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Revision Notes on Complex Numbers. So, if we multiply the numerator and denominator by the conjugate of the denominator we will be able to eliminate the $i$ from the denominator. Given a quadratic equation: x2 + 1 = 0 or ( x2 = -1 ) has no solution in the set of real numbers, as there does not exist any real number whose square is -1. ı is not a real number. We also rearranged the order so that the real part is listed first. Now, $\sqrt { - 1} $ is not a real number, but if you think about it we can do this for any square root of a negative number. Conjugate of a complex Number. But for complex numbers we do not use the ordinary planar coordinates (x,y)but All powers if $i$ can be reduced down to one of four possible answers and they repeat every four powers. Just consider complex functions as a kind of slot machine where you put in one complex number ($z$) and the machine will return you another complex number ($f(z)$). Chapter 01: Complex Numbers Notes of the book Mathematical Method written by S.M. Here are some examples of complex numbers. Study Guide for Lecture 1: The Complex Numbers. Any real number is equal to its complex conjugate. Now, if we were not being careful we would probably combine the two roots in the final term into one which can’t be done! Why is this important enough to worry about? = + Example: Z … Here, p and q are real numbers and $i=\sqrt{-1}$. There is a nice general formula for this that will be convenient when it comes to discussing division of complex numbers. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. When the real part is zero we often will call the complex number a purely imaginary number. Now, we gave this formula with the comment that it will be convenient when it came to dividing complex numbers so let’s look at a couple of examples. Complex Numbers Deﬁnitions. %�쏢 5. Algebraic Operations. 1 Complex Numbers P3 A- LEVEL – MATHEMATICS (NOTES) 1. For a complex number z = p + iq, p is known as the real part, represented by Re z and q is known as the imaginary part, it is represented by Im z of complex number z. addition, multiplication, division etc., need to be defined. POINTS TO KNOW. A complex number has a ‘real’ part and an ‘imaginary’ part (the imaginary part involves the square root of a negative number). The one difference will come in the final step as we’ll see. x��[I��A��P��F8�0Hp�f� �hY�_��ef�R��# a;X��̬�~o��zw�s)��W��=��t��4C\MR1��i��|��z�J��M�x��aXD(��:ȉq.��k�2��_F�� H�5߿�S8��>H5qn��!F��1-��M�H��{��z�N��=��%�g�tn��Jq��(��!�#C�&�,S��Y�\%�0��f��?�l)�W�� eMgf�� Show Mobile Notice Show All Notes Hide All Notes. The conjugate of the complex number $a + bi$ is the complex number $a - bi$. The complex number comprised of the symbol “i” which assures the condition i 2 = −1. The last two probably need a little more explanation. i.e., x = Re (z) and y = Im (z) Purely Real and Purely Imaginary Complex Number. So, if we just had a way to deal with $\sqrt { - 1} $ we could actually deal with square roots of negative numbers. For a second let’s forget that restriction and do the following. So, to deal with them we will need to discuss complex numbers. That can and will happen on occasion. Now, this is where the small difference mentioned earlier comes into play. It turns out that we can actually do the same thing if one of the numbers is negative. Complex Numbers Class 11 Notes. We’ll start with addition and subtraction. (Note: and both can be 0.) The same will hold for square roots of negative numbers. It is completely possible that a a or b b could be zero and so in 16 i i the real part is zero. If the complex number a+ib=x+iy, then a=xand b= 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. In the radicals section we noted that we won’t get a real number out of a square root of a negative number. A complex number is any number which can be written as a+ib where a and b are real numbers and i=√−1 is an imaginary number. ... Complex numbers and complex conjugates. Just as R is the set of real numbers, C is the set of complex numbers.Ifz is a complex number, z is of the form z = x+ iy ∈ C, for some x,y ∈ R. e.g. Rotation theorem. Next let’s take a look at multiplication. So, there is a general rule of thumb in dealing with square roots of negative numbers. Previous section Operations With Complex Numbers Next section Complex Roots. If z= a+ bithen ais known as the real part of zand bas the imaginary part. Complex numbers are often denoted by z. Don’t get excited about it when the product of two complex numbers is a real number. You will see that, in general, you proceed as in real numbers, but using i 2 =−1 where appropriate. Complex Number – any number that can be written in the form + , where and are real numbers. 