What use is there for rationalizing denominator?

  • #1
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Main Question or Discussion Point

I've been learning algebra for the past 2 years (in high school), not once have we ever had to rationalize a denominator in a radical expression. I am now relearning Algebra and Trig., what use is there? I mean, all you're doing is switching the numerator (rational) to the Denominator (irrational) in terms of rationality. (not actually switching the numbers).

Rational / Irrational → Irrational / Rational
 

Answers and Replies

  • #2
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After quite a while of thinking about this, the only conclusion I can come up with is to simplify it.

Say for example, 6 / √12 → 6√3 / √(12 ⋅ 3) → 6√3 / 6 → √3
 
  • #3
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I've been learning algebra for the past 2 years (in high school), not once have we ever had to rationalize a denominator in a radical expression. I am now relearning Algebra and Trig., what use is there? I mean, all you're doing is switching the numerator (rational) to the Denominator (irrational) in terms of rationality. (not actually switching the numbers).

Rational / Irrational → Irrational / Rational
The main rationale behind this technique is that it's easier to divide an irrational number by a rational number than the other way around, especially if you don't have a calculator or computer to do the work for you.

The technique is also used for fractions that involve complex numbers. For example, ##\frac 1 i## can be simplified by multiplying by the complex conjugate over itself; i.e., by multiplying by ##\frac{-i}{-i}## (which is 1). So ##\frac 1 i = \frac 1 i \cdot \frac{-i}{-i} = \frac{-i}{1} = -i##.
 
  • #4
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Thanks, and a small side question: How do you create the fractions? (on this forum)
 
  • #5
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  • #6
FactChecker
Science Advisor
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Regarding the format of math equations: If you see an example math equation here whose format you want to mimic, you can right-click and see the Show Math As => TeX commands. Copy it and surround it with [ tex] ... [/tex]. (Note. I had to insert a space before 'tex' to stop the formatter from parsing it). Here is an example from a post above.

[tex]\frac 1 i = \frac 1 i \cdot \frac{-i}{-i} = \frac{-i}{1} = -i[/tex]
 
  • #7
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FactChecker said:
Copy it and surround it with [ tex] ... [/tex].
I never use these any more -- I find it simpler to use $$ at the beginning of the expression and $$ at the end -- it's less to type. For stuff that I want to show inline, I use ## at the beginning and two more of them at the end. The ## pairs are equivalent to [itex] ... [/itex].
 
  • #8
pwsnafu
Science Advisor
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I personally consider ##\frac{1}{\sqrt{2}} + \frac{1}{\sqrt[3]{3}} = \frac{\sqrt2}{2} + \frac{\sqrt[3]9}{3} = \frac{3\sqrt2+2\sqrt[3]9}{6}## to be simpler than ##\frac{1}{\sqrt2} + \frac1{\sqrt[3]3} = \frac{\sqrt2+\sqrt[3]3}{\sqrt2 \sqrt[3]3}## but this is a YMMV thing.
 

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