Pdf on complex numbers

Imaginary numbers when squared give a negative result. The numbers a and b are called the real and imaginary parts of z, respectively. Complex numbers and powers of i metropolitan community college. The representation is known as the argand diagram or complex plane. General i p 1, so i2 1, i3 i, i4 1 and then it starts over again.

We can plot such a number on the complex plane the real numbers go leftright, and the imaginary numbers go updown. The complex numbers may be represented as points in the plane, with the real number 1 represented by the point 1. Another important operation on complex numbers is the complex conjugate. However, there is still one basic procedure that is missing from the algebra of complex numbers. Complex numbers are an important part of algebra, and they do have relevance to such things as solutions to polynomial equations. Notes on complex numbers university of british columbia, vancouver yuexian li march 17, 2015 1.

Convert a complex number from polar to rectangular form. 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. The answer is a combination of a real and an imaginary number, which together is called a complex number. This radical approach has fundamentally changed the capabilities of science and engineering to enhance our world through such applications as. First geometric interpretation of negative and complex. Nearly any number you can think of is a real number. Set of variable points denoted by zwhich will form an argument of. Conversely, the sin and cos functions can be expressed in terms of complex exponentials. If w is a nonzero complex number, then the equation z2 w has a so lution z. To emphasize this, recall that forces, positions, momenta, potentials, electric and magnetic. Also, radio waves, sound waves and microwaves have to travel through different media to get to their final destination.

Chalkboard photos, reading assignments, and exercises pdf 1. Complex numbers study material for iit jee askiitians. A line that bisects the cord joining complex numbers a and b in a perpendicular fashion im b re a iii argz. 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. Complex numbers of the form x 0 0 x are scalar matrices and are called. Combine this with the complex exponential and you have another way to represent complex numbers. Complex numbers are useful in representing a phenomenon that has two parts varying at the same time, for example an alternating current. A magnification of the mandelbrot setplot complex numbers in the complex plane. It is here solely to show you a different way to define complex numbers. Complex numbers are added, subtracted, and multiplied as with polynomials. Vii given any two real numbers a,b, either a b or a 0.

Here we introduce a number symbol i v1 or i2 1 and we may deduce i3 i i4 1. Re is the real axis, im is the imaginary axis, and i satisfies i2. The union of the set of all imaginary numbers and the set of all real numbers is the set of complex numbers. Complex numbers algebra all content math khan academy. Any complex number zcan be written as the sum of a real part and an imaginary part. I we add and multiply complex numbers in the obvious way. Complex numbers are ubiquitous in modern science, yet it took mathematicians a long time to accept their existence.

The set of all the complex numbers are generally represented by c. A complex number can be represented by an expression of the form. We call x the real part of z and y the imaginary part of z. Section 0104 sample quiz complex numbers multiple choice identify the choice that best completes the statement or answers the question.

Having introduced a complex number, the ways in which they can be combined, i. Electrical engineers sometimes write jinstead of i, because they want to reserve i for current, but everybody else thinks thats weird. In spite of this it turns out to be very useful to assume that there is a. Complex numbers break all the rules of traditional mathematics by allowing us to take a square root of a negative number. This includes a look at their importance in solving polynomial equations, how complex numbers add and multiply, and how they can be represented. Complex numbers and powers of i the number is the unique number for which. The real complex numbers lie on the xaxis, which is then called the real. Mathematical institute, oxford, ox1 2lb, july 2004 abstract this article discusses some introductory ideas associated with complex numbers, their algebra and geometry. The complex numbers are the only 2dimensional hypercomplex algebra that is a field. Complex number can be considered as the superset of all the other different types of number. The addition of complex numbers satisfy the following properties. The complex exponential is expressed in terms of the sine and cosine by eulers formula 9. Express the answers in the polar and rectangular forms. Youtube workbook 6 contents 6 polar exponential form 41 6.

Combine the following complex numbers and their conjugates. 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. In fact, any negative number has a complex square root. If we add or subtract a real number and an imaginary number, the result is a complex number. The complex numbers may be represented as points in the plane sometimes called the argand diagram.

Complex numbers 5 the complex exponential obeys the usual law of exponents. Complex numbers are built on the concept of being able to define the square root of negative one. The complex plane the real number line below exhibits a linear ordering of the real numbers. The real number 1 is represented by the point 1,0, and the complex number i is represented by the point 0,1.