3 + 4i is a complex number. The plane in which one plot these complex numbers is called the Complex plane, or Argand plane. For instance, $\sqrt { - 9} $ isn’t a real number since there is no real number that we can square and get a NEGATIVE 9. 9. As we noted back in the section on radicals even though $\sqrt 9 = 3$ there are in fact two numbers that we can square to get 9. Let i2 = −1. Integral Power of IOTA (i) i = √-1, i2 […] If $a$ and $b$ are both positive numbers then. This shows that, in some way, $i$ is the only “number” that we can square and get a negative value. Standard form does not allow for any $i$'s to be in the denominator. Complex Numbers and Powers of i The Number - is the unique number for which = −1 and =−1 . Here’s one final multiplication that will lead us into the next topic. ( 4 Questions) So, even if the number isn’t a perfect square we can still always reduce the square root of a negative number down to the square root of a positive number (which we or a calculator can deal with) times $\sqrt { - 1} $. Notice that to officially put the answer in standard form we broke up the fraction into the real and imaginary parts. This is termed the algebra of complex numbers. So, we need to get the $i$'s out of the denominator. In the last example (113) the imaginary part is zero and we actually have a real number. So, when taking the square root of a negative number there are really two numbers that we can square to get the number under the radical. a $\left( { - 4 + 7i} \right) + \left( {5 - 10i} \right) = 1 - 3i$, b $\left( {4 + 12i} \right) - \left( {3 - 15i} \right) = 4 + 12i - 3 + 15i = 1 + 27i$, c $5i - \left( { - 9 + i} \right) = 5i + 9 - i = 9 + 4i$. We can square both 3 and -3. Argument of a Complex Number. Traditionally the letters zand ware used to stand for complex numbers. Due to the nature of the mathematics on this site it is best views in landscape mode. So, let’s work this problem the way it should be worked. %PDF-1.4 If we follow this rule we will always get the correct answer. You also learn how to rep-resent complex numbers as points in the plane. Now, I guess in the time remaining, I'm not going to talk about in the notes, i, R, at all, but I would like to talk a little bit about the extraction of the complex roots, since you have a problem about that and because it's another beautiful application of this polar way of writing complex numbers. That is, the product rule for radicals does not hold (in general) with imaginary numbers. This is actually fairly simple if we recall that a complex number times its conjugate is a real number. ( 6 Questions) Quadratic Equations with Complex Roots. √-2, √-5 etc. There is one final topic that we need to touch on before leaving this section. stream The rule of thumb given in the previous example is important enough to make again. a + 0ı = a - 0ı = a. Notice that the conjugate of a real number is just itself with no changes. It can be done in the same manner as the previous ones, but there is a slightly easier way to do the problem. The last two probably need a little more explanation. But first equality of complex numbers must be defined. &�06Sޅ/��wS{��JLFg�@*�c�"��vRV��i��&9hX I�A�I��e�aV��gT+��KɃQ��ai��*�lE��B��` �aҧiPB��a�i�`�b��4F.-�Lg�6��+i�#2M� ��8�ϴ�sSV��,,�ӳ��+�L�TWrJ��t+��D�,�^��L� #g�Lc$��:��-��/V�MVV��*��q9�r{�̿�AF��{��W�-e��v�4=Izr0��Ƌ�x�,Ÿ�� =_{B~*-b�@�(�X�(��De�2�k�,��o�-uQ��Ly�9�{/'��) �0(R�w��/V�2C�#zD�k��\�vq$7�� a is the real part of the complex number and b is the imaginary part of the complex number. Let 2=−බ ∴=√−බ Just like how ℝ denotes the real number system, (the set of all real numbers) we use ℂ to denote the set of complex numbers. Every Shakespeare Play Summed Up in a Quote from The Office; So, let’s start out with some of the basic definitions and terminology for complex numbers. The natural question at this point is probably just why do we care about this? <> 3. Here we introduce a number (symbol ) i = √-1 or i2 = -1 and we may deduce i3 = -i i4 = 1 Now solution of x2 + 1 = 0 x2 = -1 where $a$ and $b$ are real numbers and they can be anything, positive, negative, zero, integers, fractions, decimals, it doesn’t matter. Demoivre’s Theorem. So, when we multiply a complex number by its conjugate we get a real number given by. Problem : What complex number is equal to its complex conjugate? Complex Numbers Questions for Leaving Cert Honours Level Maths; Addition, Subtraction, Multiplication of Complex Numbers ( 3 Questions) Conjugate/Division of Complex Numbers ( 4 Questions) Equality of Complex Numbers ( 5 Questions) Argand Diagram and Modulus. Complex Numbers Summary Academic Skills Advice