Exam Questions - Further complex numbers

Exam Questions - Further complex numbers

Revision Points - Further Complex Numbers

Understanding and Visualising Complex Numbers

  • Remember that a complex number is made up of a real part and an imaginary part, and is represented as a + bi.

  • Understand that the Conjugate is the mirror image of the complex number but with an opposite sign for the imaginary part, represented as a - bi.

  • Complex numbers are graphed on the complex plane, with the horizontal plane being the real part, and the vertical as the imaginary part.

Operations on Complex Numbers

  • Learn to add, subtract, multiply, and divide complex numbers, along with finding the conjugate.

  • Practice calculating the modulus and argument of complex numbers.

  • Know how to calculate the square root of a complex number and express it in the simplest form.

Polar Form and De Moivre’s Theorem

  • Gain a good understanding of the polar form of a complex number represented as r(cos θ + i sin θ), where r is the magnitude and θ is the argument.

  • Familiarise yourself with De Moivre’s Theorem, stated as (r(cos θ + i sin θ))^n = r^n(cos nθ + i sin nθ).

  • This theorem is used to raise complex numbers to a power, in trigonometric simplification, and in root extraction of complex numbers.

Roots of Complex Numbers

  • Review how to find the nth roots of complex numbers, i.e. solutions to the equation z^n = a + bi.

  • Identify the fact that if a complex number is a root of a polynomial equation, then its conjugate is another root.

  • Ensure understanding that if z is a root of a polynomial, then the conjugate of z is also a root.

Application of Complex Numbers

  • Understand the various applications of complex numbers in solving practical problems in fields such as electrical engineering and quantum physics.

  • Complex numbers can be used to solve equations that we cannot solve using just real numbers.

  • Realise the importance of these topics as they form the foundation of many concepts in Further Maths.

Remember, practice makes perfect. Applying concepts to a variety of problems helps in gaining a deep understanding and prepares you fully for the Core Pure component of the Further Maths.