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Integrating Algebra and Geometry with Complex Numbers
Complex numbers in schools are often considered only from an algebraic perspective. Yet, they have a rich geometric meaning that can support developing understanding of their use as algebraic tools. Complex numbers can be used to make connections among mathematical domains for students and teachers alike. Making these connections will enable teachers and students to see the usefulness and beauty of these numbers.
The development of complex numbers should build appropriate understanding. One goal is to nourish understanding to the highest possible level by considering students' backgrounds and abilities, thus seeking to arouse curiosity and bring satisfaction. One way to do this is to relate the geometric meaning with the algebraic notion in every step of an exploration of complex numbers. This brief is a list of reminders, not a comprehensive overview, of complex numbers. In the first part, important contextual steps are given recalling algebraic notions and their geometric meanings. In the second part, the step-‐by-‐step progression is addressed by pointing to the geometric meaning followed by the algebraic notation.
1. The Imaginary Unit a. Algebraic perspective: i = −1, and the roots of x2 + 1 = 0 are +i and –i. b. Geometric perspective: Where is (imaginary) number i with respect to the number
line?
Multiplication by –1 means a rotation of 180! on a number line (for example, (2)(–1) = –2). Multiplying a number with –1 twice returns the original to itself through a 360! rotation as seen in Figure 1.
Figure 1
Thus, Figure 1 shows that multiplication of a number, a, by (−1)! has the meaning of rotation of 360! with respect to a number line; multiplication by
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(−1)!, a rotation of 180! , and multiplication by −1!! = −1 = 𝑖 would then be
a rotation of 90! as in Figure 2.
Figure 2
2. Notion for a Complex Number
a. Algebraic perspective: Adding a real number, 𝑎, and an imaginary number, 𝑏𝑖, gives rise to the notion of the complex number 𝑎 + 𝑏𝑖, as a new entity often denoted by z.
b. Geometric perspective: The new complex number, z, is seen in the complex plane as shown in Figure 3.
Figure 3
3. Addition of Complex Numbers a. Algebraic perspective: (a + bi) + (c + di) = (a + c) + (b + d)i b. Geometric perspective: Addition of complex numbers Is pictured in Figure 4.
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Figure 4
4. Notation When the complex plane is introduced, the horizontal axis is labeled as x (real numbers) and the vertical axis labeled as yi (where y is real) . The set of complex numbers is denoted by 𝐶. For = 𝑎 + 𝑏𝑖, we write 𝑧 ∈ 𝐶, 𝑎 = 𝑅𝑒(𝑧) and 𝑏 = 𝐼𝑚(𝑧). The component 𝑎 is the real component and 𝑏 the imaginary component of 𝑧.
5. Multiplication of Complex Numbers a. Algebraic perspective: (a + bi)(c + di) = ac + adi + bci + bdi2= (ac – bd) + (ad + bc)i b. Geometric perspective: The multiplication of a complex number, a + bi, by a real
number c, represents a scaling as in Figure 5, while the multiplication of a complex number, a + bi, by the imaginary number, i, represents a rotation of 90°.
Figure 5 Multiplication can be represented as a composition of scaling, rotation and addition. For a comprehensive geometric view of multiplication, refer to multiplication in polar form later in the brief.
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6. Properties of Operations on Complex Numbers a. Algebraic perspective: Associativity, commutativity and distributivity properties
follow from properties of operations on polynomials. b. Geometric perspective: Algebraic properties can be interpreted geometrically, for
example commutativity as in Figure 6.
Figure 6
7. Conjugates and Absolute Value (Modulus) of Complex Numbers
a. Algebraic perspective: The conjugate of the complex number, z = a + bi is 𝑧 = a – bi. The modulus of a complex number z = a + bi is
𝑧 = 𝑎! + 𝑏! = 𝑎 + 𝑏𝑖 (𝑎 − 𝑏𝑖) = 𝑧 (𝑧) = 𝑧 Though complex numbers cannot be ordered as real numbers can, they can be compared by their absolute values.
b. Geometric perspective: The conjugate and modulus of a complex number, z, are seen in Figure 7.
Figure 7
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8. Complex Numbers as Vectors a. Algebraic perspective: The complex number,
𝑎 + 𝑏𝑖 may be thought of as the vector, 𝑎, 𝑏 and added as follows: 𝑎 + 𝑏𝑖 + 𝑐 + 𝑑𝑖 = 𝑎 + 𝑐 + 𝑏 + 𝑑 𝑖 ↔ 𝑎, 𝑏 + 𝑐,𝑑 = 𝑎 + 𝑐, 𝑏 + 𝑑
b. Geometric perspective: Complex numbers treated as vectors are seen in Figure 8.
Figure 8
c. To compare the meaning of the scalar (dot) product of collinear vectors and the
product of associated complex numbers, think about absolute value (modulus). Multiplication of complex numbers is not equivalent to the scalar (or vector) product of vectors.