What does a complex number mean? This one is a little different from the previous ones since the denominator is a pure imaginary number. We also won’t need the material here all that often in the remainder of this course, but there are a couple of sections in which we will need this and so it’s best to get it out of the way at this point. For instance. In fact, for any complex number z, its conjugate is given by z* = Re (z) – Im (z). The standard form of a complex number is. While this ... introducing complex numbers by Cardano, who published the famous formula for solving cubic equations in 1543, after learning of the solu- 30 0 obj INDEX. A complex number z can thus be identified with an ordered pair ((), ()) of real numbers, which in turn may be interpreted as coordinates of a point in a two-dimensional space. One important thing to remember is that i2 1 Example - w1 5 2i w2 3 5i Find w1w2 Find iw1 w1w2 (5 2i)(3 5i) Replace w1 and w2 with the associated complex numbers … We write a=Rezand b=Imz.Note that real numbers are complex – a real number is simply a complex number with zero imaginary part. 8. This can be a convenient fact to remember. Equality In Complex Number. The answer is that, as we will see in the next chapter, sometimes we will run across the square roots of negative numbers and we’re going to need a way to deal with them. What are Complex Numbers? CBSE Class 11 Maths Notes Chapter 5 Complex Numbers and Quadratic Equations Imaginary Numbers The square root of a negative real number is called an imaginary number, e.g. A number of the form z = x + iy, where x, y ∈ R, is called a complex number. Complex Numbers (NOTES) 1. Complex Number. COMPLEX NUMBER. z= a+ bi a= Re(z) b= Im(z) r θ= argz = | z| = √ a2 + b2 Figure 1. 7. Here, x is the real part ofRe(z) and y is the imaginary part or Im (z) of the complex number. j�� Z�9��w�@�N%A��=-;l2w��?>�J,}�$H��W/!e�)�]��j�T�e��|�R0L=��ز��&��^��ho^A��>��EX�D�u�z;sH��>R� i�VU6��-�tke��J�4e��.ꖉ ��JL��Sv�D��H��bH�TEمHZ��. The standard form for complex numbers does not have an ${i^2}$ in it. (i) Suppose Re(z) = x = 0, it is known as a purely imaginary number (ii) Suppose Im(z) = y = 0, z is known as a purely real number. Again, with one small difference, it’s probably easiest to just think of the complex numbers as polynomials so multiply them out as you would polynomials. Notes on Complex Numbers University of British Columbia, Vancouver Yue-Xian Li March 17, 2015 1. This however is not a problem provided we recall that. Given a quadratic equation : … View Notes - P3- Complex Numbers- Notes.pdf from MATH 9702 at Sunway University College. Although it is rarely, if ever, used in some fields of math, it comes in very handy when calculating the roots of polynomials, because the quadratics that were previously irreducible over the reals are reducible over the complex numbers. =*�k�� N-3՜�!X"O]�ER� �� In other words, we can break up products under a square root into a product of square roots provided both numbers are positive. 4. Since any complex number is speciﬁed by two real numbers one can visualize them by plotting a point with coordinates (a,b) in the plane for a complex number a+bi. These are all examples of complex numbers. 18.03 LECTURE NOTES, SPRING 2014 BJORN POONEN 7. Note that . ��Y��OIkzp�7F��5�'��0p��p��X�:��~:�ګ�Z0=��so"Y��aT�0^ ��'ù��F\Ze�4��'�4n� ��']x`J�AWZ��_�$�s��ID��0�I�!j ��=��!dP�E�d* ~�>?�0\gA��2��AO�i j|�a$k5)i`/O��'yN"��i3Y��E�^ӷSq��ZO�z�99ń�S��MN;��< Chalkboard Photos, Reading Assignments, and Exercises (PDF - 1.8MB)Solutions (PDF - 5.1MB)To complete the reading assignments, see the Supplementary Notes in the Study Materials section. But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. 6. We have created short notes of Complex Numbers for guys so that you start with your preparation! In this case we will FOIL the two numbers and we’ll need to also remember to get rid of the ${i^2}$. Consider the following example. Well the reality is that, at this level, there just isn’t any way to deal with $\sqrt { - 1} $ so instead of dealing with it we will “make it go away” so to speak by using the following definition. We next need to address an issue on dealing with square roots of negative numbers. Multiplying complex numbers – Multiplying with complex numbers is very similar to multiplying in algebra by splitting the first bracket. The last topic in this section is not really related to most of what we’ve done in this chapter, although it is somewhat related to the radicals section as we will see. When in the standard form $a$ is called the real part of the complex number and $b$ is called the imaginary part of the complex number. It will be important to remember this later on. It is