100%(2)2 out of 2 people found this document helpful
This preview shows page 1 - 3 out of 9 pages.
2Complex Numbers2.1The imaginary numberi[see Riley 3.1, 3.3]Complex numbers are a generalisation of real numbers. they occur in many branches of mathematics andhave numerous applications in physics.The imaginary number isi=√-1⇔i2=-1The obvious place to see where we have already needed this is in the solution to quadratic equation. Eg.finds the roots ofz2+ 4z+ 5 = 0(z+ 2)2+ 1 = 0(z+ 2)2=-1andz1,2=-2±√-1.So in this case we use the imaginary number and write the solutions asz1,2=-2±i .which is called a complex number. The general form of a complex number isz=x+iywherezis the conventional representation and is the sum of the real partxanditimes the imaginaryparty: these are denoted asRe(z)=xIm(z)=y ,respectively. The imaginary or real part can be zero, so if the imaginary part is, the number is real andhence real numbers are just a subset of complex numbers.Also when using the quadratic solutions formula, we had situations where there were no (real) rootsasb2-4ac <0. we could have solved the above quadratic to get the same results:z1,2=-4±√16-202=-4±√-42=-4±2√-12=-2±i .A complex number may also be written more compactly asz= (x, y) wherexandyare two real numberswhich define the complex number and may be thought of as Cartesian coordinates.1
z=x+iy q r Re(z) Im(z) Argand diagram Recall that in Cartesian coordinatesx=rcosθy=rsinθTherefore we can representzin polar coordinates asz=x+iy=r(cosθ+isinθ).The numberris called the modulus ofz, written as|z|or mod(z). This can be written in terms ofxandyas|z|=px2+y2.The angleθis called the argument ofz, written as arg(z) (or argz) and is defined asarg(z) = tan-1yx.so arg(z) is the angle that the line joining the origin tozon an Argand diagram makes with the positivex-axis. The anti-clockwise direction is taken to be positive by convention.However,θis not unique sinceθ+ 2nπ(nis zero or any integer) are also arguments for the same complexnumber.We therefore define a principal value of a complex number as that value ofθwhich satisfies-π < θ≤π. (it could also be 0< θ≤2π). Also, account must be taken of the signs of