Complex numbers Argand diagrams

𝐼𝑚 3−𝑖 2 𝑅𝑒 2+𝑖 2 𝑅𝑒 3 𝑖 𝐼𝑚 4+𝑖 (2+3𝑖) 𝑅𝑒 1 1+𝑖 𝐼𝑚 1 1+𝑖 Complex numbers KUS objectives BAT Know how to represent and use Argand diagrams Starter: Evaluate the following: 𝑅𝑒 2+𝑖 2 𝐼𝑚 3−𝑖 2 𝑅𝑒 3 𝑖 𝐼𝑚 4+𝑖 (2+3𝑖) 𝑅𝑒 1 1+𝑖 𝐼𝑚 1 1+𝑖

Real numbers can be represented on a number line WB1 Argand diagrams Real numbers can be represented on a number line Complex numbers are represented on an Argand diagram Real axis Imaginary axis 4 + 3i Named after Jean-Robert Argand, a Parisian mathematician and bookkeeper

Represent these complex numbers on an Argand diagram: WB1 (cont) Argand diagrams Real axis Imaginary axis Represent these complex numbers on an Argand diagram: Conjugate z* is a reflection of z in the real axis Argand diagram shows why, unlike real numbers on a number line, you cannot expect to use inequalities between complex numbers. On a number line greater than is to the right but there is no comparable relation between points on a plane What do you notice about the position of a complex number and its conjugate?

You can use Pythagoras’ Theorem to find the magnitude of the distances WB2 Represent the following complex numbers on an Argand diagram 𝑧 1 =2+5𝑖 𝑧 2 =3−4𝑖 𝑧 3 =−4+𝑖 Find the magnitude of |OA|, |OB| and |OC|, where O is the origin of the Argand diagram, and A, B and C are z1, z2 and z3 respectively You can use Pythagoras’ Theorem to find the magnitude of the distances y (Imaginary) z1 5i √29 5 𝑂𝐴 = 2 2 + 5 2 z3 √17 1 𝑂𝐴 = 29 2 x (Real) -5 4 3 5 𝑂𝐵 = 3 2 + 4 2 4 5 𝑂𝐵 =5 z2 𝑂𝐶 = 4 2 + 1 2 -5i 𝑂𝐶 = 17

𝒂) 𝑧=1−2𝑖, Represent 𝑤=𝑧+(3+5𝑖) on an argand diagram WB3 Add/subtract on an Argand diagrams 𝒂) 𝑧=1−2𝑖, Represent 𝑤=𝑧+(3+5𝑖) on an argand diagram Real axis Imaginary axis x x ‘Add 3 + 5i’ translates any point Z to a point W as in the diagram above Can be confusing – both the points and the arrow represent complex numbers in different ways Similarly the translation shown represents

Represent 𝑤+𝑧 on an argand diagram 𝒄) 𝑧=−2+3𝑖, 𝑤=8−3𝑖 WB3 Add/subtract on an Argand diagrams 𝒃) 𝑧=3+2𝑖, 𝑤=−4−5𝑖 Represent 𝑤+𝑧 on an argand diagram 𝒄) 𝑧=−2+3𝑖, 𝑤=8−3𝑖 Represent 𝑤−𝑧 on an argand diagram Real axis Imaginary axis Can be confusing – both the points and the arrow represent complex numbers in different ways

Show z1, z2 and z1 + z2 on an Argand diagram WB4 𝑧 1 =4+𝑖 𝑧 2 =3+3𝑖 Show z1, z2 and z1 + z2 on an Argand diagram y (Imaginary) 10i 𝑧 1 + 𝑧 2 = 4+𝑖 +(3+3𝑖) =7+4𝑖 z1+z2 z2 z1 x (Real) -10 10 -10i Notice that vector z1 + z2 is effectively the diagonal of a parallelogram

Show z1, z2 and z1 - z2 on an Argand diagram WB5 𝑧 1 =2+5𝑖 𝑧 2 =4+2𝑖 Show z1, z2 and z1 - z2 on an Argand diagram y (Imaginary) z1 5i z1-z2 𝑧 1 − 𝑧 2 z2 2+5𝑖 −(4+2𝑖) =−2+3𝑖 x (Real) -5 5 -z2 -5i Vector z1 – z2 is still the diagram of a parallelogram  One side is z1 and the other side is –z2 (shown on the diagram)

WB6 Argand diagram – multiplication by a real number 𝑧=2+𝑖 Represent w=3z on an argand diagram Real axis Imaginary axis x x Can point out that we do not have a representation of a complex number multiplied by a complex number This is similar to multiplication of real numbers – the relation between points Z and W is that W is 3 times as far from O as Z

WB7 Argand diagram – multiplication by conjugate 𝑧=2+𝑖 Represent w=z 𝑧 ∗ on an argand diagram Real axis Imaginary axis x x Can point out that we do not have a representation of a complex number multiplied by a complex number z 𝑧 ∗ = 2+𝑖 2−𝑖 = 2 2 + 1 2 =5

KUS objectives BAT Know how to represent and use Argand diagrams self-assess One thing learned is – One thing to improve is –

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