Complex numbers and polynomials - University College Cork

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Exercise 1. Prove that(a, b) ((c, d)(e, f)) = ((a, b)(c, d)) (e, f), ∀a, b, c, d, e, f ∈ R.This means that we can unambiguously define the sum, u + v + w, (respectively, theproduct uvw) of three complex numbers u, v, w as either u + (v + w) or (u + v) + w(respectively, as u(vw) or (uv)w). In particular, we can define successive powers ofa complex number in a clear way.We denote the set of complex numbers by C, and usually use the letter z as a genericcomplex number. The real numbers can be thought of as forming a subset of thecomplex numbers R: this is because pairs of the form (a, 0) have algebraic propertiesexactly similar to the real numbers, and we simply identify such pairs with a, i.e.,we write a in place of (a, 0). (Thus (an isomorphic copy of) R sits inside C: R ⊂ C.)In particular, we write 1 for (1, 0). The pair (0, 1) is also singled out for specialmention, and denoted by the letter i. It’s square is given byi 2 = (0, 1)(0, 1) = (0 2 − 1 2 , 0 × 1 + 1 × 0) = (−1, 0) = −1.Also, by the defining laws of operation,a + bi = (a, 0)(1, 0) + (b, 0)(0, 1) = (a, 0) + (0, b) = (a, b), ∀a, b ∈ R.This gives the customary expression for a complex number, in which a is its real part,and b its imaginary part. Such expressions are manipulated according to the usualoperations of real numbers with the proviso that whenever i 2 occurs it is replacedby −1. For example, treating everything as a real number,(a + bi)(c + di) = ac + a(di) + (bi)c + (bi)(di)= ac + adi + bci + bdi 2= ac + (ad + bc)i − ad= (ac − bd) + (ad + bc)i,in agreement with the definition of multiplication given at the start of this discussion.If z = a + bi, where a, b are real, its complex conjugate is defined to be ¯z = a − bi,and its modulus by |z| = √ a 2 + b 2 .Exercise 2. Suppose z, w ∈ C. Prove thatz + w = ¯z + ¯w, zw = ¯z ¯w, |z| = √ z¯z.Remark 1. Under the usual laws of addition and multiplication of matrices, complexnumbers can be viewed as 2 × 2 matrices of the form[ ] a b, a, b ∈ R.−b a2
For,[ a b−b a] [ c d+−d c] [ a + c b + d=−b − d a + c] [=a + c−(b + d)b + da + c],and[ a b−b a] [ c d−d c] [ ac − bd ad + bc=−bc − ad −bd + ac] [=ac − bd−(ad + bc)ad + bcac − bd].Remark 2. Buoyed by his success of defining complex numbers in the above manner,Hamilton tried for many years to find a method of defining operations on triplets ofreal numbers (a, b, c) so that they could be manipulated as if they were real numbers.But he didn’t succeed—in fact, no such operations can be devised. However, onOctober 16, 1843, on his way to Dunsink Observatory along the Royal Canal, hediscovered a way of multiplying quadruples of real numbers (a, b, c, d), which he wroteas a + bi + cj + dk, and called quaternions. Crucially, this was a non-commutativeoperation. He scratched the defining rules that i, j, k should obey on Broom Bridge.The year 2005 was designated the Hamilton Year by the Irish Government. TheCentral Bank of Ireland issued a special 10 euro coin in his honour, and An Poststruck a special stamp which carried the rules of the non-commutative algebra ofquaternions:i 2 = j 2 = k 2 = ijk = −1.Remark 3. Quaternions can also be thought of as 2×2 matrices of pairs of complexnumbers of the form [ ] z w, z, w ∈ C.− ¯w ¯z2 PolynomialsA polynomial is a function p defined on the complex numbers C whose value atz ∈ C is given by a linear combination of powers of z:p(z) = a 0 + a 1 z + a 2 z 2 + · · · + a n z n z ∈ C.The numbers a 0 , a 1 , . . . , a n attached to the various powers of z are independent ofz; they are called the coefficients of p, and can be real or complex numbers. Apolynomial is known when these are specified. The highest power of z present in thedisplay, by which is meant that the corresponding coefficient is non-zero, is calledthe degree of p. In the polynomial p displayed, its degree is n provided that a n ≠ 0.Polynomials of degree 0 are constants. Those of degree 1 are called linear polynomials,of degree 2, quadratic polynomials, of degree 3, cubic polynomials, of degree 4,quartic polynomials, of degree 5, quintic polynomials, and so on. For short, we mayrefer to them as lines, quadratics, cubics, etc.3
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Page 5 and 6: 3 Linear and quadratic polynomialsH
Page 7 and 8: 3.3 Heron’s problemGiven two dist
Page 9 and 10: We can also conclude that the graph
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Page 13 and 14: 2. Let p be a arbitrary quartic. Us
Page 15 and 16: with equality iff α = β = γ. Thu
Page 17 and 18: Plainly, the pair a = 1, b = 2 work
Page 19 and 20: Note especially that if a is ration
Page 21 and 22: and, for k ≥ 2,c k =Again,( mk)b
Page 23 and 24: 2. For which n ∈ N is x 2 +x+1 a
Page 25 and 26: 13. IRMO95. Suppose that a, b and c
Page 27 and 28: q = −3uv, r = u 3 + v 3 for u, v.
Page 29 and 30: To utilize this, replace x by ρ co
Page 31 and 32: Prove that b is a positive integer

Complex numbers and polynomials - University College Cork

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