# More Complex Numbers

## Scheme of work: Year 13 A-Level Further Maths: Core Pure 2: Complex Numbers

#### Prerequisite Knowledge

From AS-Level Mathematics (Complex Numbers):

1. Introduction to Complex Numbers:
• Definition of i where i2=−1
• Expressing complex numbers in the form x + iy, where x and y are real numbers.
2. Addition, Subtraction, Multiplication, and Division of Complex Numbers:
• Arithmetic operations with complex numbers.
3. Complex Conjugate:
• Definition and properties of the complex conjugate.
• Using the conjugate to divide complex numbers.
4. Modulus and Argument:
• Finding the modulus and argument of a complex number.
• Representing complex numbers in polar form.
5. Loci and Regions in the Argand Diagram:
• Basic geometric representation of complex numbers.

#### Success Criteria

1. Everything covered in AS-Level, plus:
2. Powers of i:
• Recognizing patterns with powers of i and simplifying.
3. Further Work with the Argand Diagram:
• More detailed geometric interpretations, including representing sets of complex numbers.
4. Polar Form and Multiplication/Division:
• Multiplying and dividing complex numbers using their polar forms (r, θ).
• De Moivre’s Theorem.
5. Roots of Complex Numbers:
• Finding the nth roots of complex numbers and representing them on the Argand Diagram.
6. Applications in Solving Equations:
• Using complex numbers to solve polynomial equations.
• Using complex conjugate roots.
7. Exponential Form:
• Representing complex numbers in the exponential form, re.
8. Complex Numbers in Trigonometry and Geometry:
• Solving trigonometric problems using complex numbers.
• Transformations in the complex plane.
9. Series and Sequences with Complex Numbers:
• Dealing with sequences or series that involve complex numbers.

#### Teaching Points

Modulus and Argument of Complex Numbers:

• Definition of modulus and its geometric interpretation.
• Definition of argument and its range (-π ≤ arg(z) < π).
• Polar representation of complex numbers,
r\ \left(cos\ \theta+i\ sin\ \theta\right).

Polar Form and De Moivre’s Theorem:

• Expressing complex numbers in polar form.
• Understanding and proving De Moivre’s theorem:
\left(r\ \left(cos\ \theta+i\ sin\ \theta\right)\right)^n=r^nr\ \left(cos\left(n\theta\right)+isin\left(n\theta\right)\right)
• Application of De Moivre’s theorem to find powers and roots of complex numbers.

Roots of Complex Numbers:

• Finding n th roots using De Moivre’s theorem.
• Representing principal and other roots on the Argand diagram.

Exponential Form of Complex Numbers:

• Introduction to Euler’s formula:
e^{i\theta}=\ \cos{\theta}\ +\ i\ \sin{\theta}
• Expressing complex numbers in exponential form
re^{i\theta}
• Multiplication and division in exponential form.

Applications:

• Solving polynomial equations using complex numbers.
• Recognizing the conjugate root theorem for polynomial equations with real coefficients.

Transformations in the Complex Plane:

• Geometric interpretations of multiplication by a complex number.
• Describing transformations such as rotation, dilation, etc.

Complex Numbers in Trigonometry:

• Proving trigonometric identities using complex numbers.
• Simplifying trigonometric expressions.

#### Common Misconceptions

• Recognizing patterns with powers of i and simplifying:
• Some students may think that the powers of i follow a simple pattern, such as i^n = i^(n-1) * i. This is not true, and the powers of i follow a more complex pattern.
• Students may also make mistakes when simplifying expressions involving powers of i. For example, they may simplify i^4 as i^2, which is incorrect.
• Further Work with the Argand Diagram:
• Some students may not be familiar with the Argand diagram, which is a graphical representation of complex numbers. This can make it difficult for them to visualize the powers of i and understand their properties.
• Students may also make mistakes when interpreting the geometric meaning of the powers of i on the Argand diagram. For example, they may think that i^4 lies on the positive real axis, which is incorrect.
• Polar Form and Multiplication/Division:
• Some students may not be familiar with the polar form of complex numbers, which can make it difficult for them to multiply and divide complex numbers using their polar forms.
• Students may also make mistakes when converting between rectangular and polar forms of complex numbers.
• De Moivre’s Theorem:
• De Moivre’s theorem is a powerful tool for simplifying expressions involving powers of complex numbers. However, some students may find it difficult to understand and apply.
• Students may also make mistakes when using De Moivre’s theorem to simplify expressions.
• Roots of Complex Numbers:
• Finding the roots of complex numbers can be a challenging task. Some students may make mistakes when solving equations involving roots of complex numbers.
• Students may also have difficulty visualizing the roots of complex numbers on the Argand diagram.
• Applications in Solving Equations:
• Complex numbers can be used to solve polynomial equations. However, some students may not be familiar with these applications.
• Students may also make mistakes when using complex numbers to solve polynomial equations.
• Exponential Form:
• The exponential form of complex numbers can be used to simplify expressions and solve equations. However, some students may not be familiar with this form.
• Students may also make mistakes when converting between rectangular and exponential forms of complex numbers.
• Complex Numbers in Trigonometry and Geometry:
• Complex numbers can be used to solve trigonometric problems and perform geometric transformations. However, some students may not be familiar with these applications.
• Students may also make mistakes when using complex numbers in trigonometry and geometry.
• Series and Sequences with Complex Numbers:
• Complex numbers can be used to study series and sequences. However, some students may not be familiar with these applications.
• Students may also make mistakes when working with series and sequences that involve complex numbers.

## More Complex Number Resources

### Mr Mathematics Blog

#### Planes of Symmetry in 3D Shapes

Planes of Symmetry in 3D Shapes for Key Stage 3/GCSE students.

Use isometric paper for hands-on learning and enhanced understanding.

#### GCSE Trigonometry Skills & SOH CAH TOA Techniques

Master GCSE Math: Get key SOH-CAH-TOA tips, solve triangles accurately, and tackle area tasks. Ideal for students targeting grades 4-5.

#### Regions in the Complex Plane

Explore Regions in the Complex Plane with A-Level Further Maths: inequalities, Argand diagrams, and geometric interpretations.