Write 1 + √3i in modulus-argument form

In order to understand this question we must define what modulus-argument form is. The modulus of a complex number is its distance from the origin (0,0) on the Argand Diagram. It is written as |z|. The argument of a complex number is the angle subtended anticlockwise between the x axis and a line drawn from the origin to the complex number. This is written where the angle is between -π and π radians (where the angle is below the x axis, it is written as a negative because the angle is measured clockwise around the origin). It is written as arg(z). If you are unaware of what the Argand diagram is, don’t worry, here is a brief explanation! The Argand diagram is a really useful visual aid for the use of complex numbers where they are split into their real and imaginary components. The real part is represented by a value on the x axis and the imaginary part is represented by a value on the y axis. This allows us to visualise the “size” of complex numbers, in other words the modulus. The number can be easily shown on the Argand diagram, with the x (real part) value of 1, and the y (imaginary part) value of √3. To find the modulus of this number which we will now refer to as z, we must effectively find the length of the line drawn from the origin to z. By creating a right angled triangle with the modulus as the hypotenuse, we can see that the other two lengths are 1 and √3. Pythagoras proved that a^2 +b^2=c^2 which we can use to find the length of the hypotenuse: the modulus. So the |z|^2=(1)^2 +(√3)^2 |z|^2=1+3=4 |z|=2 So we have found the modulus to be 2. We have defined that arg(z) is the angle between the x axis and the line from the origin to z. Using trigonometric properties of the right angled triangle drawn we can use the tangent function to find the argument. We know that tan(x)=Opposite/Adjacent. The opposite in this case is √3 and the adjacent is 1. Therefore tan(x)=√3/1=√3 Using the inverse tangent function we find that arg(z)=arctan(√3)= π/3 radians. And there is our final answer, that |z|=2 and arg(z)= π/3. Extension: What is the modulus-argument form of z= -1-2i Answer |z|=√5 arg(z)=-2π/3