All about complex numbers pdf

Complex numbers wont seem complicated any more with these clear, precise student worksheets covering expressing numbers in simplest form, irrational roots, decimals, exponents all the way through all aspects of quadratic equations, and graphing. Because no real number satisfies this equation, i is called an imaginary number. Despite the historical nomenclature imaginary, complex numbers are. It means the two types of numbers, real and imaginary, together form a complex, just like a building complex buildings joined together.

Multiplication and division of complex numbers is easy in polar form. Vii given any two real numbers a,b, either a b or a 0. They constitute a number system which is an extension of the wellknown real number system. Real numbers include things like zero, and one, and zero point three repeating, and pi, and e, and i could keep listing real numbers. Everyone knew that certain quadratic equations, like x2. Despite the historical nomenclatu re imaginary, co mplex numbers are. All these points lie on the circle of radius as shown in figure 9. A if a is a matrix, the product of two complex numbers. Most of your mathematical lives youve been studying real numbers. There are actually n distinct complex solutions to this equation.

Lecture 4 roots of complex numbers characterization of a. You also learn how to represent complex numbers as points in the plane. In spite of this it turns out to be very useful to assume that there is a number ifor which one has 1 i2. For division, students must be able to rationalize the denominator, which includes multiplyin.

All the examples listed here are in cartesian form. Complex numbers are an important part of algebra, and they do have relevance. In spite of this it turns out to be very useful to assume that there is a number ifor which one has. Complex numbers are added, subtracted, and multiplied as with polynomials. The complex numbers may be represented as points in the plane, with. The complex plane the real number line below exhibits a linear ordering of the real numbers. The algebra of complex numbers at this point we have considered only realvalued functions of a real variable. Re is the real axis, im is the imaginary axis, and i satisfies i2. Complex numbers of the form x 0 0 x are scalar matrices and are called real complex numbers and are denoted by. Complex numbers and powers of i metropolitan community college.

We will now examine the complex plane which is used to plot complex numbers through the use of a real axis horizontal and an imaginary axis vertical. Mathematical institute, oxford, ox1 2lb, july 2004 abstract this article discusses some introductory ideas associated with complex numbers, their algebra and geometry. The complex logarithm, exponential and power functions. More generally, it was proved at the end of the 19th century that any extension of the notion of number beyond. Lets say that z is equal to, is equal to the real number five plus the imaginary number three times i. But for complex numbers we do not use the ordinary planar coordinates x,ybut. Traditionally the letters zand ware used to stand for complex numbers.

Complex numbers algebra all content math khan academy. The real number 1 is represented by the point 1,0, and the complex number i is represented by the point 0,1. We dene addition and multiplication for complex numbers in such a way that the rules of addition and multiplication are consistent with the rules for real numbers. Complex numbers extend the idea of the onedimensional number line to the twodimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. Derivatives derivative applications limits integrals integral applications series ode laplace transform taylormaclaurin series fourier series. We dene addition and multiplication for complex numbers in such a way that the rules of addition and multiplication are consistent with the rules for. Whereas the set of all real numbers is denoted by r, the set of all complex. To see this, consider the problem of finding the square root of a complex number. He found a generalization of complex numbers, namely the quaternions cf. The complex numbers may be represented as points in the plane sometimes called the argand diagram. We call x the real part of z and y the imaginary part of z. Vii given any two real numbers a,b, either a b or a all a,b,c.

You can see the real numbers as a subset of the complex numbers therefore. It is dangerous to assume that all of the above relations are valid in the complex plane without modi. Complex numbers 17 3 complex numbers law and order life is unfair. The addition of complex numbers satisfy the following properties. In this chapter we will discuss complex numbers and the calculus of associated functions. The plane in which one plot these complex numbers is called the complex plane, or argand plane. Most people think that complex numbers arose from attempts to solve quadratic equations, but actually it was in connection with cubic equations they. A complex number can be visually represented as a pair of numbers a, b forming a vector on a diagram called an argand diagram, representing the complex plane. A scalar number is the type of mathematical object that people are used to using in everyday life. More generally, the sum of two complex numbers is a complex number. On the other hand, xn a has more solutions than just x n a. Quiz on complex numbers solutions to exercises solutions to quizzes the full range of these packages and some instructions, should they be required, can be obtained from our web. Complex numbers extends the concept of one dimensional real numbers to the two dimensional complex numbers in which two dimensions comes from real part and the imaginary part. But either part can be 0, so all real numbers and imaginary numbers are also complex numbers.

It looks like an arrow, with a starting point, a tip, a definite length, and a definite direction. An introduction to complex numbers homepages of uvafnwi. Complex numbers scavenger hunt all operations this scavenger hunt activity consists of 24 problems in which students practice simplifying, adding, subtracting, multiplying, and dividing complex numbers. The union of the set of all imaginary numbers and the set of all real numbers is the set of complex numbers. So far you have plotted points in both the rectangular and polar coordinate plane. Convert a complex number from polar to rectangular form. We would like to know which of these relations are satis. Complex numbers study material for iit jee askiitians.

Another important operation on complex numbers is the complex conjugate. A vector is a graphical representation of a complex number. To restore justice one introduces new number i, the imaginary unit, such that i2. Here we introduce a number symbol i v1 or i2 1 and we may deduce i3 i i4 1. Notation 4 we write c for the set of all complex numbers. Complex numbers often are denoted by the letter z or by greek letters like a alpha. If we add or subtract a real number and an imaginary number, the result is a complex number. Addition and subtraction of complex numbers is easy in rectangular form. The purely arithmetic theory of complex numbers as pairs of real numbers was introduced by w. A magnification of the mandelbrot setplot complex numbers in the complex plane. Can we similarly solve xn z, where z is a complex number.

Note that real numbers are complex a real number is simply a complex number with no imaginary part. This includes a look at their importance in solving polynomial equations, how complex numbers add and multiply, and how they can be represented. Before i get started on this let me first make it clear that this document is not intended to teach you everything there is to know about. Introduction to complex numbers introduction to the.

The complex numbers c are important in just about every branch of mathematics. Because no real number satisfies this equation, i is called an imaginar y numb er. Show that using these relations and calculating with the same formal rules asindealingwithrealnumbers,weobtainaskew. However, there is still one basic procedure that is missing from the algebra of complex numbers. A complex number can be represented by an expression of the form. That is, all of our work has centered on functions of the form f. These are the numbers that youre kind of familiar with. Combine the following complex numbers and their conjugates. A complex number is a mathematical quantity representing two dimensions of magnitude and direction. Complex numbers are an important part of algebra, and they do have relevance to such things as solutions to polynomial equations. R r, functions which take a real number to a real number. Postscript or pdf produced by some word processors. Graduate students in mathematics may take a semester or two of courses just devoted to the deeper properties of complex numbers.