9. Division of Complex Numbers
a. Algebraic perspective: Let z be the complex number c + di. Its multiplicative inverse 1/z is seen below:
!!= !
!!!"= !
!!!"∙ !!!"!!!"
= !!!"!!!!!
= !!!! !!
− !!!! !!
𝑖 = !! !
Division of two complex numbers, w and z, can be treated as follows where w = a + bi and z = c + di: 𝑎 + 𝑏𝑖𝑐 + 𝑑𝑖 =
𝑎 + 𝑏𝑖𝑐 + 𝑑𝑖 ∙
𝑐 − 𝑑𝑖𝑐 − 𝑑𝑖 =
𝑎𝑏 + 𝑏𝑑 + 𝑏𝑐 − 𝑎𝑑 𝑖𝑐! + 𝑑! =
(𝑎𝑏 + 𝑏𝑑)𝑐! + 𝑑! +
(𝑏𝑐 − 𝑎𝑑)𝑐! + 𝑑! 𝑖
𝑤𝑧 =
𝑤 ∙ 𝑧𝑧 !
b. The geometric interpretation of complex number division can most easily be seen using the polar form given later.
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10. Polar Form of a Complex Number a. Geometric perspective: A complex number z, is seen in polar form in Figure 9.
Figure 9
b. Algebraic perspective: For = 𝑎 + 𝑏𝑖 , we have 𝑧 = 𝑟(𝑐𝑜𝑠 𝜑 + 𝑖 𝑠𝑖𝑛𝜑) , where
r = 𝑎! + 𝑏! and tan (𝜑) = !!.
c. Notation: Considering a complex number 𝑧 = 𝑟(𝑐𝑜𝑠 𝜑 + 𝑖 𝑠𝑖𝑛 𝜑) in a polar form, we say that r is the absolute value (modulus) (see number 7 above) and 𝜑 is the argument of a complex number 𝑧.
11. Multiplication of Two Complex Numbers in Polar Form a. Geometric perspective: The multiplication of two complex numbers, w and z, is
seen in Figure 10.
Figure 10
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b. Algebraic perspective: Let z = r(cos 𝜑 + i sin 𝜑) and 𝑤 = q(cos 𝜌 + i sin 𝜌).
Thus, z𝑤 = rq(cos (𝜑 + 𝜌) + i sin(𝜑 + 𝜌)).
12. Division of Two Complex Numbers in Polar Form a. Geometric perspective: A geometric look at the division of two complex numbers is
seen in Figure 11.
Figure 11
b. Algebraic perspective: The division of two complex numbers, w and z, is given in
the following: !! = !
!𝑐𝑜𝑠 (𝜌 − 𝜑) + 𝑖 𝑠𝑖𝑛(𝜌 − 𝜑 ).
13. Power of a Complex Number a. Geometric perspective: Figure 12 shows different powers of the complex number, z.
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Figure 12
b. Algebraic perspective: If z = r(cos 𝜑 + i sin 𝜑), then 𝑧! = 𝑟!(cos (𝑛𝜑) + i sin(𝑛𝜑)).
14. Roots of a Complex Number in Polar Form
a. Geometric perspective: Figure 13 depicts the roots of a complex number, z.
Figure 13
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b. Algebraic perspective: If z = r(cos 𝜑 + i sin 𝜑), then 𝜔1= 𝑧! = 𝑟! (𝑐𝑜𝑠 !!+
𝑖 𝑠𝑖𝑛 !!. Hence, 𝜔1
3 = 𝜔23 = 𝜔3
3 =z.
For all nth roots of z: 𝑟! (𝑐𝑜𝑠 !!!!"!
+ 𝑖 𝑠𝑖𝑛 !!!!"!
, where k = 0, 1, 2, ..., n-‐1.
All nth roots give rise to a regular n-‐gon inscribed in a circle of modulus r.
15. Isometries of a Plane through Complex Numbers Compositions of reflections, rotations and translations can be represented with complex numbers as seen in Table 1. Table 1
Geometric Algebraic Reflections, conjugation 𝑧 → 𝑧
Rotations 𝑧 → 𝑢 𝑧 where 𝑢 ∈ 𝐶 and 𝑢 = 1 Translations 𝑧 → 𝑧 + 𝑎 where 𝑎 ∈ 𝐶
16. 𝑒!" Notation The usefulness and meaning of the notation 𝑒!" = cos 𝜑 + i sin 𝜑 can be explored. In particular, formulas for cos (𝜑 + 𝜌) and sin (𝜑 + 𝜌) can easily be derived.
17. 𝑒!" + 1 = 0 The formula 𝑒!" + 1 = 0, which connects five most important constants in mathematics, its meaning and historical perspectives can be explored.
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