Express in polar form

The polar form of a complex number is another way of representing complex numbers. The polar form of a complex number is represented in terms of modulus and argument of the complex number.

Complex numbers were invented by people and represent over a thousand years of continuous investigation and struggle by mathematicians such as Pythagoras, Descartes, De Moivre, Euler, Gauss, and others. Complex numbers answered questions that for centuries had puzzled the greatest minds in science. We first encountered complex numbers in the section on Complex Numbers. From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction. The first step toward working with a complex number in polar form is to find the absolute value.

Express in polar form

Cartesian coordinates are simple enough, but what about polar coordinates? We have covered this topic in the past, but we can take things a step further by finding polar forms of complex numbers. But how exactly do we do this, and why might this be an important step? Let's find out:. First of all, how are polar coordinates different compared to Cartesian coordinates? Essentially, a set of Cartesian coordinates is the result of two measures: Up and down. But what might happen if we drew a diagonal line straight from the origin to the coordinates? The endpoint might be the same, but the line now measures a single distance from the origin rather than two distances up and down. The single, diagonal line represents our polar coordinates. Recall that we express our polar coordinates with an angle and a distance. This is notably different compared to Cartesian coordinates, which we represent with x and y, in polar we instead use r radius and theta angle. While the x and y coordinates represent the distances from the axes, the distance value in a set of polar coordinates represents the distance to the origin 0,0.

All we need to do is find these two values for our complex number, and we will have our polar coordinates. Privacy Policy.

However, we need to adjust this theta to reflect the real location of the vector, which is in the 2nd quadrant a is negative, b is positive ; a represents the x-axis in the real-imaginary plane, b represents the y-axis. Express the complex number in polar form. The figure below shows a complex number plotted on the complex plane. The horizontal axis is the real axis and the vertical axis is the imaginary axis. The polar form of a complex number is.

If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Search for courses, skills, and videos. Polar form of complex numbers. About About this video Transcript. Learn how to convert a complex number from rectangular form to polar form. This video covers how to find the distance r and direction theta of the complex number on the complex plane, and how to use trigonometric functions and the Pythagorean theorem to make the conversion. Created by Sal Khan. Want to join the conversation? Log in.

Express in polar form

Online Algebra Solver ». IntMath f orum ». We can think of complex numbers as vectors , as in our earlier example. We have met a similar concept to "polar form" before, in Polar Coordinates , part of the analytical geometry section.

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Using these two relations, we get. Copyright holder you represent if other than yourself. Explanation : Remember that the standard form of a complex number is: , which can be rewritten in polar form as:. Already booked a tutor? The components of polar form of a complex number are:. The polar form of a complex number is another way of representing complex numbers. Privacy Policy. Sign in. Maths Program. Simplifying Absolute Value Expressions. Online Tutors.

However, we need to adjust this theta to reflect the real location of the vector, which is in the 2nd quadrant a is negative, b is positive ; a represents the x-axis in the real-imaginary plane, b represents the y-axis. Express the complex number in polar form. The figure below shows a complex number plotted on the complex plane.

We can now calculate r and theta. Become a problem-solving champ using logic, not rules. Let's find out:. The angle measured from the positive real axis to the line segment is called the argument of the complex number, arg z. Complex numbers answered questions that for centuries had puzzled the greatest minds in science. In other words:. While the x and y coordinates represent the distances from the axes, the distance value in a set of polar coordinates represents the distance to the origin 0,0. In order to get our values, we need to refer back to the handy Pythagorean Theorem. The angle theta now becomes Finding Products of Complex Numbers in Polar Form Now that we can convert complex numbers to polar form we will learn how to perform operations on complex numbers in polar form. The rules are based on multiplying the moduli and adding the arguments. The endpoint might be the same, but the line now measures a single distance from the origin rather than two distances up and down. Maths Formulas.