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    Introduction to Complex Numbers Frances Kirwan, based on notes by Balåzs Szendr6i and Richard Earl Michaelmas 2013 The shortest path between two truths in the real domain passes through the complex domain. Jacques Hadamard (1865-1963) 1 Complex numbers: basics Solving quadratic equations is something that mathematicians have been able to do since the time of the Babylonians: as you know, the two roots of the equation are —b b2 — 4ac (1) When b2 - 4ac > 0, these two roots are real and distinct; graphically they are where the curve y ax2 + bx + c cuts the x-axis. When b2 — 4ac = 0, then we have one real root and the curve just touches the x-axis here. But what happens when b2 — 4ac < 0? In this case there are no real solutions to the 2 equation, as no real number squares to give the negative b — 4ac. From the graphical point of view, the curve y = ax2 + br + c lies either entirely above or entirely below the x-axis. If we imagine —1 to exist, and that it behaves much like other numbers, then the two roots of the quadratic ax2 + br + c = 0 can be written in the form where A = —b/2a and B = 4ac — b2/2a are real numbers. (2) Notation 1 We shall from now on write i for —1. This is standard notation amongst mathematicians, though many books, particularly those written for engineers and physicists, use j instead. Definition 2 A complex number is a number of the form where a and b are real numbers. The real number a is known as the real part of z and b as the imaginary part. We write a = Rez and b = Imz. Two complex numbers are equal precisely when their real and = c and b = d. This is called imaginary parts are equal; that is, a + bi = c + di if and only if a comparing real and imaginary parts'. 1
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    Note that we can regard real numbers as complex: a real number is simply a complex number with zero imaginary part. Notation 3 We write C for the set of all complex numbers. Thus 2 Operations on complex numbers We add, subtract, multiply and divide complex numbers much as one would expect. We add and subtract complex numbers by adding their real and imaginary parts: (a + bri) + (c + We can multiply complex numbers by expanding the brackets in the usual fashion and using = (JC + t)Ci + + To divide complex numbers, we note firstly that (c -F di) (c — di) = c2 +d2 is always real and non-negative, and it is strictly positive if c + di # 0. Then x bc — ad i. c2 + (12 c2 + d2 The number c — di which we used in this calculation has a special name in relation to c + di. Definition 4 Let z = a + bi where a, b e IR. The (complex) conjugate of z is the number a — bi, and this is denoted Z (or in some books z* ). Let us see some useful algebraic properties of the conjugate function. Proposition 5 Let z, w e C. Then zw zw. if w # 0. (3) (4) (5) (6) PROOF Let us prove one of these statements; the remaining ones are left as exercises. (5): let z = a + bi and w = c + di. Then zw ac — bd) — (bc + ad) i (a — bi) (c — di) Note from equation (2) that when the real quadratic equation aa;2 + br + c = 0 has complex roots which are not real, then these roots are conjugates of each other. More generally, we have 2
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    that ak (R/ ) k + a/c—I (F/)k 1 + • • • + all + ao O [since is a root] o. We needed a special symbol i for —1, but we proceed to show that no further symbols are needed to find the square root of i. Suppose that z = a + bi, where a and b are real. Then z2 = i if and only if = (a + bi)2 — t)2) 2abi. i Comparing real and imaginary parts, this is equivalent to 2 b2 = 0 and 2ab a So b = 1/2a from the second equation, and substituting for b into the first equation gives a4 = 1/4, which has real solutions a = l//v/ä or a = —l/v/ä. So the two complex numbers z which satisfy z2 = i (i.e. the two square roots of i) are I—i and Similarly any non-zero complex number has exactly two complex square roots. In the same spirit, the quadratic formula (1) is also valid for any complex coefficients a, b, c with a # 0, provided that appropriate sense is made of the square roots of the complex number b2 Example 7 We can use the quadratic formula (1) to find the two solutions of Corollary 6 The complex (non-real) roots of a real polynomial come in complea; conjugate pairs. Equiv- alently, if e C satisfies the polynomial equation (1k z + (1k —1 z + • • • + ao = 0, where each (Ji is real, then is also a root of this equation. PROOF Note from the algebraic properties of the conjugate function, proved in the previous proposition, = 2i. • all -l- (JO • + al h/ + ao [since each at is real] We take a — 2 z 1 b = —3 — i, and c = 2+ i in (1). Then 2 b2 3
