About this topic
Before you begin
Calculate with complex numbers and conjugate roots, then interpret modulus, argument and transformations on an Argand diagram.
Learning goals
- Add complex numbers
- Subtract complex numbers
- Multiply complex numbers
- Divide by a nonzero complex number
- Find the conjugate of a complex number
- Find every distinct root of a real quadratic over the complex numbers
- Solve a unique linear complex equation, including a conjugate term
- Recover the remaining distinct roots of a real cubic or quartic from a non-real root
- Explain complex equality, the number system and real-coefficient root properties
- Calculate the modulus of a complex number
- Calculate a complex number's principal argument
- Represent and explain complex operations on an Argand diagram
Syllabus statements covered
- Include: extending real numbers to complex numbers; complex roots of quadratic equations; modulus, argument and conjugate; four operations; equality; conjugate roots of real-coefficient polynomials; Argand diagrams; geometrical effects of conjugation, negation, addition, subtraction and multiplication by i. Exclude polar or modulus-argument and exponential forms.
- Include: extending real numbers to complex numbers; complex roots of quadratic equations; modulus, argument and conjugate; four operations; equality; conjugate roots of real-coefficient polynomials; Argand diagrams; geometrical effects of conjugation, negation, addition, subtraction and multiplication by i. Exclude polar or modulus-argument and exponential forms.
Choose a lesson
Work through them in order.
- Not startedComplex arithmetic and polynomial rootsNextUse i² = −1, compare real and imaginary parts and retain conjugate roots for real-coefficient polynomials.
- Not startedArgand diagrams, modulus and argumentNextRead complex numbers as points and distinguish reflection, negation, translation and multiplication by i.