HoOrAyy...
2 more days all of us will goin back home...
n celebrates our hari raya in our hometown....
arghh...when r back here,we have to all out again to sit 4 our final exam
Tuesday, September 23, 2008
Tuesday, September 9, 2008
the number i
Dave's Short Course on The Number iAlthough the Fundamental Theorem of Algebra was still not proved in the 18th century, and complex numbers were not fully understood, the square root of minus one was being used more and more.
Analysis, especially calculus and the theory of differential equations, was making great headway. Certain functions, including the trigonometric functions and exponential functions, appear in solutions to integrals and differential equations. Euler (1707-1783) made the observation, here written in modern notation, that
eix = cos x + i sin x
where i denotes √–1. This is an equation which allows you to interpret the exponentiation of an imaginary number ix as having a real part, cos x, and an imaginary part, i sin x. This was an especially useful observation in the solution of differential equations. Because of this and other uses of i, it became quite acceptable for use in mathematics. Euler, a very influential mathematician, recommended the general use of these imaginary numbers in his Introduction to Algebra.
By the end of the 18th century numbers of the form x + yi were in fairly common use by research mathematicians, and it became common to represent them as points in the plane. The standard convention now in use to display them is to place the real numbers, that is, those numbers of the form x + 0i, on the horizontal x-axis, with positive numbers to the right and negative ones to the left. Also, imaginary numbers, that is, those numbers of the form 0 + yi, on the vertical y-axis, where positive values of y are up, and negative ones down. Thus, i is located one unit above 0 (the origin, where the axes meet), and –i is located one unit below 0.
This particular display of numbers of the form x + yi is attributed to various individuals including Wessel, Argand, and Gauss. It was easy to come by, since the usual (x,y)-coordinates for the plane had been used for over a century. Nonetheless, it is a very useful way to understand these numbers.
The Fundamental Theorem of Algebra–proved! Still, at nearly the end of the 18th century, it wasn't yet known what form all the solutions of a polynomial equation might take. Gauss published in 1799 his first proof that an nth degree equation has n roots each of the form a + bi, for some real numbers a and b. Once he had done that, it was known that complex numbers (in the sense of solutions to algebraic equations) were the numbers a + bi, and it was appropriate to call the xy-plane the "complex plane".
In some sense all the historical discussion before Gauss was prehistory of complex numbers. But that's just the history that is useful in understanding the need for complex numbers. Although there are other concepts of numbers that either go beyond complex numbers or include something other than complex numbers, we know that at least no other "numbers" are needed to solve polynomial equations. The use of complex numbers pervades all of mathematics and its applications to science.
Analysis, especially calculus and the theory of differential equations, was making great headway. Certain functions, including the trigonometric functions and exponential functions, appear in solutions to integrals and differential equations. Euler (1707-1783) made the observation, here written in modern notation, that
eix = cos x + i sin x
where i denotes √–1. This is an equation which allows you to interpret the exponentiation of an imaginary number ix as having a real part, cos x, and an imaginary part, i sin x. This was an especially useful observation in the solution of differential equations. Because of this and other uses of i, it became quite acceptable for use in mathematics. Euler, a very influential mathematician, recommended the general use of these imaginary numbers in his Introduction to Algebra.
By the end of the 18th century numbers of the form x + yi were in fairly common use by research mathematicians, and it became common to represent them as points in the plane. The standard convention now in use to display them is to place the real numbers, that is, those numbers of the form x + 0i, on the horizontal x-axis, with positive numbers to the right and negative ones to the left. Also, imaginary numbers, that is, those numbers of the form 0 + yi, on the vertical y-axis, where positive values of y are up, and negative ones down. Thus, i is located one unit above 0 (the origin, where the axes meet), and –i is located one unit below 0.
This particular display of numbers of the form x + yi is attributed to various individuals including Wessel, Argand, and Gauss. It was easy to come by, since the usual (x,y)-coordinates for the plane had been used for over a century. Nonetheless, it is a very useful way to understand these numbers.
The Fundamental Theorem of Algebra–proved! Still, at nearly the end of the 18th century, it wasn't yet known what form all the solutions of a polynomial equation might take. Gauss published in 1799 his first proof that an nth degree equation has n roots each of the form a + bi, for some real numbers a and b. Once he had done that, it was known that complex numbers (in the sense of solutions to algebraic equations) were the numbers a + bi, and it was appropriate to call the xy-plane the "complex plane".
In some sense all the historical discussion before Gauss was prehistory of complex numbers. But that's just the history that is useful in understanding the need for complex numbers. Although there are other concepts of numbers that either go beyond complex numbers or include something other than complex numbers, we know that at least no other "numbers" are needed to solve polynomial equations. The use of complex numbers pervades all of mathematics and its applications to science.
