My math

الاثنين، 1 أبريل 2013

The Midrange

The midrange is infrequently used and hardly known. It is a relatively straight-forward measure of central tendency.

It is merely the arithmetic mean amongst the data subset including only the minimum and maximum values of the larger set, ignoring all intermediate values of the set. In other words, it is the arithmetic average of strictly the minimum and maximum values of the set.

Midrange = $\frac{\mathrm{Min(x)} + \mathrm{Max(x)}}{2}$

Complex number

A complex number is a number that can be expressed in the form

a + bi

, where

a

and

b

are real numbers and

i

is the imaginary unit, where

i 2 = -1

.^[1]In this expression,

a

is the real part and

b

is the imaginary part of the complex number. Complex numbers extend the idea of the one-dimensionalnumber line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number

a + bi

can be identified with the point

(a, b)

in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone.

Complex numbers are used in many scientific and engineering fields, including physics, chemistry, biology, economics, electrical engineering,mathematics, and statistics. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th century,^[2] but complex numbers are no more or less "fictitious" or "imaginary" than any other kind of number.

Completing the square

completing the square is a technique for converting a quadratic polynomial of the form

$ax^2 + bx + c\,\!$

to the form

$a(\cdots\cdots)^2 + \mbox{constant}.\,$

In this context, "constant" means not depending on x. The expression inside the parenthesis is of the form (x − constant). Thus one converts ax² + bx + c to

$a(x - h)^2 + k\,$

and one must find h and k.

Quadratic factorization

The term

$x - r\,$

is a factor of the polynomial

$ax^2+bx+c, \$

if and only if r is a root of the quadratic equation

$ax^2+bx+c=0. \$

It follows from the quadratic formula that

$ax^2+bx+c = a \left( x - \frac{-b + \sqrt {b^2-4ac}}{2a} \right) \left( x - \frac{-b - \sqrt {b^2-4ac}}{2a} \right).$

In the special case ( $b^2 = 4ac$ ) where the quadratic has only one distinct root (i.e. the discriminant is zero), the quadratic polynomial can be factored as

$ax^2+bx+c = a \left( x + \frac{b}{2a} \right)^2.\,\!$

Quadratic formula

A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real.

Having

$ax^2+bx+c=0\,$

the roots are given by the quadratic formula^[1]

$x=\frac{-b \pm \sqrt {b^2-4ac}}{2a}$

where the symbol "±" indicates that both

$x=\frac{-b + \sqrt {b^2-4ac}}{2a}\quad\text{and}\quad x=\frac{-b - \sqrt {b^2-4ac}}{2a}$

عجائب الرياضيات *

من هذه العجائب انك إذا ضربت العدد 37 في العدد 3 فإنك تحصل على عدد مكون من ثلاثة أر قام متشابهة وإذا ضربته بمضاعفات العدد 3 فإنك تحصل على متشابهة أيضا .

1*3*37=111
2*3*37=222
3*3*37=333
4*3*37=444
5*3*37=555
6*3*37=666
7*3*37=777
8*3*37=888
9*3*37=999

السبت، 30 مارس 2013

القيمه المطلقه لعدد مركب

يمكن إعادة تعريف القيمة المطلقة لعدد مركب رياضيا من العلاقة

$|a| = \sqrt{a^2}$

والذي يمكن تعميمه كما يلي:

لاي عدد مركب

$z = x + iy,\,$

حيث x وy أعداد حقيقية, القيمة المطلقة لـ z ورمزها |z| تعرف بـ

$|z| = \sqrt{x^2 + y^2}.$