[table][br][br][tr][br][td]Complex numbers[/td][br][td]複素数[/td][br][td]복소수[/td][br][td]复数[/td][br][/tr][br][tr][br][td]Real part[/td][br][td]実部[/td][br][td]실부[/td][br][td]实部[/td][br][/tr][br][tr][br][td]Imaginary part[/td][br][td]虚部[/td][br][td]허부[/td][br][td]虚部[/td][br][/tr][br][tr][br][td]Modulus[/td][br][td]絶対値[/td][br][td]모듈러스[/td][br][td]模[/td][br][/tr][br][tr][br][td]Argand diagram[/td][br][td]アルガン図[/td][br][td]아르간 도표[/td][br][td]阿尔冈图[/td][br][/tr][br][tr][br][td]Complex conjugates[/td][br][td]共役複素数[/td][br][td]복소 공액[/td][br][td]共轭复数[/td][br][/tr][br][tr][br][td]Imaginary unit (i)[/td][br][td]虚数単位[/td][br][td]허수 단위(i)[/td][br][td]虚数单位[/td][br][/tr][br][tr][br][td]Geometric transformations[/td][br][td]幾何変換[/td][br][td]기하학적 변환[/td][br][td]几何变换[/td][br][/tr][br][/table][br]

[table][br][tr][br] [td][b]Factual Inquiry Questions[/b][br] [list][br] [*]What defines a complex number?[br] [*]How is the real part of a complex number represented on the complex plane?[br] [*]What is the result of adding two complex numbers algebraically?[br] [*]Can you list the steps to multiply two complex numbers?[br] [*]How do we find the modulus of a complex number?[br] [/list][br] [/td][br] [td][b]Conceptual Inquiry Questions[/b][br] [list][br] [*]How does the concept of complex conjugates aid in the division of complex numbers?[br] [*]In what way does multiplying a complex number by i (the imaginary unit) affect its position on the Argand diagram?[br] [*]What is the significance of the angle a complex number makes with the positive real axis, and how is it related to the number’s multiplication and division?[br] [*]Can you explain the relationship between the operations of addition, subtraction, multiplication, and division of complex numbers and the geometric transformations on the complex plane?[br] [/list][br] [/td][br] [td][b]Debatable Inquiry Questions[/b][br] [list][br] [*]Is there a situation in which the division of complex numbers would give an unexpected result, perhaps not conforming to the usual patterns observed on the complex plane?[br] [*]Can the behavior of complex numbers under various operations be considered analogous to physical movements or transformations in space? How might this analogy break down?[br] [*]To what extent do the operations on complex numbers challenge our traditional understanding of dimensions in mathematics?[br] [*]Are complex numbers truly "complex" when their behavior is quite predictable and follows geometric patterns, or is the term misleading?[br] [*]How might the concept of complex numbers be extended or modified to better model phenomena in advanced physics or other sciences?[br] [/list][br] [/td][br][/tr][br][/table][br]

Complex numbers are written in the form [math]a+bi[/math], where:[br][br]- [math]a[/math] is the real part[br]- [math]bi[/math] is the imaginary part[br]- i is the imaginary unit with the property that [math]i^2=-1[/math][br][br]For example, [br][br][math]z=\frac{\sqrt{8}-2i}{3}[/math] has [math]\text{Re}(w)=\frac{\sqrt{8}}{3}[/math] and [math]\text{Im}(w)=-\frac{2i}{3}[/math]

Real numbers can be visualized on a number line. Each point on the line represents one real number.[br][br]In a similar way, complex numbers can be represented in a two-dimensional coordinate plane, where the horizontal axis represents the real part of the number, and the vertical axis represents the imaginary part of the number.[br][br]Each complex number [math]z=x+iy[/math] , [math]x,y\in\mathbb{R}[/math], is represented by a point [math]P(x,y)[/math] in the plane and the coordinates are the real and imaginary parts of the complex number itself.[br][br]Purely real numbers lie on the x-axis, and purely imaginary numbers lie on the y-axis.[br]

In this section we are exploring what the results of the operations of addition, subtraction, multiplication and division have on the numbers geometrically. We will look at much of this in more depth and for now it's just to see what we can see.

Consider these discussion questions[br][br]How does adding a real number to a complex number affect its position on the complex plane?[br]What is the result of adding two complex numbers geometrically in terms of movement along the axes?[br]Can you predict the sum of a real and an imaginary number by their positions on the complex plane?[br]What patterns emerge when adding complex numbers along the same line in the complex plane?

Consider these discussion questions[br][br]How does the result of subtracting[math]z_2[/math] from [math]z_1[/math], as shown by [math]z_3[/math], reflect their relative positions on the complex plane?[br]If [math]z_1[/math] and [math]z_2[/math] were points on a vector field, what would the vector represented by [math]z_3[/math] indicate?[br]How can we interpret the subtraction of two complex numbers in terms of movement along the real and imaginary axes?[br]What would be the geometric interpretation if [math]z_2[/math] were subtracted from [math]z_1[/math], instead of [math]z_1[/math] from [math]z_2[/math]?[br]If [math]z_1[/math] is fixed and[math]z_2[/math] varies, how does [math]z_3[/math] map out a path on the complex plane?

Consider these discussion questions [br][br]How does the division of complex numbers alter their position on the complex plane?[br]What happens to the magnitude and direction of a complex number when it is divided by another complex number?[br]How can you use the concept of conjugates to simplify the division of complex numbers?[br]Can you predict the quotient of two complex numbers by examining their positions on the complex plane?[br]How does the division of a complex number by a real number compare to division by an imaginary number on the complex plane?[br]If [math]z_1[/math] and [math]z_2[/math] are both on a of the form [math]y=mx[/math], what do you notice?[br]

Consider these discussion questions[br][br]How does the multiplication of a complex number by a real number affect its position on the complex plane?[br]What type of number does multiplying two purely imaginary numbers together give?[br]What geometric transformation occurs when multiplying a complex number by i?[br]Can you predict the product of a complex number with a purely imaginary number based on their initial positions?[br]What is the result of multiplying two complex numbers that are each other's conjugates?[br]

Consider these discussion questions[br][br]What happens to the magnitude of a complex number when it is raised to higher powers?[br]How does the angle of a complex number with the positive real axis change with successive powers?[br]Can you describe the pattern formed by the powers of a complex number on the complex plane?[br]What does the image suggest about the convergence or divergence of the powers of the complex number?[br]How would changing the initial complex number affect the pattern of its powers on the complex plane?[br]Can you predict the n-th power of a given complex number without actual calculation, using its position on the complex plane?[br]Take a complex number with a magnitude of 1. For example, [math]\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i[/math] or [math]i[/math] or what do you notice about powers of numbers of this form?

Here we will explore how to add, subtract, multiply and divide complex numbers algebraically.

Here we will explore how to add, subtract, multiply and divide complex numbers algebraically.

Check your understanding with these exam style questions

Practice questions 1-6[br]Section A - style short response style questions 7-19[br]Challenge questions 20-21

Part 4 - Optional extension challenge

Can you find the required [math]z_2[/math] for any given [math]z_1[/math] and [math]n[/math] ?