11/25/15 Calculus And Analytical geometry

Calculus and Analytical Geometry.

Topic: Complex Number:

Submitted to: Sir Azeem Inam

Dated:11/29/15

Group Members

1. Aqib Mohammad Usman

2. Jahanzaib Usman

3. Daniyal Ali Channa

4. Mohammad Hamza

5. Syed Asad Ali6. Kamran Ul Hassan Bhatti.

What is COMPLEX NUMBER: - COMPLEX NUMBER IS BASICALLY THE COMBINATION OF TWO PARTS ONE IS REAL AND THE OTHER IS AN IMAGINARY NUMBER.THE COMPLEX NUMBER IS APPLIED ON AN INTRICATE MATH SUCH AS ALGEBRA BUT IT ALSO CAN APPLY IN OUR REALL LIFE TOO.ESPECIALLY IT MAINLY USED IN ELCTRONICS AND ELCTROMECHANISM.THE STANDARD FORMAT TO WRITE THE COMPLEX UMBER IS C+ID IN IT C IS REAL AND THE OTHER I=ONE IS IMAGINARY PART.THE ACTUALL MEANING OF COMPLEX NUMBER IS THAT IN WHICH TWO TYPES OF NUMBER COMBINE TOGETER TO FORM COMPLEX NUMBER .

SIMPLY.IT LOOKS LIKE AN BUILDING IN IT WHERE WE NEED SO MANY SMALL BLOCKS TO BUILD IT.THOSE NUMBER WHICH ARE PLOTTED ON HORIZONTAL LINE ARE KNOWNS AS REAL NUMBER LIKE FRACTION,INTEGER AND THE OTHER WHICH WE ONLY JUST IMAGINE THROUGH IMAGINATION FOR EXAMPLE THE (-) IN THE √

Parts of complex number: - AS WE STUDY MOSTLY ADDITON, SUBTRACTION, MULTIPLICATION AND DIVISION THESE ALL ARE THE PARTS OF COMLEX NUMBER.

NOW THE CONCEPT OF ABSOLUTE ZERO VALUE IN COMPLEX NUMBER

THE DISTANCE FROM ZERO ON THE NUMBER LINE BECAUSE IN COMPLEX NUMBER THERE IS REAL AND IMAGINARY PART BOTH ARE EXISTS.HOWER THE DISTANCE MESAURED FROM THE ZERO IS ABSOLUTE VALUE ON X –AXIS IN HORIZONTAL AND VERTICAL Y-AXIS

WHY WE USE COMPLEX NUMBER IT IS MOSTLY USED IN QUADRATIC FORMULA

a x2+bx+c=0.so if there is an (-) in the √ so we have to apply complex

number to simply that.

The plane with all the representations of the complex numbers is called the Gauss-plane.With the complex number a + bi corresponds just one vector OP or P.

The image points of the real numbers 'a' are on the x-axis. Therefore we say that the x-axis is the real axis.

The image points of the 'pure imaginary numbers' 'bi' are on the y-axis. Therefore we say that the y-axis is the imaginary axis.

Complex number: A complex number is that number that can be expressed in the form ofa+bi in which a is the real part of a complex number and b is the imaginary part of the complex number where a and b are real numbers.If b=0 then the number a+bi=a is real number and if b is not equal to 0 then the number a+bi is an imaginary number.The imaginary number I is used to Two complex numbers are equal if their real parts are equal and their imaginary parts are equal if a+bi = c+di then a=c and b=d.The set of complex numbers is two

dimensional and a coordinate plane is required to show in the graphical form.The complex number a+bi is identified with the point (a,b) in the complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.A complex number can be represented as a pair of numbers(a,b) they form a diagram called an argand diagram that representing the complex plane.A complex number a+bi is can be written in the form of z=r (costhita + i sinthita) where r is the modulus and thita is the argument of the complex number.

DIAGRAM OF COMPLEX PLANE:

USES OF COMPLEX NUMBER: Complex numbers have many uses in science, mathematics, engineering, statics etc. A complex number are used in solving different equations with function of complex root.

