complex numbers

1 Francisco Ortiz | Marcos Martínez | Sergio Sánchez

The complex

numbers

Francisco Ortiz Hernández

Sergio Sánchez Linares

Marcos Martínez Baños


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


1. Origin of the complex numbers

Everything started, in 50 A.D. when Heron of Alexandria tried to solve √ . He gave

up because he thought that it was impossible. In the 1500’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 “Ars Magna.” He solves an equation + 10x – 40 = 0. This answer

was 5±√ . He said that work with complex numbers was “as subtle as it would be

useless” and “a mental torture.” In 1637, Rene Descartes came up with the standard

form for complex numbers, which is a+b𝑖. He disliked the complex numbers and he

said that if the complex numbers were in a problem, you couldn´t solve it. Isaac

Newton agreed with Descartes.

In 1777, Euler put the symbol 𝑖 for √ . 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+b𝑖, where a and b are real numbers. This expression has two parts:

a→ real part b→ imaginary part

·If b=0 the complex number is reduced to a real number, so a+0𝑖=a ·If 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: •C={a+b𝑖/a,b€R}

The complex numbers a+b𝑖 and –a-b𝑖 are called opposites.

The complex numbers z=a+b𝑖 and ̅=a-b𝑖 are called conjugates.

Two complex numbers are equals when they have the same real component and the same imaginary component.

Examples: 3+8𝑖, 9+6𝑖, 5+3𝑖, 4𝑖…


Imaginary unit: The imaginary unit is the symbol 𝑖. If 𝑖 is equal to -1, 𝑖 is equal to

√ .

Imaginary number: The imaginary numbers are the complex numbers, which

imaginary part isn´t zero. Example: 3+2𝑖

Purely Imaginary Number: The purely imaginary numbers are the numbers which

real part is zero. Example: 3𝑖

Interesting property:

The imaginary unit 𝑖 has an interesting property. ‘’Flip’’ it cross by four different values when you multiplicity it:

So, 𝑖*𝑖=-1, after -1*𝑖=-𝑖, after –𝑖*𝑖=1 and finally 1*𝑖=𝑖.

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+b𝑖, then the conjugate of “Z” is z=a-b𝑖.

The opposite of a complex number is its

symmetrical respect of the origin.

Examples:

Complex number Opposite Conjugate 3+2𝑖 -3-2𝑖 3-2𝑖

-5+6𝑖 5-6𝑖 -5-6𝑖

-9-3𝑖 9+3𝑖 -9+3𝑖


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+b𝑖, 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

2𝑖

4

-3-3𝑖

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+b𝑖) + (c+d𝑖) = (a+c) + (b+d) 𝑖

Examples:

(2 + 3𝑖) + (-4 + 5𝑖) = (2 - 4) + (3 + 5) 𝑖 = - 2 + 8𝑖 (6 - 5𝑖) + (2 - 𝑖) = (6 + 2) + (- 5 - 1) 𝑖 = 8 - 6𝑖 (3+2𝑖) + (-7 - 𝑖) = 3 - 7 + 2𝑖 - 𝑖 = -4 + 𝑖 (-7 - 𝑖) + (3 + 2𝑖) = -7 + 3 – 𝑖 + 2𝑖= -4 + 𝑖


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+b𝑖) - (c+d𝑖) = (a-c) + (b-d) 𝑖

Examples:

(2 - 5𝑖) - (-4 - 5𝑖) = (2 - (-4)) + (-5 - (-5)) 𝑖 = 6

(9 - 6𝑖) – (5 + 7𝑖) = (9 – 5) + (- 6 - 7) 𝑖 = 4 -13𝑖

(8 - 6𝑖) - (2𝑖 - 7) = 8 - 6𝑖 - 2𝑖 + 7 = 15 - 8𝑖

5.3. Multiply Complex Numbers

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

(a + b𝑖) · (c + d𝑖) = (ac - bd) + (ad + bc) 𝑖

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 𝑖 = -1.

Examples:

(3 + 2𝑖) · (3 - 3𝑖) = (9 - (- 6)) + (- 9𝑖 + 6𝑖) = 15 - 3𝑖

(5 + 2𝑖) · (2 − 3𝑖) = 10 − 15𝑖 + 4𝑖- 6𝑖 = 10 − 11𝑖 + 6 = 16 − 11𝑖

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:

=

( ) ( )

( ) ( ) =

=

=


5.5. Power of Complex Numbers.

We can do the power of a complex number using the Tartaglia's triangle. For example:

(2 ) = + 3· ·3𝑖 + 3·2·( 𝑖) + ( 𝑖) = 8 + 36𝑖 + 54𝑖 + 27𝑖 = 8 + 36𝑖 – 54 - 27𝑖= -46+9𝑖

The power of the imaginary unit is periodic with period 4. To calculate 𝑖 , we have to divide m by 4 and we have to see the rest of the division.

𝑖 = 1 𝑖 = 𝑖 𝑖 = -1 𝑖 = - 𝑖

𝑖 = 1 𝑖 = 𝑖 𝑖 = -1 𝑖 = - 𝑖

𝑖 = 1 𝑖 = 𝑖 𝑖 = -1 𝑖 = - 𝑖

𝑖 = 𝑖 = 1 𝑖 = 𝑖 = 𝑖

𝑖 = 𝑖 = -1 𝑖 = 𝑖 = - 𝑖

Exercises: 1) Solve and represent the solution:

- 4x + 13=0

=-4

3 +27=0

+ 6x + 10 = 0 2) Calculate a and b for that ( 𝑖) =3+4𝑖 3) Calculate a for that ( 𝑖) is a purely imaginary number.

4) Calculate (4 - 3𝑖) · (4 + 3𝑖) - ( – 𝑖) . 5) Calculate this operation graphically:

(2 + 3𝑖) + (5 + 2𝑖)

(3 + 4𝑖) - (7 + 2𝑖)

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 𝑖 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

numbers…


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

complex numbers

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