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    Knowing Vi = ± (1 + i) // v/5, from the previous problem, we have —b b2 — 4ac 2 2 2 2 2 or 2 3 The Argand Plane The real numbers are often represented on the real line, increasing as we move from left to right. The complex numbers, having two components, their real and imaginary parts, can be represented as points on a plane; we call it the complea; plane or Argand plane or Argand diagram. The point (a, b) represents the complex number a + bi so that the horizontal axis contains all the real numbers, and thus is termed the real aa;is, while the vertical axis contains all those complex numbers which are purely imaginary (i.e. have no real part), and thus is called the imaginary axis. i 0 real axis a + bi 1 imaginary axis Figure 1: Complex numbers on the Argand plane A complex number z in the complex plane can be represented by Cartesian coordinates (that is, its 4
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    real and imaginary parts), but equally useful is the representation of z by polar coordinates. If we let r be the distance of z from the origin, and if, for z # 0, we define 0 to be the angle that the line connecting the origin to z makes with the positive real axis (measured in the clockwise direction), then we can write z = a + bi = r cos 0 + ir sin 0. The relationship between the Cartesian and polar coordinates of z is simple a r cos0 and b = r sin 0, b a2 + b2 and tan 0 a Definition 8 The number r is called the modulus of z = a+bi and is written Iz . the argument of z and is written arg z. The argument of 0 is undefined. For the complex number z = a + bi, we have the formulas b a2 + b2 and sin arg z cos arg z a2 + b2 In particular, complex numbers of the form (7) we see that The number 0 is called a have modulus z = cos0 + i sin 0 cos + (sin Note also that arg z is defined only up to the addition of integer multiples of 2m. For example, the argument of 1 + i could be taken as T //4 or 9T/4 or —7T/4, etc. For simplicity, in these notes we shall give all arguments in the range 0 < 0 < 2m, so that T //4 would be the preferred choice here. We now come to some useful algebraic properties of the modulus and argument functions. Proposition 9 Let z, w e C. Then I zw Iz/w zz Izl Iw 2 Moreover, up to the addition of integer multiples of 2m, the following equations also hold for z, w # 0 arg (zw arg (z/ w) arg z arg z + arg w. arg z — arg w. — arg z. (8) (9) (10) (11) (12) (13) (14) PROOF A selection of the above statements is proved here; the remaining ones are left as exercises. 5
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    (8): let z (12): let and Then = a + bi and w = c + di. Then zw = (ac — bd) + (bc + ad) i, so that I zw a2c2 + b2d2 + b2c2 + a2d2 (a2 + b2) (c2 + (12) (12 + b2 c2 + d2 z = r (cos0 + i sin 0) w = R (cos + i sin e ) . zw rR (cos 0 + i sin 0) (cos e + i sin e ) rR ((cos 0 cos e — sin 0 sin e ) + i (sin 0 cos e + cos 0 sin 9)) rR (cos (0 + 6) + i sin (0 + 6)) . We can read off that Izwl = r R = I zl IWI which gives us a second proof of (8), and also that arg (zw) = 0 + e = arg z + arg w up to the addition of integer multiples of 2T. An important geometric property of the modulus is Proposition 10 (The Triangle Inequality) Let z, w e C. Then < Iz—w PROOF Let us discuss the inequality (15) (16) Note that the shortest distance between 0 and z + w is the modulus of z + w. This is shorter in length than the path which goes from 0 to z to z + w. The total length of this second path is I zl + Iw . For an algebraic proof, note that for any complex number z + Z = 2 Rez and Rez < I zl . So for z,wec, 2 Then = Re (zü)) < Izl Iw 2 6
