Rationalization (fractions)

As rationalization (also: Streamline or rational Read ) is referred to in the elementary algebra a technique, an irrational number (for example, a root or a complex number ) in the numerator or in the denominator of a fraction to eliminate, that is, by an equivalent expression with only rational To replace numbers .

The irrational number itself to be eliminated is usually a monomial or binomial . In the basic procedure , the fraction is expanded with a suitably selected factor, ie both the numerator and the denominator of the fraction are multiplied by this factor, which does not change the value of the fraction.

Sample calculations

In the following examples, the variable stands for any real number . ${\displaystyle r}$

Rationalization of a monomial denominator

Take the break as an example

${\displaystyle {\frac {r}{\sqrt[{n}]{x^{m}}}}}$

with a common root in the denominator and with given. If you expand with , you get ${\displaystyle n>m}$ ${\displaystyle {\sqrt[{n}]{x^{n-m}}}}$

${\displaystyle {\frac {r}{\sqrt[{n}]{x^{m}}}}\cdot {\frac {\sqrt[{n}]{x^{n-m}}}{\sqrt[{n}]{x^{n-m}}}}={\frac {r{\sqrt[{n}]{x^{n-m}}}}{\sqrt[{n}]{x^{n}}}}={\frac {r{\sqrt[{n}]{x^{n-m}}}}{x}}.}$

Rationalization of a binomial denominator

In general, the fraction is expanded with the conjugation of the binomial denominator. The multiplication of a binomial by the conjugated binomial also occurs in the third binomial formula .

Example with a binomial root

The break is given

${\displaystyle {\frac {r}{{\sqrt {x}}+{\sqrt {y}}}}.}$

The extension with the conjugated binomial gives

${\displaystyle {\frac {r}{{\sqrt {x}}+{\sqrt {y}}}}\cdot {\frac {{\sqrt {x}}-{\sqrt {y}}}{{\sqrt {x}}-{\sqrt {y}}}}={\frac {r({\sqrt {x}}-{\sqrt {y}})}{{\sqrt {x}}^{2}-{\sqrt {y}}^{2}}}={\frac {r({\sqrt {x}}-{\sqrt {y}})}{x-y}}.}$

Application to complex numbers

A complex number in the denominator of a fraction can also be avoided. Let's take the reciprocal of a complex number as an example . You get ${\displaystyle z=a+ib}$

${\displaystyle {\frac {1}{z}}={\frac {1}{a+ib}}.}$

By expanding this fraction with the complex conjugate of , dh , one obtains ${\displaystyle z}$ ${\displaystyle {\overline {z}}=a-ib}$

${\displaystyle {\frac {1}{a+ib}}={\frac {a-ib}{(a+ib)(a-ib)}}={\frac {a-ib}{a^{2}+b^{2}}}={\frac {a}{a^{2}+b^{2}}}-i{\frac {b}{a^{2}+b^{2}}}.}$

Rationalization of a counter

Although rationalizing the denominator is much more important, rationalizing the numerator is often useful in calculus . Particularly when determining limit values , this often allows indefinite expressions to be calculated.

As an example, consider the expression

${\displaystyle \lim _{x\to 0}{\frac {{\sqrt {r+x}}-{\sqrt {r}}}{x}}}$

given. In this case, inserting returns the indefinite expression . By rationalizing the counter, using the third binomial formula analogously to the above, one obtains: ${\displaystyle x=0}$ ${\displaystyle {\tfrac {0}{0}}}$

${\displaystyle \lim _{x\to 0}{\frac {{\sqrt {r+x}}-{\sqrt {r}}}{x}}\cdot {\frac {{\sqrt {r+x}}+{\sqrt {r}}}{{\sqrt {r+x}}+{\sqrt {r}}}}=\lim _{x\to 0}{\frac {\cancel {(r+x)-r}}{{\cancel {x}}\cdot ({\sqrt {r+x}}+{\sqrt {r}})}}={\frac {1}{2{\sqrt {r}}}}.}$

Applications

• Some mathematical conventions provide for a fraction to be represented without roots in the denominator, if possible.
• The method can be useful to simplify the numerical calculation of such fractions.
• In many cases it is not possible to continue calculating without rationalizing the denominator or even the numerator.

literature

• George Chrystal: Introduction to algebra: For the use of secondary schools and technical colleges . 4th edition. Elibron, 2002, ISBN 1-4021-5907-2 .

• BF Caviness, R. Fateman: Simplification of Radical Expressions . In: Proceedings of 1976 AMC Symposium on Symbolic and Algebraic Computation . ( Online copy (PDF; 1.0 MB)).