Inverse function by maple
today our class is using maple software again....which we learn about inverse function,by doing this task we can know how to solve tutorial questions no.7......
The important step that have to know is by clicking at tools button and we click again at tutos button,then we click at calculus-single variable again......lastly we click at function inverse......
by doing this step,we can know how to solve complicated inverse function and its graphs....
The important step that have to know is by clicking at tools button and we click again at tutos button,then we click at calculus-single variable again......lastly we click at function inverse......
by doing this step,we can know how to solve complicated inverse function and its graphs....
complex number:conjugate
Complex conjugate
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Geometric representation of z and its conjugate in the complex plane.
In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number
(where a and b are real numbers) is
The complex conjugate is also very commonly denoted by z * . Here is chosen to avoid confusion with the notation for the conjugate transpose of a matrix (which can be thought of as a generalization of complex conjugation). Notice that if a complex number is treated as a matrix, the notations are identical.
For example,
Complex numbers are often depicted as points in a plane with a cartesian coordinate system (see diagram). The x-axis contains the real numbers and the y-axis contains the multiples of i. In this view, complex conjugation corresponds to reflection at the x-axis.
In polar form, however, the conjugate of reiφ is given by re − iφ. This can easily be verified by using Euler's formula.
Pairs of complex conjugates are significant because the imaginary unit i is qualitatively indistinct from its additive and multiplicative inverse − i, as they both satisfy the definition for the imaginary unit: x2 = − 1. Thus in most "natural" settings, if a complex number provides a solution to a problem, so does its conjugate, such as is the case for complex solutions of the quadratic formula with real coefficients.
Contents[hide]
1 Properties
2 Use as a variable
3 Generalizations
4 See also
//
[edit] Properties
These properties apply for all complex numbers z and w, unless stated otherwise.
if w is non-zero
if and only if z is real
for any integer n
Idempotence; i.e the conjugate of the conjugate of a complex number, z is again that number
if z is non-zero
The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.
if z is non-zero
In general, if is a holomorphic function whose restriction to the real numbers is real-valued, and is defined, then
Consequently, if p is a polynomial with real coefficients, and p(z) = 0, then as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs. (See the complex conjugate root theorem article.)
The function from to is a homeomorphism (where the topology on is taken to be the standard topology). Even though it appears to be a "tame" well-behaved function, it is not holomorphic; it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and compatible with the arithmetical operations, and hence is a field automorphism. As it keeps the real numbers fixed, it is an element of the Galois group of the field extension . This Galois group has only two elements: φ and the identity on . Thus the only two field automorphisms of that leave the real numbers fixed are the identity map and complex conjugation.
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Geometric representation of z and its conjugate in the complex plane.
In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number
(where a and b are real numbers) is
The complex conjugate is also very commonly denoted by z * . Here is chosen to avoid confusion with the notation for the conjugate transpose of a matrix (which can be thought of as a generalization of complex conjugation). Notice that if a complex number is treated as a matrix, the notations are identical.
For example,
Complex numbers are often depicted as points in a plane with a cartesian coordinate system (see diagram). The x-axis contains the real numbers and the y-axis contains the multiples of i. In this view, complex conjugation corresponds to reflection at the x-axis.
In polar form, however, the conjugate of reiφ is given by re − iφ. This can easily be verified by using Euler's formula.
Pairs of complex conjugates are significant because the imaginary unit i is qualitatively indistinct from its additive and multiplicative inverse − i, as they both satisfy the definition for the imaginary unit: x2 = − 1. Thus in most "natural" settings, if a complex number provides a solution to a problem, so does its conjugate, such as is the case for complex solutions of the quadratic formula with real coefficients.
Contents[hide]
1 Properties
2 Use as a variable
3 Generalizations
4 See also
//
[edit] Properties
These properties apply for all complex numbers z and w, unless stated otherwise.
if w is non-zero
if and only if z is real
for any integer n
Idempotence; i.e the conjugate of the conjugate of a complex number, z is again that number
if z is non-zero
The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.
if z is non-zero
In general, if is a holomorphic function whose restriction to the real numbers is real-valued, and is defined, then
Consequently, if p is a polynomial with real coefficients, and p(z) = 0, then as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs. (See the complex conjugate root theorem article.)
The function from to is a homeomorphism (where the topology on is taken to be the standard topology). Even though it appears to be a "tame" well-behaved function, it is not holomorphic; it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and compatible with the arithmetical operations, and hence is a field automorphism. As it keeps the real numbers fixed, it is an element of the Galois group of the field extension . This Galois group has only two elements: φ and the identity on . Thus the only two field automorphisms of that leave the real numbers fixed are the identity map and complex conjugation.
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