represented as z. Consider the following. However, if BOTH numbers are negative this won’t work anymore as the following shows. However, that is not the only possibility. This number is NOT in standard form. Mobile Notice. Let’s just take a look at what happens when we start looking at various powers of $i$. These notes are about complex analysis, the area of mathematics that studies analytic functions of a complex variable and their properties. Next Section . complex numbers In this chapter you learn how to calculate with complex num-bers. i.e., Im (z) = 0. A complex number is an element $(x,y)$ of the set $$ \mathbb{R}^2=\{(x,y): x,y \in \mathbb{R}\} $$ obeying the … So, thinking of numbers in this light we can see that the real numbers are simply a subset of the complex numbers. Here are some examples of complex numbers. When faced with them the first thing that you should always do is convert them to complex number. The easiest way to think of adding and/or subtracting complex numbers is to think of each complex number as a polynomial and do the addition and subtraction in the same way that we add or subtract polynomials. A complex number z is a purely real if its imaginary part is 0. So all that we need to do is distribute the 7$i$ through the parenthesis. A … Complex numbers are mentioned as the addition of one-dimensional number lines. Example for a complex number: 9 + i2. They constitute a number system which is an extension of the well-known real number system. We can summarize this up as a set of rules. and so if we square -3$i$ we will also get -9. = + ∈ℂ, for some , ∈ℝ However, we will ALWAYS take the positive number for the value of the square root just as we do with the square root of positive numbers. Complex numbers are often denoted by z. ∴ i = √ −1. Deﬁnition 2 A complex number3 is a number of the form a+ biwhere aand bare real numbers. Note that the parentheses on the first terms are only there to indicate that we’re thinking of that term as a complex number and in general aren’t used. In other words, it is the original complex number with the sign on the imaginary part changed. 10. So, in each case we are really looking at the division of two complex numbers. If we were to multiply this out in its present form we would get. The next topic that we want to discuss here is powers of $i$. Note that if we square both sides of this we get. Few Examples of Complex Number: 2 + i3, -5 + 6i, 23i, (2-3i), (12-i1), 3i are some of the examples of complex numbers. Here are some examples of complex numbers and their conjugates. From the section on radicals we know that we can do the following. For instance. Now we need to discuss the basic operations for complex numbers. z = x+ iy ↑ real part imaginary part. Using this definition all the square roots above become. When i is raised to any whole number power, the result is always 1, i, –1 or – i. complex number. eSaral helps the students by providing you an easy way to understand concepts and attractive study material for IIT JEE which includes the video lectures & Study Material designed by expert IITian Faculties of KOTA. As we saw earlier $\sqrt { - 9} = 3\,i$. 3+5i √6 −10i 4 5 +i 16i 113 3 + 5 i 6 − 10 i 4 5 + i 16 i 113. Having introduced a complex number, the ways in which they can be combined, i.e. Representation Of A Complex Number. As we started by defining the algebraic operations on complex numbers, all functions that are compositions of these basic algebraic operations are complex functions. As with square roots of positive numbers in this case we are really asking what did we square to get -9? Paul's Online Notes Home / Complex Number Primer. Complex Numbers Class 11 – A number that can be represented in form p + iq is defined as a complex number. Now we also saw that if $a$ and $b$ were both positive then $\sqrt {ab} = \sqrt a \,\sqrt b $. 1. Can you see the pattern? It is completely possible that $a$ or $b$ could be zero and so in 16$i$ the real part is zero. The numbers x and y are called respectively real and imaginary parts of complex number z. Complex numbers are built on the concept of being able to define the square root of negative one. Take a Study Break. Imaginary Number – any number that can be written in the form + , where and are real numbers and ≠0. Multiplication of complex numbers will eventually be de ned so that i2 = 1. The main idea here however is that we want to write them in standard form. Modulus of a Complex Number. 1 Complex numbers and Euler’s Formula 1.1 De nitions and basic concepts The imaginary number i: i p 1 i2 = 1: (1) Every imaginary number is expressed as a real-valued multiple of i: p 9 = p 9 p 1 = p
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