Added content to stub (#33043)
Gave definition, explained how to simplify them and then some examples.pull/30728/head^2
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@ -3,13 +3,53 @@ title: Complex Fractions
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---
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## Complex Fractions
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This is a stub. <a href='https://github.com/freecodecamp/guides/tree/master/src/pages/mathematics/complex-fractions/index.md' target='_blank' rel='nofollow'>Help our community expand it</a>.
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<a href='https://github.com/freecodecamp/guides/blob/master/README.md' target='_blank' rel='nofollow'>This quick style guide will help ensure your pull request gets accepted</a>.
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<!-- The article goes here, in GitHub-flavored Markdown. Feel free to add YouTube videos, images, and CodePen/JSBin embeds -->
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#### More Information:
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<!-- Please add any articles you think might be helpful to read before writing the article -->
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Complex fractions are fractions that contain fractions in the numerator or denominator, e.g., (1/2)/3 and 1/(2/3) as well as (1/2)/(3/4) are complex fractions. To work with/simplify these fractions there is really only one rule to keep in mind:
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```
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(a/b)/(c/d) = (a/b)*(d/c)
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```
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when `b`, `c` and `d` are not 0. Namely, dividing by a fraction is the same thing as multiplying by it's [reciprocal](https://en.wikipedia.org/wiki/Multiplicative_inverse). To see this, note that multiplying both sides of our expression above by `c/d` gives
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```
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[(a/b)/(c/d)]*(c/d) = (a/b)
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```
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one the left, and
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```
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[(a/b)*(d/c)]*(c/d) = [(a*d)/(b*c)]*(c/d)
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= (a*d*c)/(b*d*c*) = a/b
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```
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on the right, so going backwards and dividing `a/b` by `c/d` indeed gives the product `(a/b)*(d/c)`.
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Using this, our examples above can be simplified to
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```
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(1/2)/3 = (1/2)*(1/3) = 1/6,
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1/(2/3) = 1*(3/2) = 3/2,
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(1/2)/(3/4) = (1/2)*(4/3)
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= 4/6 = 2/3.
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```
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This extends to [real numbers](https://en.wikipedia.org/wiki/Real_number), [complex numbers](https://en.wikipedia.org/wiki/Complex_number) and [algebraic expressions](https://en.wikipedia.org/wiki/Algebraic_expression) as well. For example,
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```
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(pi/2)/2 = pi/4,
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1/(sqrt(2)/2) = 2/sqrt(2)
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= 2*sqrt(2)/2
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= sqrt(2).
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```
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and
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```
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(1 + 1/x^2)/(2x + 2/x^3) = [(x^2 + 1)/x^2]/[(2x^4 + 2)/x^3]
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= [(x^2 + 1)/x^2)]*[x^3/(2x^4 + 2)]
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= [(x^2 + 1)*x^3]/[x^2*(2x^4 + 2)]
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= [(x^2 + 1)*x]/[(2x^4 + 2)].
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```
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Note that in the example above we needed to get the numerator and denominator into a fractional form before being able to use the rule where division by a fraction is equivalent to multiplying by a denominator. Similarly, if we want to simplify
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```
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(1 + 1/2)/(2 + 2/3)
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```
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we must first simplify this into a fraction divided by another fraction to be able to use our rule above. We have
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```
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1 + 1/2 = (2 + 1)/2 = 3/2,
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2 + 2/3 = (2*3 + 2)/3 = 8/3,
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```
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so we have
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```
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(1 + 1/2)/(2 + 2/3) = (3/2)/(8/3) = (3/2)*(3/8) = 9/16.
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```
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