6
Z = 2 sqrt(3) at an angle of 30 degrees.
"a + bi" is a common way to write a complex number. Here, "a" and "b" are real numbers.Another common way to write a complex number is in polar coordinates - basically specifying the distance from zero, and an angle.
A complex number (z = x + iy) can be plotted the x-y plane if we consider the complex number the point (x,y) (where x is the real part, and y is the imaginary part). So once you plot the complex number on the x-y plane, draw a line from the point to the origin. The Principle Argument of z (denoted by Arg z) is the measure of the angle from the x-axis to the line (made from connecting the point to (0,0)) in the interval (-pi, pi]. The difference between the arg z and Arg z is that arg z is an countably infinite set. And the Arg z is an element of arg z. Why? : The principle argument is needed to change a complex number in to polar representation. Polar representation makes multiplication of complex numbers very easy. z^2 is pretty simple: just multiply out (x+iy)(x+iy). But what about z^100? This is were polar represenation helps us, and to get into this representation we need the principle argument. I hope that helped.
The complex formula log z defines a polar grid. The real part is (xx+yy)^.5 and the imaginary part is atany/x. Plotting this will give you concentric circles and lines radiating from (0,0) at right angles to the circles.
Polar coordinates are another way to write down a location on a two dimensional plane. The first number in a pair of coordinates is the distance one has to travel. The second number in the pair is the angle from the origin.
You should write this in polar notation, i.e., with an angle. Visualize the imaginary axis as being 90° from the real axis. Thus, 6i = 6 (angle) 90°, that is, it has an absolute value of 6, and is at an angle of 90°. The main square root of that is equal to the square root of 6 (angle) 45°. To get the other square root, add 180° degrees to that angle (same absolute value). Now, use your calculator's polar-->rectangular conversion to separate that into real and imaginary parts (if that's what you want).
This is best done if the complex number is in polar coordinates - that is, a distance from the origin, and an angle. Take the square root of the argument (the absolute value) of the complex number; and half the angle.
A COMPLEX NUMBER CAN BE CONVERTED INTO A POLAR FORM LET US TAKE COMPLEX NUMBER BE Z=a+ib a is the real number and b is the imaginary number THEN MOD OF Z IS SQUARE ROOT OF a2+b2 MOD OF Z CAN ALSO BE REPRESENTED BY r . THEN THE MOD AMPLITUDE FORM IS r(cos@Very interesting, but -i is not a complex no. it is a simple (imaginary) no. with no real part.
"a + bi" is a common way to write a complex number. Here, "a" and "b" are real numbers.Another common way to write a complex number is in polar coordinates - basically specifying the distance from zero, and an angle.
For a complex number in polar form with Magnitude, and Angle: (Magnitude)*(cos(angle) + i*sin(angle)) will give the form: a + bi
Yes. Also, for finding any other root (cubic root, fourth root, etc.). The main square root of a complex number can be found easily if it is expressed in polar notation. For example: the square root of 5 at an angle of 46 degrees) the complex number that has the absolute value 5 and an angle of 46 degrees) is equal to the square root of 5, at an angle of 46/2 = 23 degrees.
"a + bi" is a common way to write a complex number. Here, "a" and "b" are real numbers.Another common way to write a complex number is in polar coordinates - basically specifying the distance from zero, and an angle.
If the polar coordinates of a complex number are (r,a) where r is the distance from the origin and a the angle made with the x axis, then the cartesian coordinates of the point are: x = r*cos(a) and y = r*sin(a)
2sqrt2(cos45 + i * sin45)
False apex
A complex number can be thought of as a vector with two components, called the "real part" (usually represented on the horizontal axis), and the "imaginary part" (usually represented on the vertical axis). You can also express the complex number in polar form, that is, with a a length and an angle.
You just plug in the coefficients, and do the normal operations. Of course you have to know how to calculate with complex numbers. Assuming the coefficients are real, you may at some moment get the root of a negative number. Say, for instance, you have the square root of minus 2, then the solution of that part is the square root of plus 2, multiplied by i.If the original coefficients are complex, you may have to calculate the root of a complex number. This is a little more complicated. For this, you convert the complex number to polar coordinates - that is, to a length and an angle. Then, to actually take the square root, you take half the angle, and the square root of the distance - and convert back to rectangular coordinates (separating the real and the imaginary part). (For the second solution, add 180 degrees to the angle.)
From the question, it seems you already calculated the square root, or know how to get it. You can cube complex numbers just like you cube normal numbers: multiply them by themselves; the number must appear three times as a factor. For example, the cube of (2 + i) is (2+i) x (2+i) x (2+i). Another method - usually faster - to calculate any power is to express the complex number in polar form (absolute value and angle). For the specific case of a cube, the cube of such a number is the cube of the absolute value, at an angle that is three times the angle of the original number.