two games is 4/9 (= .444), calculated as $\frac{3 + 1}{5 + 4}$.

Yes, thank you, I was thinking scoring % in basketball, but same thing here. In some sense, 20/30 would simplify to 2/3, but not always, not necessarily: having 2 hits in 3 at bats is not the same as having 20 hits in 30 at bats.