Full teaching notes for a2 complex numbers teaching. Most of the theorems contained in these notes are likewise contained in churchill and. Cook liberty university department of mathematics and physics spring 2010. Note that real numbers are complex a real number is simply a complex number with zero imaginary part. Most people think that complex numbers arose from attempts to solve quadratic equations, but actually it was in connection with cubic equations they. Alevel mathematicsocrfp1complex numbers wikibooks, open. The product of complex conjugates is always a real number. I hope you read last night by way of preparation for that, but since thats something were going to have to do a lot of a differential equations, so. Read all the important points and chapterwise notes on iitjee. A complex number zis an ordered pair of real numbers a,b. Graphically the absolute value of complex number is the distance from the origin to the complex point in the complex plane. The product of a real number and an imaginary number is an imaginary number.
Here we introduce a number symbol i v1 or i2 1 and we may deduce i3 i i4 1. Hence, product of the conjugate pair of complex numbers is a real number. Complex number system study material for iit jee askiitians. Download englishus transcript pdf i assume from high school you know how to add and multiply complex numbers using the relation i squared equals negative one. More precisely, it can be shown exercise 22 that if and only if z is a real number.
Complex numbers and powers of i the number is the unique number for which. Chalkboard photos, reading assignments, and exercises pdf 1. Complex numbers are represented geometrically by points in the plane. The number i is often called the imaginary unit of c although nowadays it is hard to see anything imaginary. These are techniques that do not involve anything deep in algebra or analysis. These are my teaching notes for the cie a2 pure complex numbers unit. To read the notes, click the links below for connection to the appropriate pdf files. The product of two imaginary numbers is a real number. Note that real numbers are complex a real number is. Algebra revision notes on complex numbers for iit jee. Crystallographic data centre upon quoting the deposition number.
Now we need to discuss the basic operations for complex numbers. In spite of this it turns out to be very useful to assume that there is a number ifor which one has 1 i2. In other words, if c and d are real numbers, then exactly one of the. Im a little less certain that you remember how to divide them. Definition the modulus of a complex number, denoted by, is defined by 1 if, then is a real number, and. Use pythagorean theorem to determine the absolute value of this point. A complex number z is a purely real if its imaginary part is 0. In fact, it just implies that a complex number is a combination of a real and imaginary number. Full teaching notes for a2 complex numbers tes resources. Hence, they would supply you only with the theory part i covered. In fact, for any complex number z, its conjugate is given by z rez imz.
Now, notice when i look at the at any complex number, so, in terms of this, the polar form of a complex number, to draw the little picture again, if here is our complex number, and here is r, and here is the angle theta, so the nice way to write this complex number is r e to the i theta. The argument of a complex number is defined as the angle it makes with the real x axis. We now consider another method of representing complex numbers. The argument for a complex number z, denoted argz, is given by. View notes test 2 notes to print from math 1111 at georgia southern university. A series of lecture notes on the elementary theory of algebraic numbers, using only. It uni es the mathematical number system and explains many mathematical phenomena. In spite of this it turns out to be very useful to assume that there is a. You are normally asked for the principal argument, between and. Weidentify arealnumber x with the complex number x,0. It also enables them to understand the majority of math jokes and memes. It turns many of their previous no solutions into answers, how exciting is that.
Consider a complex number z 1 1 re i if it is multiplied by another complex number w 2 2 rei. Complex numbers and imaginary numbers the set of all numbers in the form a bi, with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. I will try to add examples to expand on what is already in the text. Express your radical solutions in the simplest form. You can click on the pic to view it in another tab to get an enlarged view of the same. If b 0, then the complex number is called an imaginary number figure 10.
Nnn complex catalysts for transfer hydrogenation of ketones. How to prove the real part of a complex number is zero 28 3. The complex plane the real number line below exhibits a linear ordering of the real numbers. If an example below seems vague to you, it is safe to ignore it. The numbers x and y are called respectively real and imaginary parts of complex number z. Here is a way to understand some of the basic properties of c using our knowledge of linear algebra. Complex numbers are built on the concept of being able to define the square root of negative one. This is normally worked out by finding the triangle on the argand diagram with the angle in and using trigonometry. Files available from the acs website may be downloaded for. Files available from the acs website may be downloaded for personal use only. The dirutheniumiinnn complex assembled through 4,4.
Adding, multiplying, dividing, subtracting in rectangular form finding the modulus and argument of a complex number converting between rectangular and polar form finding the square root of a complex number loci of complex numbers ive also included a. This section provides a complete set of lecture notes for the course. Notice that the conjugate of a real number is just itself with no changes. This helps us distinguish it from iused as a variable or index. Also, i will try to give comments about connections to other elds of mathematics where appropriate. The real complex numbers x and y are respectively called the real part and imaginary part of the complex number. I am enclosing a scanned version of my notes on complex numbers. Notes on complex numbers university of british columbia, vancouver yuexian li march 17, 2015 1. Flexible learning approach to physics eee module m3. We would like to show you a description here but the site wont allow us.
We introduce an imaginary number, i defined by now that you know about i, we can solve the ones on the previous page using this complex number system. Traditionally the letters zand ware used to stand for complex numbers. A complex number is any expression that is a sum of a pure imaginary number and a real number. Since many people are unwilling to cede the existence of mathematical objects merely on the basis of. If a complex number z is viewed as a vector in, then the norm or length of the vector is called the modulus of z. Dimeric rutheniumiinnn complex catalysts bearing a pyrazolyl. The argument of a complex number is the angle in radians of the complex number point from the real axis the xaxis, measured counterclockwise. After having given a correct formula of the determination of the volume of a frustum of a pyramid with square base and applied it successfully to the case where the side of the lower base is 10, of the.