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.
Geometric invariance
In geometry, the invariant properties of points, lines, angles, and various planar and solid objects are all understood in terms of the invariant properties of these objects under such operations as translation, rotation, reflection, and magnification. For example, the area of a triangle is invariant under translation, rotation and reflection, but not under magnification. On the other hand, the interior angles of a triangle are invariant under magnification, and so are the proportionalities of the lengths of its sides.
The Pythagorean Theorem states that the square of the hypotenuse of any right triangle is equal to the sum of the squares of its legs. In other words, the relationship expressing the length of the hypotenuse in terms of the lengths of the other two sides is an invariant property of right triangles, under magnification, or any other operation that results in another right triangle.
Very recently, geometric figures called fractals have gained popularity in the scientific community. Fractals are geometric figures that are invariant under magnification. That is, their fragmented shape appears the same at all magnifications.
Algebraic invariance
Algebraic invariance refers to combinations of coefficients from certain functions that remain constant when the coordinate system in which they are expressed is translated, or rotated. An example of this kind of invariance is seen in the behavior of the conic sections (cross sections of a right circular cone resulting from its intersection with a plane). The general equation of a conic section is ax2 + bxy + cy2 + dx + ey + f = 0. That is, each of the equations of a circle, or an ellipse, a parabola, or hyperbola represents a special case of this equation. One combination of coefficients, (b2-4ac), from this equation is called the discriminant. For a parabola, the value of the discriminant is zero, for an ellipse it is less than zero, and for a hyperbola is greater than zero. However, regardless of its value, when the axes of the coordinate system in which the figure is being graphed are rotated through an arbitrary angle, the value of the discriminant (b2-4ac) is unchanged. Thus, the discriminant is said to be invariant under a rotation of axes. In other words, knowing the value of the discriminant reveals the identity of a particular conic section regardless of its orientation in the coordinate system. Still another invariant of the general equation of the conic sections, under a rotation of axes, is the sum of the coefficients of the squared terms (a+c).
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