# What use is there for rationalizing denominator?

• B
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

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

Mark44
Mentor
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##.

• leighflix
Thanks, and a small side question: How do you create the fractions? (on this forum)

jedishrfu
Mentor
• leighflix
FactChecker
Science Advisor
Gold Member
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.

$$\frac 1 i = \frac 1 i \cdot \frac{-i}{-i} = \frac{-i}{1} = -i$$

• leighflix
Mark44
Mentor
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 $...$.

• leighflix and FactChecker
pwsnafu
Science Advisor
I personally consider ##\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} = \frac{\sqrt2}{2} + \frac{\sqrt9}{3} = \frac{3\sqrt2+2\sqrt9}{6}## to be simpler than ##\frac{1}{\sqrt2} + \frac1{\sqrt3} = \frac{\sqrt2+\sqrt3}{\sqrt2 \sqrt3}## but this is a YMMV thing.