Are complex numbers and complex fractions related?

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cjcapta

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I know that a complex number is any number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit. A is called the real part and b is called the imaginary part. Is a complex fraction: (a/b) / (c/d) ? if so, are they related at all or do they both just share the name complex?
 
The similarity in the names is a pure accident of the history of mathematics education. If I were to bet, I would wager that the name “complex fraction” is the newer of the two usages.
 
A complex fraction can be written as \(\displaystyle (a+bi)/(c+di)\). Multiply the numerator and denominator by \(\displaystyle c-di\).
 
royhaas said:
A complex fraction can be written as \(\displaystyle (a+bi)/(c+di)\). Multiply the numerator and denominator by \(\displaystyle c-di\).
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So is there a name for (a/b) / (c/d) ?
 
I think that you have something like this in mind: \(\displaystyle \frac{{1 + \frac{a}{b}}}{{\frac{{1 + c}}{{1 - d}}}}\)
In fact, mathematics educators call such "complex fractions".
 
pka said:
I think that you have something like this in mind: \(\displaystyle \frac{{1 + \frac{a}{b}}}{{\frac{{1 + c}}{{1 - d}}}}\)
In fact, mathematics educators call such "complex fractions".
Click to expand...
So is royhaas wrong?
 
cjcapta said:
So is royhaas wrong?
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Absolutely not! He simply read your question as referring to complex numbers.
In fact, there is a notion of fractions in the complex field. That is what he said.
Your question is about terminology: “complex fractions”.
That term is used by mathematics educators to mean one thing and by mathematicians to mean what Prof Haas said.
That is far from an unusual occurrence. Almost all terms have several meanings.
 
And that is one of the reasons we ask:

What does your text-book/class-note say?
 
cjcapta said:
... is there a name for (a/b) / (c/d) ?
Click to expand...


I learned the term for dividing by a fraction as a "compound" fraction.

I agree with the earlier post that a complex fraction is a number involving complex numbers.

The following are examples of compound fractions.

\(\displaystyle \frac{3}{\frac{13}{4}} \; \frac{\frac{x}{3y}}{\frac{7x}{2w}}\)

 


(The ways in which mathematicians name things could comprise an entire field of study. I claim that such a study should be an adjunct to formal linguistics. Sometimes, the choice of terms boils down to laziness and ambiguity -- similar to what we do with speech and notation when there is a lot to say or write.)

Mathematicians call many things "standard"; some of these things are anything but.

Mathematicians call things "normal", where I don't see normalcy, either.

Dividing by 3/2 is not, as far as I can tell, a complex operation, but clearly we name it so, regardless.

Defining a number to have an imaginary component is not really complex, either.

But, here, I am interpreting "complex" to mean "difficult". Shame on me for my narrowmindedness!

I learned a name for the set of non-negative integers: the set of Whole Numbers. Yet, I've met several mathematicians who claim that there is no such thing as a set of whole numbers.

Add general ambiguity inherent within contemporary American english, and the tracks are set for derailment.

Are complex numbers and complex fractions related?

No. But their names are.

One dictionary states the following partial definitions.

COMPLEX: ... having more than one part, compound, ...

COMPOUND: ... having more than one part, ... complex ...

At the end of the day, we can say and write whatever we like. If we want to achieve successful communication from our head to others', then we need to wait for the reaction so as to correct misinterpretations or differences of perception. This process is slow when the communication is not interpersonal, and there is a lot of mathematical communication which is not interpersonal.

(I want everybody to know that I'm currently restraining myself from bringing up the term "numeral" ...)

 
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