![]() As these examples illustrate, the complex numbers share very many of the properties of the real numbers. For any x + yi not equal to 0, there is another complex number u + vi such that ( x + yi)( u + vi) = 1, specifically u + vi = x/( x 2 + y 2) - yi/( x 2 + y 2). Thus the product of two complex numbers is another expression of the ( x + yi)( u + vi) = ( xu + yui + xvi + yvi 2) = ( xu - yv) + ( yu + xv) i Product of x + yi by the real number c is cx + ( cy) i.īut it is also possible to define a multiplication of one complex number with another, giving as the product of two complex numbers The sum of x + yi and u + vi is ( x + u) + ( u + v) i, and the To rules for addition and scalar multiplication for complex To add number pairs and multiply them by real scalars, and this leads Part and the imaginary part of the complex number. The two coordinates of the pair ( x, y) are called the real Plane, so we may say that the complex numbers form a two-dimensionalĬollection. They called this collection the complex numbers.Įach complex number x + yi corresponds to a number pair ( x, y) in the Their system all numbers of the form x + yi, with Numbers and multiply them by real numbers, they had to include in Mathematicians introduced a new symbol, i, with the property ToĬonstruct a larger number system where this equation can be solved, Square of any real number is never negative, it is impossible to findĪ real number that solves the equation x 2 = -1. Inadequate for solving some simple equations. Sufficient for very many purposes in algebra and geometry, they are N-tuples of real numbers, starting with the points on a Up to this point, we have considered pairs, triples, and ![]() All you're left with is a six times I.Complex Numbers as Two-Dimensional Numbers Complex Numbers as Two-Dimensional Numbers Cosine of 90 degrees is equal to zero and sine of 90 degrees is equal to one. Now we know what the cosineĪnd sine of 90 degrees is. And so we can rewrite this here, or we can rewrite the product as w sub one times w sub two is equal to its modulus six times cosine And so if we wanted to give it an angle between zero and 360 degrees, if we just subtract 360 from that, that is going to be equal to 90 degrees. So this is equal to 450 degrees, which is more than a complete rotation. To add these two angles, that gets you to 450 degrees. W sub one times w sub two, if we start at w sub two'sĪrgument, 120 degrees and then we rotate itīy w sub one's argument, well then you're going Scale up w two's modulus by w one's modulus. Modulus of w one times w two? Well, we're just going to So what is going to happen? Well, let me write it here. So let's imagine that weĪre transforming w two by multiplying it by w one. ![]() ![]() Now, when you multiply complex numbers you could view as one And that the argument of w sub two is going to be equal to, we can see that right And by the same line of reasoning, we know that the modulus of w And we know that the argument of w sub one is equal to 330 degrees. That it's written here that the modulus of w sub ![]() Out what is the product? Pause this video and see Two different complex numbers here, and we want to figure Multiply both the numerator and denominator by (cos(y) - i*sin(y)) I'll show here the algebraic demonstration of the multiplication and division in polar form, using the trigonometric identities, because not everyone looks at the tips and thanks tab.īut I also would like to know if it is really correct.Ī*(cos(x) + i*sin(x)) * b*(cos(y) + i*sin(y))Ī*b*(cos(x) + i*sin(x))*(cos(y) + i*sin(y))Īb*(cos(x) + i*sin(x))*(cos(y) + i*sin(y))Īb*(cos(x) cos(y) + cos(x)*sin(y)*i + cos(y)*sin(x)*i + sin(x)*sin(y)*i^2)Īb(cos(x) cos(y) - sin(x)*sin(y) + i(cos(x) sin(y) + cos(y)*sin(x)) ![]()
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