Armando Martinez-Cruz Kip Hurwitz Department of Mathematics CSU Fullerton Presented at 2013 CMC Conference Palm Springs, CA

Agenda Welcome CCSS Intro to Software Complex Number Arithmetic Geometric Meaning of Operations Questions

Complex Numbers and CCSS High School, Number and Quantity – the Complex Number System Mathematics » Represent complex numbers and their operations on the complex plane. » 5 (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i) 3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.

The complex number a + bi Re(a + bi) = a Im(a + bi) = b

Introduction to the software To plot a complex number a + bi, use the INPUT box at the bottom. For 0 use 0 + 0i. EXAMPLE: Try typing “1+i”. Plot two other complex numbers yourself. Notice that the software labels the numbers with subscripts automatically. However it is much easier not to use subscripts, so try typing “a=1+3i”.

Introduction to the software, cont. To connect points with segments, click on the third icon from the left. If the icon looks like, then click on the little arrow and select Connect your three complex numbers with segments to construct a triangle.

Click on if you would like to shade your triangle. Notice that the software automatically computes the length of segments, and the area of a shaded polygon (in this case, a triangle).

Complex Arithmetic

Addition and Subtraction (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) – (c + di) = (a – c) + (b – d)i

Parallelograms

Geometric effects of complex addition Adding w to z shifts z to the right by Re(w) units and up Im(w) units. It follows that translations of polygons can be defined by this idea.

Geometric effects of complex addition, cont. Definition: A translation of a polygon is the uniform shift of each of the vertices of the polygon.

Multiplication and Division (a + bi)(c + di) = (ac – bd) + (bc + ad)i If c + di ≠ 0, then

Multiplication and Division Let m = |a + bi| and n = |c + di|. Also, let θ = arg(a + bi) and φ = arg(c + di). (a + bi) × (c + di) = m∙n cis(θ + φ) (a + bi) ÷ (c + di) = m/n cis(θ – φ)

Geometric effects of complex multiplication Multiplying z by w scales the modulus of z by a factor of |w| and increases the argument of z by arg(w) radians (counterclockwise). Definition: A dilation of a polygon is the uniform scaling of the sides of a polygon. Definition: A rotation is a uniform radial reorientation of the vertices of a polygon.

Similar Triangles |z| = 2; |a′| = 2|a| and |b′| = 2|b| arg(z) = 90°; arg(a′) = arg(a) + 90° arg(b′) = arg(b) + 90°

Other geometric effects of complex number operations Definition: The conjugate of a + bi is a – bi. This is a reflection about the x-axis. Similarly, the relation between a + bi & –a + bi and a + bi & –a – bi is reflection.

Geometric effects of complex arithmetic Definition: Two geometric shapes are similar provided that one can be obtained from the other by dilation, possibly with additional translation, rotation and reflection. Definition: Two geometric shapes are congruent provided that one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections, i.e. no dilation allowed.

Geometric effects of complex addition and multiplication. Let shape 1 have vertices a 1, a 2, …, a n ∈ ℂ. Let shape 2 have vertices b 1, b 2, …, b n ∈ ℂ. If there exists w, z ∈ ℂ, z ≠ 0 such that b k = w + za k for each k = 1, 2, …, n, then the two shapes are similar. If there exists w, z ∈ ℂ such that b k = w + za k for each k = 1, 2, …, n and |z| = 1, then the two shapes are congruent. This cannot be a definition.

Roots of Unity The seventh roots of unity give a regular heptagon, normally not constructible. DeMoivre’s Formula Go back to CCSS slide……CCSS slide

Roots of Unity

Questions?