Essential Mathematics: Complex Numbers

47 flashcards · Mathematics · by lakesare

1. Why is i not defined to be √−1?

   Because √a √ b = √ ab doesn't work for imaginary numbers.

2. Square roots of 3: √3, −√3. Square roots of −3: ___? ___? A square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x.For example, 4 and −4 are square roots of 16, because 42 = (−4)2 = 16.

   Square roots of −3: i √3 − i √3

3. What's the complex conjugate of a + bi?

   a - bi

4. [Finding complex conjugates] What's the complex conjugate of 4 + 2i?

   4 - 2i

5. [Finding complex conjugates] What's the complex conjugate of 9i?

   -9i

6. [Finding complex conjugates] What's the complex conjugate of 5?

   5

7. [Properties of complex conjugates] How would you go about proving the properties of complex conjugation?

   By writing z = a + bi and w = c + di.

8. [Properties of complex conjugates] Why is this true?

   Because (A + B)(A − B) = A2 − B2

9. [Multiplying complex numbers by their complex conjugates] Multiply this number by its complex conjugate: 2 + 3i.

   (2 + 3i)(2 − 3i) = 22 + 32 = 4 + 9 = 13

10. [Multiplying complex numbers by their complex conjugates] Multiply this number by its complex conjugate: 5i.

    (5i) × (−5i) = 52 = 25

11. [Dividing complex numbers] How would you solve this? How would you the fraction as a single complex number.

    Multiply the top and bottom by 5 + i (the complex conjugate of the denominator).

12. [Multiplying complex numbers by their complex conjugates] What's good about multiplying a complex number by its conjugate?

    The result is always a noncomplex (real) number.

13. [Complex plane] Where's −i?

14. [Complex plane] Mark z + w by thinking geometrically.

15. [Complex plane] Mark this number by thinking geometrically.

16. [Complex plane] Mark this number by thinking geometrically.

17. [Complex plane] Mark 2z by thinking geometrically.

18. [Modulus of complex number] Find the modulus of 17i.

    17.

19. Argument VS principal argument of a complex number?

    argument - all possible angles between z and positive x-axis In fact, you can see that each non-zero complex number has infinitely many arguments, because you can add any integer multiple of 2π to an argument to obtain another argument.principal argument - angle that lies in the interval (−π, π]. Denoted as Arg(z). In some other texts, the principal argument is called the principal value of the argument, and sometimes the interval [0, 2π) is used rather than (−π, π].

20. What does Arg(z) denote?

    Principal argument of the complex number z.

21. Find the principal argument of −4i.

    From the diagram, Arg(−4i) = −π/2.

22. [Polar vs Cartesian form of a complex number] Write down a polar form of a complex number z using modulus of z argument of z

    z = |z| (cos(Arg(z)) + i sin(Arg(z))) Unless there’s a good reason not to do so, you should use the principal argument when writing a complex number in polar form.

23. [Polar vs Cartesian form of a complex number] How would you go about writing this complex number in a cartesian form?

    Just by expanding the brackets.

24. [Polar vs Cartesian form of a complex number] How would you go about writing this complex number in a polar form?

    By finding modulus |z|, finding a principle argument Arg(z), and writing z as |z|(cos(Arg(z)) + i sin(Arg(z)))

25. [Polar vs Cartesian form of a complex number] What's z = −4i in a polar form?

    modulus: |−4i| = 4 argument: Arg(−4i) = −π/2

26. [Polar vs Cartesian form of a complex number] What operations are easy to do in a cartesian form VS polar form of a complex number?

    cartesian form - addition, subtraction The Cartesian form of a complex number is convenient for adding and subtracting complex numbers, because to add or subtract complex numbers you just add or subtract their real and imaginary parts separately.polar form - multiplication, division, powers Polar form is more suited for multiplication and division.

27. [Polar form of a complex number] How would you go about proving this?

    This shows that to multiply two complex numbers in polar form, you multiply their moduli and add their arguments.

28. [Polar form of a complex number] This is the formula of zw. What's the formula of z/w?

29. [Visualising multiplication by 2i geometrically] Describe the geometric effect of multiplying some z by 2i. Polar form of 2i for convenience:Product of complex numbers for convenience: Using the polar form of complex numbers you can now visualise in the complex plane how to multiply one complex number by another.

    Multiplying z by 2i corresponds to an rotation of z through π/2 radians, and scaling of z by 2.

30. [De Moivre's formula] Given the formula for zw - what's the formula for zn (De Moivre's formula)?

31. [Working out powers of complex numbers] How would you go about solving this?

    Write z5 in polar form Use De Moivre's formula: zn = rn(cos nθ + i sin nθ). Write z5 in cartesian form

32. [Finding a quadratic equation with a given pair of solutions] How would you go about finding a quadratic equation that has solutions 2 ± i.

33. What's meant by roots of unity?

    Roots of unity are z in the equation of the form zn = 1. The word 'unity' refers to the number 1. solutions of z2 = 1 are the square roots of unity, solutions of z3 = 1 are the cube roots of unity, solutions of z4 = 1 are the fourth roots of unity. The 3 cube roots of unity.

34. [Finding roots of complex numbers] How would you go about finding z? z5 = −1 − i

35. Name 2 interesting consequences of the Euler's formula.

    Euler's equation: eiπ + 1 = 0 (take θ = π) Exponential form of a complex number: z = reiθ

36. How do we go from Euler's formula (derived via the Taylor series): to Euler's equation:

    By setting θ = π.

37. How do we go from Euler's formula (derived via the Taylor series): to the exponential form of a complex number:

    By superseding the polar form's cos(θ) + i sin(θ) with eiθ.

38. Name 3 ways to represent a complex number.

    cartesian form: z = a + bi polar form: z = r(cos θ + i sin θ) exponential form: z = reiθ

39. [Converting from exponential form to Cartesian form] How would you go about writing the complex number z = 4e−5πi/6 in Cartesian form?

    Write z in a polar form, and then just multiply the values out.

40. [Converting from Cartesian form to exponential form] How would you go about writing the complex number z = 1- i in exponential form?

    Write z in polar form, and then convert it to the exponential form.

41. Write product of complex numbers in exponential form.

42. Write De Moivre's formula in exponential form.

43. What's z? A square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x.For example, 4 and −4 are square roots of 16, because 42 = (−4)2 = 16.

44. What's the fundamental theorem of arithmetic VS fundamental theorem of algebra?

    fundamental theorem of arithmetic - every integer can be written as a product of its prime factors uniquely. Also called unique factorisation theorem. fundamental theorem of algebra - every polynomial of degree n has n complex solutions, some of which may be equal to each other.

45. What's meant by the insolvability of the quintic equations? Solving quintic equations in terms of radicals was a major problem in algebra from the 16th century, when quadratic, cubic and quartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with the Abel–Ruffini theorem.

    Polynomials of degree 5: have 5 solutions, however it's impossible to obtain these solutions in terms of their coefficients. E.g. this quintic has 5 solutions:but it's impossible to obtain these solutions from the coefficients 1, 3 and 6 by using only the operations +, −, ×, ÷ and n√ .Same goes for >5 degree polynomials. Polynomials of degree 2, 3, and 4 have solutions in terms of their coefficients.General solutions for the quadratic formula:General solutions for the cubic formula:

46. What do solutions look like for polynomial equations: with real coefficients with any coefficients? A polynomial equation.

    Polynomial equations with real coefficients have the solutions in the form of complex conjugate pairs!

47. De Morgan VS De Moivre formula?

    De Morgan's formula is about logic De Moivre's formula is about taking powers of complex numbers