Other Free Encyclopedias » Science Encyclopedia » Science & Philosophy: Intuitionist logic to Kabbalah

Invariant - Geometric Invariance, Algebraic Invariance

unchanged magnitude operation axes

In mathematics a quantity is said to be invariant if its value does not change following a given operation. For instance, multiplication of any real number by the identity element (1) leaves it unchanged. Thus, all real numbers are invariant under the operation of "multiplication by the identity element (1)." In some cases, mathematical operations leave certain properties unchanged. When this occurs, those properties that are unchanged are referred to as invariants under the operation. Translation of coordinate axes (shifting of the origin from the point (0,0) to any other point in the plane) and rotation of coordinate axes are also operations. Vectors, which are quantities possessing both magnitude (size) and direction, are unchanged in magnitude and direction under a translation of axes, but only unchanged in magnitude under rotation of the axes. Thus, magnitude is an invariant property of vectors under the operation of rotation, while both magnitude and direction are invariant properties of a vector under a translation of axes.

An important objective in any branch of mathematics is to identify the invariants of a given operation, as they often lead to a deeper understanding of the mathematics involved, or to simplified analytical procedures.

Invasive Species - Survival Of Exotic Species, Effects Of Invasive Species, Examples Of Invasions, Management [next]

User Comments

Your email address will be altered so spam harvesting bots can't read it easily.
Hide my email completely instead?

Cancel or

Vote down Vote up

11 months ago