The complex numbers

Francisco Ortiz Hernández Sergio Sánchez Linares Marcos Martínez Baños 1 Francisco Ortiz | Marcos Martínez | Sergio Sánchez

Index

1. Origin of the complex numbers……………………………………………………………………………… 3 2. Complex numbers…………………………………………………………………………………………………. 3 3. Conjugate and opposite………………………………………………………………………………………… 4 4. Graphic representation of the complex numbers………………………………………………….. 5 5. Operation with complex numbers…………………………………………………………………………. 5 5.1. Addition of a complex number…………………………………………………………………… 5 5.2. Subtraction of Complex Numbers………………………………………………………………. 6 5.3. Multiply Complex Numbers……………………………………………………………………….. 6 5.4. Divide two Complex Numbers……………………………………………………………………. 6 5.5 Power of Complex Numbers..……………………………………………………………………… 7 6. What are complex numbers used for......................................................................... 7 Bibliography………………………………………………………………………………………………………………. 8 2 Francisco Ortiz | Marcos Martínez | Sergio Sánchez

1. Origin of the complex numbers Everything started, in 50 A.D. when Heron of Alexandria tried to solve â&#x2C6;&#x161; . He gave up because he thought that it was impossible. In the 1500â&#x20AC;&#x2122;s, the square root of negative numbers and formulas for solving 3rd and 4th degree polynomial equations were discovered. In 1545, Girolamo Cardano wrote the first major work with imaginary numbers titled â&#x20AC;&#x153;Ars Magna.â&#x20AC;? He solves an equation + 10x â&#x20AC;&#x201C; 40 = 0. This answer was 5Âąâ&#x2C6;&#x161; . He said that work with complex numbers was â&#x20AC;&#x153;as subtle as it would be uselessâ&#x20AC;? and â&#x20AC;&#x153;a mental torture.â&#x20AC;? In 1637, Rene Descartes came up with the standard form for complex numbers, which is a+bđ?&#x2018;&#x2013;. He disliked the complex numbers and he said that if the complex numbers were in a problem, you couldnÂ´t solve it. Isaac Newton agreed with Descartes. In 1777, Euler put the symbol đ?&#x2018;&#x2013; for â&#x2C6;&#x161; . Robert Argand wrote how to represent the complex numbers in a plane, and this plane is called Argand diagram. After that, many mathematicians studied the theory of complex numbers. Augustin Louis and Niels Henrik made a general theory about the complex numbers and it was accepted. August Mobius try to apply the complex numbers in geometry. All of these mathematicians helped the world better understand complex numbers, and why they are useful.

2. Complex numbers in binomial form The complex numbers in binomial form are the expression a+bđ?&#x2018;&#x2013;, where a and b are real numbers. This expression has two parts: aâ&#x2020;&#x2019; real part

bâ&#x2020;&#x2019; imaginary part

ÂˇIf b=0 the complex number is reduced to a real number, so a+0đ?&#x2018;&#x2013;=a ÂˇIf a=0 the complex number is reduced to bi, and we say that the number is a pure imaginary number.

The set of the complex numbers appoint by C is: â&#x20AC;˘C={a+bđ?&#x2018;&#x2013;/a,bâ&#x201A;ŹR} The complex numbers a+bđ?&#x2018;&#x2013; and â&#x20AC;&#x201C;a-bđ?&#x2018;&#x2013; are called opposites. The complex numbers z=a+bđ?&#x2018;&#x2013; and Ě&#x2026;=a-bđ?&#x2018;&#x2013; are called conjugates. Two complex numbers are equals when they have the same real component and the same imaginary component. Examples: 3+8đ?&#x2018;&#x2013;, 9+6đ?&#x2018;&#x2013;, 5+3đ?&#x2018;&#x2013;, 4đ?&#x2018;&#x2013;â&#x20AC;Ś 3 Francisco Ortiz | Marcos MartĂ­nez | Sergio SĂĄnchez

Imaginary unit: The imaginary unit is the symbol đ?&#x2018;&#x2013;. If đ?&#x2018;&#x2013; is equal to -1, đ?&#x2018;&#x2013; is equal to â&#x2C6;&#x161;

.

Imaginary number: The imaginary numbers are the complex numbers, which imaginary part isnÂ´t zero. Example: 3+2đ?&#x2018;&#x2013;

Purely Imaginary Number: The purely imaginary numbers are the numbers which real part is zero. Example: 3đ?&#x2018;&#x2013;

Interesting property: The imaginary unit đ?&#x2018;&#x2013; has an interesting property. â&#x20AC;&#x2DC;â&#x20AC;&#x2122;Flipâ&#x20AC;&#x2122;â&#x20AC;&#x2122; it cross by four different values when you multiplicity it: So, đ?&#x2018;&#x2013;*đ?&#x2018;&#x2013;=-1, after -1*đ?&#x2018;&#x2013;=-đ?&#x2018;&#x2013;, after â&#x20AC;&#x201C;đ?&#x2018;&#x2013;*đ?&#x2018;&#x2013;=1 and finally 1*đ?&#x2018;&#x2013;=đ?&#x2018;&#x2013;.

