Complex Numbers

APA Style Sheet Running head: NUMBER SYSTEMS AND

Number Systems and Algebraic Structures Hermilio D. Maia ID # 65156 Western Governors University

Objective: 202.1.1-02

07/29/2008

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APA Style Sheet

Number Systems and Algebraic Structures Complex Numbers A. 1. The discovery of complex numbers is attributed to Girolamo Cardano, an Italian mathematician, who called them “fictitious numbers” while working on the solution of general cubic equations. Complex numbers are an extension of the set of real numbers R, considered a subset of the complex number set C. 2. A complex number has the format a + bi, where a: is the real part and b: is the imaginary part. Every real number can be considered a complex number where the complex part b=0. Complex numbers with a real part a=0 are called imaginary numbers, and represented by bi. Complex numbers are defined by the imaginary unit expressed as i = − 1 where i 2 = −1 . Examples of complex numbers: z = 2 + 3i; z = -5i; z = 3 (b=0); z = 7 − 4i Graphically, complex numbers are represented using the

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APA Style Sheet complex or Argand plane. This is a Cartesian plane where the y_axis represents the imaginary part of the complex number, and the x_axis specifies the real part. The complex numbers are position vectors or points in the Cartesian plane. We have here the geometric representation of the complex number z = x+yi and its complex conjugate z=x-yi.

3. Addition and Multiplication of complex numbers. Initially, let’s start with the definition of absolute value of a complex number, and equality of complex numbers. Since we are representing complex numbers as position vectors, all operations with complex numbers can be performed as operations with vectors. The absolute value of a complex number z = a = bi is defined as |z| = sqrt( a 2 + b 2 ). Two complex numbers a = bi = c + di, if and only if a = c and b = d. 3.1.

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APA Style Sheet Addition Rule to add or subtract complex numbers, we add or subtract their real number parts together, and then add or subtract their imaginary parts together. Example # 1. Add z1=5+3i and z2=1-6i Rule: Add the real number parts and then the imaginary number parts, z1+z2 = (5+1)+(3+(-6))i= 6 – 3I Following we have the calculations performed by Maple9, including the graph of the sum. > restart; > z1:=5+3*I;

> z2:=1-6*I;

> z3:=z1+z2;

> with(plottools): L1 := arrow([0,0], [5,3], .2, .5, .1, color=green): L2 := arrow([0,0], [1,-6], .2, .5, .1, color=blue): L3 := arrow([0,0],

[6,-3],.2,.5,.1,color=red):

> plots[display](L1,L2,L3, axes=normal,view=[-10..10,-10..10]);

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APA Style Sheet

The green colored vector is the complex number z1 = 5 + 3i. The blue colored vector is the complex number z2 = 1 - 6i The green red vector is the complex number z1 + z2 = 6 – 3i. The parallelogram rule for vector addition was used to determine the vector sum. The red vector is the diagonal of the parallelogram formed by the green and blue vectors. 3.2

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APA Style Sheet Subtraction Using the same vectors as in the addition example we have: z1 – z2 = (5 – 1) + (3 – (-6))I = 4 + 9i. The Maple9 calculations and graph of the vector difference are shown bellow: > restart; > z1:=5+3*I;

> z2:=1-6*I;

> z3:=z1-z2;

> with(plottools): L1 := arrow([0,0], [5,3], .2, .5, .1, color=green): L2 := arrow([0,0], [1,-6], .2, .5, .1, color=blue): L3 := arrow([0,0],

[4,9],.2,.5,.1,color=red):

> plots[display](L1,L2,L3, axes=normal,view=[-10..10,-10..10]);

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APA Style Sheet

The same color-coding is also used here. 3.3 Multiplication. We treat multiplication of complex numbers the same way as the multiplication of binomials. When a complex number is multiplied by an imaginary number, we use the distributive property of multiplication.

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APA Style Sheet When we multiply two complex numbers, the same property is again used, but employing FOIL. 3.3.1 3i (5 - 2i) = (3i x 5)+ (3i x (-2i)) = 15i – 6i2 = 15i – 6(-1) = 15 I + 6. In standard form 6 = 15i. Maple9 confirmation and graph, (same color-coding as the other examples): > restart; > z1:=3*I;

> z2:= 5-2*I;

> z1*z2;

> with(plottools): L1 := arrow([0,0], [0,3], .2, .5, .1, color=green): L2 := arrow([0,0], [5,-2], .2, .5, .1, color=blue): L3 := arrow([0,0],

[6,15],.2,.5,.1,color=red):

> plots[display](L1,L2,L3, axes=normal,view=[-10..10,-10..10]);

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APA Style Sheet

3.4 Multiply (3 + 2i)(5 – 4i) F

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(3 x 5)+ (3 x (-4i))+(2i x 5)+(2i x (-4i)) = 15 – 12i + 10i – 8i2 = 15 –2i – 8(-1)=

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APA Style Sheet

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15 –2i + 8 = 23 – 2i Maple9 calculation and graph: > restart; > z1:= 3+2*I;

> z2:=5-4*I;

> z1*z2;

> with(plottools): L1 := arrow([0,0], [3,2], .2, .5, .1, color=green): L2 := arrow([0,0], [5,-4], .2, .5, .1, color=blue): L3 := arrow([0,0],

[23,-2],.2,.5,.1,color=red):

> plots[display](L1,L2,L3, axes=normal,view=[-10..25,-10..10]);

APA Style Sheet

4. Keeping in mind the two cases of addition and multiplication, the main differences in the way these two operations are performed are the following. 4.1. To add or subtract complex numbers, we make a clear distinction during the process in how we add or subtract first the real number parts and then add or subtract the imaginary

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APA Style Sheet

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number parts. To multiply complex numbers, we make use of the distributive property of multiplication; making sure that when we multiply an imaginary number by another the product is a real number because i2 = -1. In the case of multiplication, we do separate the real and imaginary numbers before we multiply, because the operation falls under scalar multiplication of vectors.

APA Style Sheet References No references.

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