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    w Figure 2: Triangle inequality on the Argand plane to give the required result. To conclude this section, we introduce a new notation for non-zero complex numbers of modulus r and argument 0: write z = r cos 0 + ir sino = reto where eto = cos 0 + i sin 0. In this notation, complex numbers of unit length (length 1) are simply those of the form eto for some 0 e IR. More generally we write = eC(cos d + i sin d) for any c, d e IR. Our new notation is consistent with earlier uses of the exponential function, as the following proposition shows: i(0+d,) = reto and w = seid' then zw = r se Proposition 11 If z positive integer n. w z w and in particular zn = r e for any n ino PROOF The first identity is simply a re-statement of (8) and (12) of Proposition 9. The second follows by repeated application of the first. Finally, here is a useful result summarising multiple angle formulas: Corollary 12 (De Moivre's Theorem) For a real number 0 and integer n we have that cos no + i sin no = (cos 0 + i sin 0) n 7
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    PROOF Using the last formula in Proposition 11, (COS 0 + i sin 0) n = cos no + i sin no. 4 Transformations of the Argand plane We can think of complex numbers z a + bi as points in an Argand plane, but it can often be useful to think of them as vectors as well. Addition of a complex number z = a + bi then becomes a transformation of the complex plane: adding z to another complex number w translates that number by the vector That is, the map represents a translation by a units to the right and b units up in the complex plane. In a similar vein, we can also think of conjugation as a transformation. The conjugate of a point w is its mirror image in the real axis. So represents reflection in the real axis. translation by z w reflection w Figure 3: Translation and reflection on the Argand plane Turning to multiplication, write a non-zero complex number in polar coordinates as z = x + iy = reto Then the transformation of the complex plane given by multiplication by z, can be decomposed as the composition of two maps 8
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    and w rw. The latter, with r positive real, is very easy to understand: it is just dilation by magnitude r. Turning to the former, recall that eto is a complex number with modulus one, and, using the formula in Proposition 11 W = = SC So the modulus of etow is the same as that of w, whereas the argument is increased by 0. Thus this transformation is a rotation of the complex plane by angle 0 in the anti-clockwise direction around the origin. eco w rotation w rw dilation Figure 4: Dilation and rotation on the Argand plane 5 The Fundamental Theorem of Algebra From the quadratic formula (1), we know that a real quadratic polynomial can be solved using complex numbers. Here is a much more general result. Theorem 13 (The Fundamental Theorem of Algebra) Any polynomial equation with complea; coefficients ai e C has a complex root 0/ e C. The proof of this theorem is far beyond the scope of these notes. Note that for any given polynomial with complex coefficients the theorem only guarantees the existence of the a root somewhere in C, unlike the quadratic formula, which gives us a formula for the two roots. The Fundamental Theorem gives no hint as to where in C a root is to be found. Corollary 14 Given any polynomial equation a 0 + a 1 + (12 x • 9
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    with complex coefficients at e C and an # 0, there are n (not necessarily distinct) complex numbers . , In such that 71, • = an @ — 71) @ — 72) • • (1,0 (J I X • • • an a; n In particular, this shows that any complex polynomial of degree n has, counting repetitions, exactly n roots in C. This statement is not hard to derive from the Fundamental Theorem, although the Fundamental Theorem itself is very hard to prove. 