Triple fractions (and more complex fractions)Usually a b c = a c b i.e. b...

Wisniewool

Wisniewool

Answered

2022-07-06

Triple fractions (and more complex fractions)
Usually
a b c = a c b
i.e. b / c is seen as the denominator, and a is the numerator.
If you have a / b / c / d, what do you choose to take as the denominator?
b c d , c d ,  or  d ?
and why?
And what about for a / b / c / d / e?

Answer & Explanation

karburitc

karburitc

Expert

2022-07-07Added 7 answers

The trouble is
a / ( b / c ) ( a / b ) / c
since a / ( b / c ) = a c b and ( a / b ) / c = a b c . So, without brackets, the notation is ambiguous. Introducing brackets, for example, we could choose between
a / b / c / d = a ( b / c / d ) 1
and
a / b / c / d = ( a / b / c ) d 1
The first leads to
a ( b ( c / d ) 1 ) 1 = a ( b c 1 d ) 1 = a c b d
The second leads to
( a / b ) c 1 d 1 = a b 1 c 1 d 1 = a b c d
which is less interesting, it seems to me. The first expression leads to
a / b / c / d / e = a ( b ( c / d / e ) ) 1 = a ( b d c e ) 1 = a c e b d
Collecting results, the following interesting pattern emerges:
a / b = a b 1 = a b
a / b / c = a b 1 c = a c b
a / b / c / d = a b 1 c d 1 = a c b d
a / b / c / d / e = a b 1 c d 1 e = a c e b d
a / b / c / d / e / f = a b 1 c d 1 e f 1 = a c e b d f . . .
and so on.

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