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

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?=?m?+?ni?(n?≠ 0)?is a root of a quadratic with real coefficients then?z* =?m?-?ni?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?∈ ?,?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?= (z?– (p?+?qi))(?z?– (p?-?qi))
  • This can be rearranged into the form (z?–?p?–?qi)(z?–?p?+?qi)