6 Roots of Unity Problem 15 Let n be a natural number. Find all those complex numbers z such that zn 1 We know from the Fundamental Theorem of Algebra that there are (counting repetitions) n solutions: these are known as the nth roots of unity. Let us first solve zn = 1 directly for n = 2, 3, 4. When n = 2 we have 2 —1 and so the square roots of 1 are ± 1. When n = 3 we can factorise as follows: So 1 is a root and completing the square we see 0 = z2 z +1 which has roots 2 -1/2 + v9i/2, and -1/2 So the cube roots of 1 are 1, • When n = 4 we can factorise as follows: so that the fourth roots of 1 are 1, —1, i and —i. 2 1 2 3 4 v/äi/2. Plotting these roots on the Argand plane we can see a pattern developing. = r (cos0 + i sin 0) satisfies z n = 1. Then by Proposi- Returning to the general case, suppose that z tion 9, zn has modulus r n and has argument no. As 1 has modulus 1 and argument 0, we can compare = 1 giving r = 1 (remember r is a non-negative real number). Comparing argu- their moduli to find r n ments, we see no = 0 up to the addition of integer multiples of 2m. Thus no = 2kT for some integer k, giving 0 = 2kT/n. So the roots of z n = 1 are 2kT 2kT where k is an integer. + i Sin z = cos n n 10
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    2 —1 2 i —i 1 Figure 5: Roots of unity on the Argand plane for n = 2, 3, 4 At first glance there seem to be an infinite number of roots, but since cos and sin have period 27T these roots repeat with period n. So the nth roots of unity are the n complex numbers 2kT where k = 0, 1,2 . + i Sin z = cos n n in accordance with Corollary 14. Plotted on an Argand plane, the nth roots of unity form a regular n-gon inscribed within the unit circle with a vertex at 1. Figure 6: Roots of unity on the Argand plane for large n Problem 16 Find all the solutions of the cubic z 11
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    —2 + 2i in its polar form, we have If we write 37T 37T 8 cos + i Sin 4 4 So if z3 = —2 + 2i, and z has modulus r and argument 0, then up to the addition of integer multiples of 27T, 2kT 0 for some integer k. 4 3 As before, we need only consider k = 0, 1, 2 (as other integer values of k lead to repeats) and we see the 3 = V'ß and 30 which gives three roots are 2 2 3m 4 and 2 cos 4 11T + i sin 11T + i sin cos 12 12 19T 19T + i Sin cos 12 12 2 2 2 2 2 2 2 1 2 Historical notes It is only comparatively recently that mathematicians have become comfortable with the roots of the quadratic equation aa;2 -k bat -F c = 0 when b2 — 4ac < 0. During the Renaissance the quadratic would have been considered unsolvable, or its roots would have been called imaginary. The term 'imaginary' was first used by the French Mathematician René Descartes (1596-1650). Whilst he is known more as a philosopher, Descartes made many important contributions to mathematics and helped found coordinate geometry — hence the naming of Cartesian coordinates. But what meaning can 'imaginary' roots have? This philosophical point pre-occupied mathematicians until the start of the 19th century; afterwards these 'imaginary' numbers started proving so useful (especially in the work of Cauchy and Gauss) that these philosophical concerns were essentially forgotten. The i notation was first introduced by the Swiss mathematician Leonhard Euler (1707-1783). Much of our modern notation is due to him including e and T. Euler was a giant in 18th century mathematics and the most prolific mathematician ever. His most important contributions were in analysis (eg. on infinite series, calculus of variations). The study of topology arguably dates back to his solution of the Königsberg Bridge Problem. The term 'complex number' is due to the German mathematician Carl Gauss (1777-1855). Gauss is considered by many the greatest mathematician ever. He made major discoveries in almost every area of mathematics from number theory and non-Euclidean geometry, to astronomy and magnetism. His name precedes a multitude of theorems and definitions throughout mathematics. In 1799, he proved the Fundamental Theorem of Algebra, one of the first major results concerning complex numbers, which conclusively demonstrated their usefulness. The Argand plane is named after the Swiss mathematician Jean-Robert Argand (1768-1822). Abraham De Moivre (1667-1754) was a French protestant who moved to England. He is best remembered for this formula, but his major contributions were in probability and appeared in his The Doctrine Of Chances (1718). 12


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