In
elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entail ...
, root rationalisation is a process by which
radicals in the
denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
of an
algebraic fraction
In algebra, an algebraic fraction is a fraction (mathematics), fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same laws as arithmet ...
are eliminated.
If the denominator is a
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
in some radical, say
with , rationalisation consists of multiplying the numerator and the denominator by
and replacing
by (this is allowed, as, by definition, a
th root of is a number that has as its th power). If , one writes with (
Euclidean division
In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
), and
then one proceeds as above by multiplying by
If the denominator is
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
in some square root, say
rationalisation consists of multiplying the numerator and the denominator by
and expanding the product in the denominator.
This technique may be extended to any algebraic denominator, by multiplying the numerator and the denominator by all
algebraic conjugate
In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element , over a field extension , are the roots of the minimal polynomial of over . Conjugate elements are commonly called conju ...
s of the denominator, and expanding the new denominator into the
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
of the old denominator. However, except in special cases, the resulting fractions may have huge numerators and denominators, and, therefore, the technique is generally used only in the above elementary cases.
Rationalisation of a monomial square root and cube root
For the fundamental technique, the numerator and denominator must be multiplied by the same factor.
Example 1:
:
To rationalise this kind of
expression
Expression may refer to:
Linguistics
* Expression (linguistics), a word, phrase, or sentence
* Fixed expression, a form of words with a specific meaning
* Idiom, a type of fixed expression
* Metaphorical expression, a particular word, phrase, o ...
, bring in the factor
:
:
The
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
disappears from the denominator, because
by definition of a square root:
:
which is the result of the rationalisation.
Example 2:
:
To rationalise this radical, bring in the factor
:
:
The cube root disappears from the denominator, because it is cubed; so
:
which is the result of the rationalisation.
Dealing with more square roots
For a
denominator
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
that is:
:
Rationalisation can be achieved by multiplying by the
conjugate:
:
and applying the
difference of two squares
In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity
:a^2-b^2 = (a+b)(a-b)
in elementary algebra.
P ...
identity, which here will yield −1. To get this result, the entire fraction should be multiplied by
:
This technique works much more generally. It can easily be adapted to remove one square root at a time, i.e. to rationalise
:
by multiplication by
:
Example:
:
The fraction must be multiplied by a quotient containing
.
:
Now, we can proceed to remove the square roots in the denominator:
:
Example 2:
This process also works with
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
with
:
The fraction must be multiplied by a quotient containing
.
:
Generalizations
Rationalisation can be extended to all
algebraic numbers and
algebraic function In mathematics, an algebraic function is a function that can be defined
as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additi ...
s (as an application of
norm form In mathematics, a norm form is a homogeneous form in ''n'' variables constructed from the field norm of a field extension ''L''/''K'' of degree ''n''. That is, writing ''N'' for the norm mapping to ''K'', and selecting a basis ''e''1, ..., ''e ...
s). For example, to rationalise a
cube root, two linear factors involving
cube roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important ...
should be used, or equivalently a quadratic factor.
References
This material is carried in classic algebra texts. For example:
*
George Chrystal
George Chrystal FRSE FRS (8 March 1851 – 3 November 1911) was a Scottish mathematician. He is primarily know for his books on algebra and his studies of seiches (wave patterns in large inland bodies of water) which earned him a Gold Meda ...
, ''Introduction to Algebra: For the Use of Secondary Schools and Technical Colleges'' is a nineteenth-century text, first edition 1889, in print ({{isbn, 1402159072); a trinomial example with square roots is on p. 256, while a general theory of rationalising factors for surds is on pp. 189–199.
Elementary algebra
Fractions (mathematics)