complex numbers notes

Study Guide for Lecture 1: The Complex Numbers. Given a quadratic equation : … 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. A complex number z is a purely real if its imaginary part is 0. Complex Numbers and Powers of i The Number - is the unique number for which = −1 and =−1 . ∴ i = √ −1. Chapter 01: Complex Numbers Notes of the book Mathematical Method written by S.M. 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 i is raised to any whole number power, the result is always 1, i, –1 or – i. However, if BOTH numbers are negative this won’t work anymore as the following shows. 1 Complex Numbers P3 A- LEVEL – MATHEMATICS (NOTES) 1. A complex number is an element $(x,y)$ of the set $$ \mathbb{R}^2=\{(x,y): x,y \in \mathbb{R}\} $$ obeying the … 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} $. Here, x is the real part ofRe(z) and y is the imaginary part or Im (z) of the complex number. 3+5i √6 −10i 4 5 +i 16i 113 3 + 5 i 6 − 10 i 4 5 + i 16 i 113. The last two probably need a little more explanation. You also learn how to rep-resent complex numbers as points in the plane. ( 4 Questions) 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. Complex Number. In fact, for any complex number z, its conjugate is given by z* = Re (z) – Im (z). The same will hold for square roots of negative numbers. However, that is not the only possibility. There is one final topic that we need to touch on before leaving this section. 7. COMPLEX NUMBER. We next need to address an issue on dealing with square roots of negative numbers. 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. But first equality of complex numbers must be defined. Complex Numbers Deﬁnitions. 9. This shows that, in some way, $i$ is the only “number” that we can square and get a negative value. A number of the form z = x + iy, where x, y ∈ R, is called a complex number. Standard form does not allow for any $i$'s to be in the denominator. Here are some examples of complex numbers and their conjugates. 6. The quantity √-1 is an imaginary unit and it is denoted by ‘i’ called Iota. Show Mobile Notice Show All Notes Hide All Notes. 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. Demoivre’s Theorem. Equality In Complex Number. you are probably on a mobile phone). √-2, √-5 etc. It is completely possible that a a or b b could be zero and so in 16 i i the real part is zero. Complex Numbers Class 11 Notes. Yusuf, A. Majeed and M. Amin, published by Ilmi Kitab Khana, Lahore - PAKISTAN. 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. For instance. So, let’s start out with some of the basic definitions and terminology for complex numbers. POINTS TO KNOW. 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. When faced with them the first thing that you should always do is convert them to complex number. Why is this important enough to worry about? z = x+ iy ↑ real part imaginary part. 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! When the real part is zero we often will call the complex number a purely imaginary number. These are all examples of complex numbers. where $a$ and $b$ are real numbers and they can be anything, positive, negative, zero, integers, fractions, decimals, it doesn’t matter. Previous section Operations With Complex Numbers Next section Complex Roots. 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. In the last example (113) the imaginary part is zero and we actually have a real number. All powers if $i$ can be reduced down to one of four possible answers and they repeat every four powers. Here’s one final multiplication that will lead us into the next topic. i.e., x = Re (z) and y = Im (z) Purely Real and Purely Imaginary Complex Number. Here, p and q are real numbers and $i=\sqrt{-1}$. 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. A complex number can be noted as a + ib, here “a” is a real number and “b” is an imaginary number. Note that if we square both sides of this we get. (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. 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. 3 + 4i is a complex number. Every Shakespeare Play Summed Up in a Quote from The Office; We also rearranged the order so that the real part is listed first. 4. 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. i.e., Im (z) = 0. The plane in which one plot these complex numbers is called the Complex plane, or Argand plane. Mobile Notice. 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. 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