And we therefore see that the sum of a complex number with its complex conjugate is simply two □. And this makes a lot of sense if we consider the general form. Notice how the sum of a complex number and its conjugate is just a real number. And we add these by adding their real parts and separately adding their imaginary parts, which we often think of like collecting like terms. And this also means that the sum of the two numbers is eight plus two □ plus eight minus two □. This means its complex conjugate is eight minus two □. Our complex number has a real part of eight and an imaginary part of two. We can say that the complex conjugate of a number □, given by □ plus □□, is □ star equals □ minus □□. Remember, □ star is the conjugate of the complex number □, given by eight plus two □. If □ equals eight plus two □, what is □ plus □ star? And we’re going to see now what that might look like. It’s distributive over addition and multiplication. Another beauty of the conjugate is that it shares all of the same properties as any other complex number. So the conjugate of the real number is simply that number. Since we change the sign of the imaginary part to find the conjugate, the conjugate of this number will be □ minus zero □. But it’s one of the form □ plus zero □. In fact, we can say that a real number is of the form □. It’s □.Īnd what about a purely real number? Will this have a conjugate? Well yes. And we can see that we can generalize this and say that the complex conjugate of the conjugate is simply the original number. And in fact, the complex conjugate of the conjugate is four minus six □. Similarly, a complex number, four minus six □ - we’ll have a conjugate of four plus six □. In layman’s terms, the conjugate of a complex number is found by changing the sign for the imaginary part of the number.įor example, a complex number given as three plus two □ - we’ll have a complex conjugate of three minus two □. For a complex number of the form □ plus □□, its conjugate denoted by □ bar or □ star is □ minus □□. Outside of mathematics, it can mean to juxtapose or join together, indicating that a complex number and its conjugate have a special relationship. The definition of the word conjugate is having features in common but opposite or inverse in some particular. Throughout this video, we will look to, where possible, derive general results that can be used in more complicated complex number problems.Įvery single complex number has associated with it another complex number, known as its conjugate. We will begin by defining what we mean by the complex conjugate, before considering their properties and how these can be exploited to help us solve equations involving complex numbers. In this video, we will learn how to use the conjugate of a complex number to evaluate expressions.
0 Comments
Leave a Reply. |