  |
|
|
|
Over the course of one's mathematical education, they inevitably are forced to memorize a great many mathematical identities, maybe without even realizing it.
Mathematical identities are frequently taught in math classes as short, highly simplified equations which demonstrate the relations between various elements and which demonstrate fundamental mathematical principles in easily memorized forms (and it is memorization for which most identities are intended). While it can be easy for a student of mathematics to become frustrated by what seems to be a tedious and nearly endless series of identities to remember throughout math education, the importance of these most basic, general, mathematical notions cannot be stressed enough. Essentially, mathematical identities come in two varieties: Identity Relations and Identity Elements. Identity ElementsElemental identities are those which signify the number or operator which acts upon an element without changing it. For example, the identity element for addition/subtraction is zero, because: X + 0 = X, and X - 0 = X. Likewise, the identity element for multiplication/division is one, because: Y x 1 = Y, and Y/1 = Y. These are clearly the simplest of all identity elements, but there are certainly others, including identities for exponentials, for matrices, for sets, for functions, for derivatives, etc. Identity RelationsIdentity relations are surely the most common form of identity used in math classes everywhere. These are the equalities which students are forced to memorize, one after another, because they offer deep insight into the relations between various objects, giving key insight into solving a wide variety of problems. These identities, as the name implies, are all relational. For instance, if one is solving a problem in trigonometry, if given sin(x) and asked to find the derivative of this, they have the choice of either going through the long, arduous process of solving the problem from scratch, or they might simply remember the extremely simple identity d/dx sin(x) = cos(x). A good student of mathematics, then, will not have to think much about this problem at all, for a decent knowledge of mathematical identities will provide an immediate answer. Other identities which are important to remember are those for exponential equations, such as log(AB) = logA + logB, or for polynomial equations, such as a2 + b2 = (a+b)(a-b). One of the most famous identities, though not necessarily one the average student of math is likely to come across for some time, is that which is known as Euler's Identity, and which is simply: eip + 1 = 0 The internet is a great place to search for mathematical identities, and for anyone who is interested in continuing to learn mathematics, memorization of these key tools is a great place to start.
The copyright of the article Mathematical Identities in Math/Chaos Theory is owned by Isaac M. McPhee. Permission to republish Mathematical Identities in print or online must be granted by the author in writing.
|
|
|
|
