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Introduction to Argand Diagrams (DP IB Maths: AI HL)
Revision Note
Argand Diagrams
What is the complex plane?
- The complex plane, sometimes also known as the Argand plane, is a two-dimensional plane on which complex numbers can be represented geometrically
- It is similar to a two-dimensional Cartesian coordinate grid
- The x-axis is known as the real axis (Re)
- The y-axis is known as the imaginary axis (Im)
- The complex plane emphasises the fact that a complex number is two dimensional
- i.e it has two parts, a real and imaginary part
- Whereas a real number only has one dimension represented on a number line (the x-axis only)
What is an Argand diagram?
- An Argand diagram is a geometrical representation of complex numbers on a complex plane
- A complex number can be represented as either a point or a vector
- The complex number x + yi is represented by the point with cartesian coordinate (x, y)
- The real part is represented by the point on the real (x-) axis
- The imaginary part is represented by the point on the imaginary (y-) axis
- Complex numbers are often represented as vectors
- A line segment is drawn from the origin to the cartesian coordinate point
- An arrow is added in the direction away from the origin
- This allows for geometrical representations of complex numbers
Examiner Tip
- When setting up an Argand diagram you do not need to draw a fully scaled axes, you only need the essential information for the points you want to show, this will save a lot of time
Worked example
a)
Plot the complex numbers z1 = 2 + 2i and z2 = 3 – 4i as points on an Argand diagram.
b)
Write down the complex numbers represented by the points A and B on the Argand diagram below.
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Complex Roots of Quadratics
What are complex roots?
- A quadratic equation can either have two real roots (zeros), a repeated real root or no real roots
- This depends on the location of the graph of the quadratic with respect to the x-axis
- If a quadratic equation has no real roots we would previously have stated that it has no real solutions
- The quadratic equation will have a negative discriminant
- This means taking the square root of a negative number
- Complex numbers provide solutions for quadratic equations that have no real roots
How do we solve a quadratic equation when it has complex roots?
- If a quadratic equation takes the form ax2 + bx + c = 0 it can be solved by either using the quadratic formula or completing the square
- If a quadratic equation takes the form ax2 + b = 0 it can be solved by rearranging
- The property i = √-1 is used
- If the coefficients of the quadratic are real then the complex roots will occur in complex conjugate pairs
- If z = p + qi (q ≠ 0) is a root of a quadratic with real coefficients then z* = p - qi is also a root
- The real part of the solutions will have the same value as the x coordinate of the turning point on the graph of the quadratic
- When the coefficients of the quadratic equation are non-real, the solutions will not be complex conjugates
- To solve these you can use the quadratic formula
How do we factorise a quadratic equation if it has complex roots?
- If we are given a quadratic equation in the form az2 + bz + c = 0, where a, b, and c ∈ ℝ, a ≠ 0 we can use its complex roots to write it in factorised form
- Use the quadratic formula to find the two roots, z = p + qi and z* = p - qi
- This means that z – (p + qi) and z – (p – qi) must both be factors of the quadratic equation
- Therefore we can write az2 + bz + c = a(z – (p + qi))( z – (p - qi))
- This can be rearranged into the form a(z – p – qi)(z – p + qi)
Examiner Tip
- Once you have your final answers you can check your roots are correct by substituting your solutions back into the original equation
- You should get 0 if correct! [Note: 0 is equivalent to ]
Worked example
Solve the quadratic equation z2 - 2z + 5 = 0 and hence, factorise z2 - 2z + 5.
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