3. Conjugate and opposite The conjugate of a complex number is defined as its symmetrical respect of the real axis, that is to say, if z=a+bđ?&#x2018;&#x2013;, then the conjugate of â&#x20AC;&#x153;Zâ&#x20AC;? is z=a-bđ?&#x2018;&#x2013;. The opposite of a complex number is its symmetrical respect of the origin.

Examples:

Complex number

Opposite

Conjugate

3+2đ?&#x2018;&#x2013; -5+6đ?&#x2018;&#x2013; -9-3đ?&#x2018;&#x2013;

-3-2đ?&#x2018;&#x2013; 5-6đ?&#x2018;&#x2013; 9+3đ?&#x2018;&#x2013;

3-2đ?&#x2018;&#x2013; -5-6đ?&#x2018;&#x2013; -9+3đ?&#x2018;&#x2013;

4 Francisco Ortiz | Marcos MartĂ­nez | Sergio SĂĄnchez

4. Graphic representation of the complex numbers The graphic representation of a complex number is very easy. We need the Cartesian axes in order to represent a complex number. The axis X is called real axis and the axis Y is called the imaginary axis. In the expression a+bđ?&#x2018;&#x2013;, we use (a, b) as the coordinate point. When we find the point, we draw a vector from (0, 0) until the point (a, b). Finally, we obtain the vector that represents the complex number.

Examples:

Complex number

Graphic representation

2đ?&#x2018;&#x2013; 4 -3-3đ?&#x2018;&#x2013;

5. Operation with complex numbers. 5.1. Addition of a complex number Complex numbers are added by adding the real part with the real part and the imaginary parts with the imaginary parts.

(a+bđ?&#x2018;&#x2013;) + (c+dđ?&#x2018;&#x2013;) = (a+c) + (b+d) đ?&#x2018;&#x2013; Examples: (2 + 3đ?&#x2018;&#x2013;) + (-4 + 5đ?&#x2018;&#x2013;) = (2 - 4) + (3 + 5) đ?&#x2018;&#x2013; = - 2 + 8đ?&#x2018;&#x2013; (6 - 5đ?&#x2018;&#x2013;) + (2 - đ?&#x2018;&#x2013;) = (6 + 2) + (- 5 - 1) đ?&#x2018;&#x2013; = 8 - 6đ?&#x2018;&#x2013; (3+2đ?&#x2018;&#x2013;) + (-7 - đ?&#x2018;&#x2013;) = 3 - 7 + 2đ?&#x2018;&#x2013; - đ?&#x2018;&#x2013; = -4 + đ?&#x2018;&#x2013; (-7 - đ?&#x2018;&#x2013;) + (3 + 2đ?&#x2018;&#x2013;) = -7 + 3 â&#x20AC;&#x201C; đ?&#x2018;&#x2013; + 2đ?&#x2018;&#x2013;= -4 + đ?&#x2018;&#x2013; 5 Francisco Ortiz | Marcos MartĂ­nez | Sergio SĂĄnchez

5.2. Subtraction of Complex Numbers To subtract complex numbers, we add the subtraction of the real parts with the subtraction of the imaginary parts.

(a+bđ?&#x2018;&#x2013;) - (c+dđ?&#x2018;&#x2013;) = (a-c) + (b-d) đ?&#x2018;&#x2013; Examples: (2 - 5đ?&#x2018;&#x2013;) - (-4 - 5đ?&#x2018;&#x2013;) = (2 - (-4)) + (-5 - (-5)) đ?&#x2018;&#x2013; = 6 (9 - 6đ?&#x2018;&#x2013;) â&#x20AC;&#x201C; (5 + 7đ?&#x2018;&#x2013;) = (9 â&#x20AC;&#x201C; 5) + (- 6 - 7) đ?&#x2018;&#x2013; = 4 -13đ?&#x2018;&#x2013; (8 - 6đ?&#x2018;&#x2013;) - (2đ?&#x2018;&#x2013; - 7) = 8 - 6đ?&#x2018;&#x2013; - 2đ?&#x2018;&#x2013; + 7 = 15 - 8đ?&#x2018;&#x2013;

5.3. Multiply Complex Numbers The multiplication of two complex numbers a + b i and c + d i is defined as follows.

(a + bđ?&#x2018;&#x2013;) Âˇ (c + dđ?&#x2018;&#x2013;) = (ac - bd) + (ad + bc) đ?&#x2018;&#x2013; However, you do not need to memorize the above definition; the multiplication can be carried out using properties similar to those of the real numbers and the added property đ?&#x2018;&#x2013; = -1. Examples: (3 + 2đ?&#x2018;&#x2013;) Âˇ (3 - 3đ?&#x2018;&#x2013;) = (9 - (- 6)) + (- 9đ?&#x2018;&#x2013; + 6đ?&#x2018;&#x2013;) = 15 - 3đ?&#x2018;&#x2013; (5 + 2đ?&#x2018;&#x2013;) Âˇ (2 â&#x2C6;&#x2019; 3đ?&#x2018;&#x2013;) = 10 â&#x2C6;&#x2019; 15đ?&#x2018;&#x2013; + 4đ?&#x2018;&#x2013;- 6đ?&#x2018;&#x2013; = 10 â&#x2C6;&#x2019; 11đ?&#x2018;&#x2013; + 6 = 16 â&#x2C6;&#x2019; 11đ?&#x2018;&#x2013;