A complex numbers could be used to represent the position of an object in a two dimensional plane.

The complex equation is a basic formula for designing air foils, airplane wings and figuring out flow forces around a circular object .

xxxxxxxxxxxxxxxxxxxxxxxxxxx

Complex Number:A complex number is a number that can be expressed in the form of a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation i2 = −1.In this expression, a is the real part and b is the imaginary part of the complex number. They are used in a variety of computations and situations. Complex numbers are useful for our purposes because they allow us to take the square root of a negative number and to calculate imaginary roots.

A complex number is any number that contains both a real and imaginaryterm.

Add/subtract the real terms together, and add/subtract the imaginary termsof each complex number together. The result will be a complex number.

Treat each complex number as an algebraic term and multiply/divideusing rules of algebra. The result will be a complex number

Put division in fraction form and multiply numerator and denominator bythe conjugate of the denominator

A complex number has a representation in a plane.Simply take an x-axis and an y-axis (orthonormal) and give the complex number a + bi the representation-point P with coordinates (a,b).The point P is the image-point of the complex number (a,b).

Complex Number:COMPLEX NUMBER IS BASICALLY THE COMBINATION OF TWO PARTS ONE IS REAL AND THE OTHER IS AN IMAGINARY NUMBER.THE COMPLEX NUMBER IS APPLIED ON AN INTRICATE MATH SUCH AS ALGEBRA BUT IT ALSO CAN APPLY IN OUR REALL LIFE TOO.ESPECIALLY IT MAINLY USED IN ELCTRONICS AND ELCTROMECHANISM.THE STANDARD FORMAT TO WRITE THE COMPLEX UMBER IS C+ID IN IT C IS REAL AND THE OTHER I=ONE IS IMAGINARY PART.THE ACTUALL MEANING OF COMPLEX NUMBER IS THAT IN WHICH TWO TYPES OF NUMBER COMBINE TOGETER TO FORM COMPLEX NUMBER .

SIMPLY.IT LOOKS LIKE AN BUILDING IN IT WHERE WE NEED SO MANY SMALL BLOCKS TO BUILD IT.THOSE NUMBER WHICH ARE PLOTTED ON HORIZONTAL LINE ARE KNOWNS AS REAL NUMBER LIKE FRACTION,INTEGER .

Application of Complex Number:IT IS MOSTLY USED IN QUADRATIC FORMULA a x2+bx+c=0.so if there is an (-) in the √ so we have to apply complex number to simply that.

The plane with all the representations of the complex numbers is called the Gauss-plane.With the complex number a + bi corresponds just one vector OP or P.

The image points of the real numbers 'a' are on the x-axis. Therefore we say that the x-axis is the real axis.

The image points of the 'pure imaginary numbers' 'bi' are on the y-axis. Therefore we say that the y-axis is the imaginary axis.

COMPLEX NUMBERA complex number is the combination of real and imaginary parts which is represented as a+bi where a is the real part or real number and b is also a real part or real number but due to iota it becomes imaginary part. All the numbers are real but imaginary numbers are those which give negative number when we square it .it is just because we need it here in complex number. Like 1 is the real number but in imaginary it is i (iota) and if we square root of 1 it gives

The above wording is simply express in the diagram to understand easily

If we conjugate the complex equation a+bi the sign of the imaginary part will be changed

For example:

You can see in the picture first we have a sign positive with imaginary part and when we want conjugate of this the sign is changed of imaginary part into negative this process is reversible in this picture

Complex number can also be plotted on a plane with two perpendicular axes x-axis and y-axis. The x-axis is representing the real axis and y-axis representing the imaginary axis. If we want to know distance we use the formula |z|=√ x2+y2 according to this picture |z|=r an if you want

to find angle we use the formula θ=tan−1( y / x¿)¿ so can easily know

the angle of complex plane.

In polar form of complex number we can easily fin distance and angle the common representation of polar complex number is

z = r cos θ + i r sin θ = r (cos θ + i sin θ)

z=cisθ

We use de moivre’s formula for exponents.

[ r (cos θ + i sin θ) ¿n = rn(cos nθ + i sin nθ)