Complex numbers

• i = root(-1).
• An imaginary number only contains imaginary parts and is in the form b i.
• A complex number has both real and imaginary parts and is in the form z = a+bi.
• When using arithmetic with complex numbers, collect real and imaginary parts and solve separately.
• i^2 = -1.
• i^3 = -i.
• i^4 = 1.
• i^5 = i.
• The complex conjugate (Z*) of a complex number a+bi is a-bi.
• If roots of a polynomial are complex, they must exist as a complex conjugate pair.
• On an argand diagram, the x axis is the real axis and the y axis is the imaginary axis.
• The modulus of a complex number z=a+bi, denoted by |z|, is the distance from the origin to the point represented by that number on an argand diagram and the equation is |z| = root(x^2+y^2).
• The argument of a complex number z=a+bi, denoted by arg z, is the angle between the positive real axis and the line joining the point represented by z to the origin. The angle is tan^-1(y/x).
• For a complex number z with |z| = r and arg z = theta, the modulus argument form of z is z = r(cos theta + i sin theta).
• For any two complex numbers a and b, |ab| = |a||b|, |a/b| =|a|/|b|.
• For any two complex numbers a and b, arg(ab) = arg(a)+arg(b), arg(a/b) = arg(a)-arg(b).
• For a complex number a = x+iy, the locus of points z on an argand diagram such that |z-a| = r, is a circle with centre (x, y) and radius r.
• Given two complex numbers a and b, the locus of points z on an argand diagram such that |z-a|=|z-b| is the perpendicular bisector of the segment of a line joining a and b. When = is replaced with >, then it is all the points closer to whichever number the sign is pointing to.
• Given a complex number a, the locus of points z such that arg(z-a) = theta is a half line from a that makes an angle theta with a line from a that is parallel to the real axis.
• Euler's formula is e^(i theta) = cos theta + i sin theta.
• For a complex number z with r = |z| and arg z = theta, the exponential form is z = re^(i theta).
• From Euler's formula, sin theta can be written in terms of exponentials as 1/2i(e^(i theta) - e^(-i theta)).
• From Euler's formula, cos theta can be written in terms of exponentials as 1/2(e^(i theta) + e^(-i theta)).
• For two complex numbers a and b in exponential form, ab = |a| |b| e^(i(arg a + arg b)). and a/b = |a|/|b| e^(i(arg a - arg b)).
• De Moivre's theorem is used to calculate powers of complex numbers and in cartesian form is (r(cos theta + i sin theta))^n = r^n (cos (ntheta) + i sin (ntheta)) and in exponential form is (re^(i theta))^n = r^n e^(i ntheta).
• If z = cos theta + i sin theta, 2cos(n theta) = z^n + 1/z^n and 2i sin(n theta) = z^n - 1/z^n.
• Finding the n roots of a complex number w is equivalent to solving the equation w = z^n. Use De Moivre along with the fact that r(cos(theta) + i sin(theta)) = r(cos(theta + 2kpi) + i sin(theta + 2kpi) and then try different integer values of k.
• The n roots of a complex number z lie at the vertices of a regular n -gon with centre O.
• An nth root of unity is a solution to the equation z^n = 1.
• If you know one root of a complex number with n roots, you can find the others by multiplying by an nth root of unity.
• The equation for roots of unity is given by w = e^(2pi i/n) where n is the nth root of unity.