5.4. Divide two Complex Numbers We use the multiplication property of complex number and its conjugate to divide two complex numbers. For example: We first multiply the numerator and denominator by the conjugate of the denominator and then we simplify. (

=(

)(

)

)(

)

=

( (

(

) ( )) (

) (

))

=

=

Other example: (

=(

)( )(

) )

=

=

= 6

Francisco Ortiz | Marcos MartĂ­nez | Sergio SĂĄnchez

5.5. Power of Complex Numbers. We can do the power of a complex number using the Tartaglia's triangle. For example: (2 ) = -46+9đ?&#x2018;&#x2013;

+ 3Âˇ

Âˇ3đ?&#x2018;&#x2013; + 3Âˇ2Âˇ( đ?&#x2018;&#x2013;) + ( đ?&#x2018;&#x2013;) = 8 + 36đ?&#x2018;&#x2013; + 54đ?&#x2018;&#x2013; + 27đ?&#x2018;&#x2013; = 8 + 36đ?&#x2018;&#x2013; â&#x20AC;&#x201C; 54 - 27đ?&#x2018;&#x2013;=

The power of the imaginary unit is periodic with period 4. To calculate đ?&#x2018;&#x2013; , we have to divide m by 4 and we have to see the rest of the division. đ?&#x2018;&#x2013; =1 đ?&#x2018;&#x2013; =1 đ?&#x2018;&#x2013; =1 đ?&#x2018;&#x2013; =đ?&#x2018;&#x2013; =1 đ?&#x2018;&#x2013; =đ?&#x2018;&#x2013; =đ?&#x2018;&#x2013;

đ?&#x2018;&#x2013; =đ?&#x2018;&#x2013; đ?&#x2018;&#x2013; =đ?&#x2018;&#x2013; đ?&#x2018;&#x2013; =đ?&#x2018;&#x2013;

đ?&#x2018;&#x2013; = -1 đ?&#x2018;&#x2013; = -1 đ?&#x2018;&#x2013; = -1

đ?&#x2018;&#x2013; =-đ?&#x2018;&#x2013; đ?&#x2018;&#x2013; =-đ?&#x2018;&#x2013; đ?&#x2018;&#x2013; =-đ?&#x2018;&#x2013;

đ?&#x2018;&#x2013; = đ?&#x2018;&#x2013; = -1 đ?&#x2018;&#x2013; =đ?&#x2018;&#x2013; =-đ?&#x2018;&#x2013;

Exercises: 1) Solve and represent the solution: ď&#x201A;ˇ - 4x + 13=0 ď&#x201A;ˇ =-4 ď&#x201A;ˇ 3 +27=0 ď&#x201A;ˇ + 6x + 10 = 0 2) Calculate a and b for that ( đ?&#x2018;&#x2013;) =3+4đ?&#x2018;&#x2013; 3) Calculate a for that ( đ?&#x2018;&#x2013;) is a purely imaginary number. 4) Calculate (4 - 3đ?&#x2018;&#x2013;) Âˇ (4 + 3đ?&#x2018;&#x2013;) - ( â&#x20AC;&#x201C; đ?&#x2018;&#x2013;) . 5) Calculate this operation graphically: ď&#x201A;ˇ (2 + 3đ?&#x2018;&#x2013;) + (5 + 2đ?&#x2018;&#x2013;) ď&#x201A;ˇ (3 + 4đ?&#x2018;&#x2013;) - (7 + 2đ?&#x2018;&#x2013;)

6. What are complex numbers used for? The complex numbers are used for maths, engineering, signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, business administration, accounting, vibration analysis... Thanks to the use of functions with complex numbers, we can study the performance of the wings of a plane and understand this kind of problems. Imaginary numbers are useful because they allow the construction of non-real complex numbers, which have essential concrete applications in a variety of scientific and related areas. The number đ?&#x2018;&#x2013; can be used for describing the rise or descent of inhabitants of a town or city. In mathematic we use the complex numbers to resolve quadratic equations, the square root of negative numbersâ&#x20AC;Ś 7 Francisco Ortiz | Marcos MartĂ­nez | Sergio SĂĄnchez

Bibliography http://rossroessler.tripod.com/ The book: “Anaya 1º Bachillerato Matematicas I” The book: “La variable compleja de Murray R.” Editorial: Danieluser. http://www.analyzemath.com/complex/complex_numbers.html www.disfrutadelasmatematicas.com www.ditutor.com

8 Francisco Ortiz | Marcos Martínez | Sergio Sánchez

Complex Numbers

Monográfico 4º

Complex Numbers

Monográfico 4º