![]() ![]() De Moivre’s TheoremĬomplex numbers can also be written in polar form. We've been able to simplify the fraction by applying the complex conjugate of the denominator. As stated earlier, the product of the two conjugates will simplify to the sum of two squares.Ģ(1 - 3 j) / (1 + 3 j)(1 – 3 j) = 2(1 - 3 j) / (12 + 32) The denominator of the fraction is now the product of two conjugates. So we will multiply the complex fraction 2 / (1 + 3 j) by (1 – 3 j) / (1 – 3 j) where (1 – 3 j) is the complex conjugate of (1 + 3 j). When dealing with fractions, if the numerator and denominator are the same, the fraction is equal to 1.Īlso, when a fraction is multiplied by 1, the fraction is unchanged. The concept of conjugates would come in handy in this situation. As it is, we can't simplify it any further except if we rationalized the denominator. The above expression is a complex fraction where the denominator is a complex number. To illustrate the concept further, let us evaluate the product of two complex conjugates. They are important in finding the roots of polynomials. However, it has the opposite sign from the imaginary unit.įor example, if x and y are real numbers, then given a complex number, z = x + yj, the complex conjugate of z is x – yj.Ĭomplex conjugates are very important in complex numbers because the product of complex conjugates is a real number of the form x2 + y2. The conjugate of a complex number would be another complex number that also had a real part, imaginary part, the same magnitude. Simply put, a conjugate is when you switch the sign between the two units in an equation. ![]() So j23 = j3 = -j …… as already shown above. A simple shortcut to simplify an imaginary unit raised to a power is to divide the power by 4 and then raise the imaginary unit to the power of the reminder.įor example: to simplify j23, first divide 23 by 4.Ģ3/4 = 5 remainder 3. It always simplifies to -1, - j, 1, or j. Following the examples above, it can be seen that there is a pattern for the powers of the imaginary unit. Understanding the powers of the imaginary unit is essential in understanding imaginary numbers. Hence the square of the imaginary unit is -1. The imaginary unit, j, is the square root of -1. We'll consider the various ways you can simplify imaginary numbers. And since imaginary numbers are not physically real numbers, simplifying them is important if you want to work with them. The nature of problems solved these days has increased the chances of encountering complex numbers in solutions. So when the negative signs can be neutralized before taking the square root, it becomes wrong to simplify to an imaginary number. However, this does not apply to the square root of the following,Īnd not sqrt(-4) * sqrt(-3) = 2 j * sqrt(3)j From this representation, the magnitude of a complex number is defined as the point on the Cartesian plane where the real and the imaginary parts intersect.Ĭare must be taken when handling imaginary numbers expressed in the form of square roots of negative numbers. The x-axis represents the real part, with the imaginary part on the y-axis. For example, a + bj is a complex number with a as the real part of the complex number and b as the imaginary part of the complex number.Ĭomplex numbers are sometimes represented using the Cartesian plane. An imaginary number can be added to a real number to form another complex number. The square of an imaginary number, say bj, is ( bj)2 = - b2. So the square of the imaginary unit would be -1. The imaginary unit is defined as the square root of -1. The difference is that an imaginary number is the product of a real number, say b, and an imaginary number, j. An imaginary number is essentially a complex number - or two numbers added together. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |