REAL AND COMPLEX NUMBERS

The union of racional and irracional numbers forms a set called real numbers. If we remember the different types of numerical sets, we see that we have:

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Rational numbers also contain integers numbers that have exacto decimal expression (0,123) and those that have periodic decimal expression (7,01252525…). If the denominador of the irreducible fracción only has prime factors which are powers of 2 or 5, the decimal expression is esact. If the denominator of the irreducible fraction has a prime factor that is neither 2 nor 5, the fraction will have a periodic decimal expression. Therefore, all fractions have exact or periodic decimal expression; and any exact or periodic decimal expression can be written as a fraction.

There are other numbers that are not rational, as it is the case of √2, o π,which cannot be put as a fraction, so these numbers, together with the rationals numbers, form the ret of real numbers. Real numbers that are not rational numbers are called irrational numbers, whose decimal expression is infinite non-periodic numbers.

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Number line

  The real numbers are dense, that is, between every two real numbers there are infinite numbers. Every real number occupies a position on the number line and conversely, every point of the line can be matched with a real number.  

  Chosen the origin of coordinates and the size of the unit (or what is the same, if we put the 0 and 1) every real number occupies a position on the number line and backwards, any point near the line can be matched with a real number.

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Absolute value

  The absolute valué of modules os a number, is equivalente to the value of that number ignoring the sign. For example, the absolute value for -1 is 1, and the absolute blue of +1 is also 1.

  In formal language, the absolute value is defined by the symbol │x│and its representation on a coordinate axis results in:

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Intervals

  An interval of real numbers is a set of numbers corresponding to a part of the Number Line, consequently, an interval is a subset of the set of real numbers.There are several types of intervals:

  • Open interval: is that in which ends are not part of it, that is, all points on the line between two ends are part of the interval, except the ends themselves. Graphically we represent it on the real line as follows:
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  • Closed interval: is that in which extremes are part of it, that is, all points on the line between ends, including these, are part of the interval. In this case the inequalities are not strict.  
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  • Half-open interval: is that in which only one end is part of it, that is, all points on the line between ends, including one of these, are part of the interval.  
  • Half-open interval on the left: the lower end is not part of the interval, but the upper on yes, in other words; the end that is outside the interval is a strict inequality. 
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  • Half-open intervalo on the right: the upper end is not part of the interval, but the lower end yes, in other words, the end that is outside the interval is associated with a strict inequality.
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Real half-lines

   The whole Line of real numbers is: R= (-∞, ∞) = (S+) U (S-) U (0) but within it we can differentiate two real half-lines:  

  • Half-line of positive numbers: S+ = (0,∞) that is, from zero to infinity.  
  • Half-line of negative numbers: S- = (-∞,0)
    that is, from the least infinity, the negative infinity until zero. 

Setting

   It is a special way of expressing open intervals. The Setting a and radios r are defined and denoted by E(a,r) as the set of numbers that are at a distance of a less than r.  

For example: 

  • The surroundings of center 5 and radios 2 re the numbers that are of 5 a distance smaller than 2. If we think about it a little, it will be the numbers between 5-2 and 5+2, that is, the interval (3,7).
    E(5,2) = (3,7) 

 

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Complex numbers

A complex number is defined as an expression of the form:  

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The type of expression z = x + iy is called the binomial form where the real part is the real number x, that is denoted Re(z), and the imaginary part is the real number y, which is denoted by Im(z). Therefore we have: z = Re(z) + iIm(z).

The set of complex numbers is, therefore;

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This construction allows to consider the real numbers as a subset of the complex numbers, being real that complex number whiose imaginary part is null. Thus, the complex numbers of the form z = x + i0 are real numbers and are called imaginary numbers to those of the form 0 + iy, that is, with their real part null.  

Two complex numbers: 

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Operations in binomial form

   The addition and product operations defined in the real numbers can be extended to complex numbers. For the sum and the product of two complex numbers written in the binomial form: x + iy, u + iv the usual properties of the Algebra are taken into account with what they are defined:  

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It is verified, again, that the square of the complex number i is a negative real number, -1, therefore:  

(0 + i) · (0 + i) = -1 + i·(0) = -1 

   If the complex numbers are real numbers, that is, complex numbers with their imaginary part null, these operations are reduced to the usual ones among the real numbers because:  

(x + i0) + (u + i0) = (x + u) + i(0)      (x + i·0) (u + i0) = (x·u) + i(0)

   This allows to consider the body of the real numbers R as a subset of the complex numbers, C. The set of the complex numbers also has an algebraic structure of the body.  

   The conjugate of the complex number z = x + yi, is defined as: z’ = x – yi

Representation of complex numbers in the plane

   The modern development of the complex numbers began with the discovery of its geometric interpretation that was exposed indiscriminately by John Wallis (1685) and already completely satisfactory by Caspar Wessel (1799). Wessel’s work received no attention, and the geometric interpretation of complex numbers was rediscovered by Jean Robert Argand (1806) and again by Carl Friedrich Gauss (1831).  

   The set of complex numbers with addition operations and the product by a real number has vector space structure of dimension two, and is, therefore, isomorphic to R2. A base of this space is formed by the set {1, i}.

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Just as real numbers represent points on a line, complex numbers can be put in one-to-one correspondence with points on a plane. Real numbers are represented on the abscissa axis or real axis, and multiples of  i = √-1 they are represented as points on the imaginary axis, perpendicular to the real axis at the origin. This geometric representation is known as the Argand diagram. The y = 0 axis is called the real axis and the x = 0 is the imaginary axis. 

   As the necessary and sufficient condition for x + iy to coincide with u + iv is that x = u, y = v, the set of complex numbers is identified with R2 and the complex numbers can be represented as points of the “complex plane” . The complex number z = x + iy corresponds to the abscissa and the ordinate of the point of the plane associated with the pair (x, y). Sometimes the complex number z is referred as the z-point and in others it is referred as the z-vector.  

   The sum of complex numbers corresponds graphically with the sum of vectors. However, the product of complex numbers is neither the scalar product of vectors nor the vector product.  

   The conjugate of z, z’, is symmetric to z with respect to the abscissa axis.  

Trigonometric form of complex numbers

   We can differentiate between:  

  • Module: The module of a complex number is defiende as: |z| = √x2 + y2, and represesents the distance from z to the origin, that is, the length of the free vector (x,y) of R2. Therefore the module can never be a negative real number. The module of a real number coincides with its absolute value.
    Another way to express the module of a complex number is by the expression: |z| = √z.z’, where z’ is the conjugate of z, being the product of a complex number by its conjugate equal to:  (x + iy)(x – iy) = x2 + y2 a real and positive number. 
  • Argument: The argument of a complex number, when   z ≠ 0, represents the angle, in radians, that forms the vector of position in the positive abscissa.   It is therefore any real number θ such that:  
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       We have that every non-non-pero complex number has an infinity of positive and negative argument, which differ from each other in integer multiples of 2π. 

       If z is equal to zero, its module is zero, but its argument is not defined. 

       If we want to avoid the multiplicity of arguments, we can select for a semi-open interval of length 2π, which is called choosing a weapon of the argument; for example, if it is required θ ⍷ [0,2π), the main argument of z is obtained, denoted by Arg (z). If z is a negative real number, its main argument is π. Sometimes it is preferable to use multivalued arguments:
     

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    Where Z representa the set of integers.  

       If Arg (z) is defined as arctg (y / x) there is a new ambiguity, due to there are two angles in each interval of length 2π from which only one is valid. Therefore, statements with arguments should be made with a certain caution, for example the expression:  

    arg (z·w) = arg (z) + arg (w)

    is true if the arguments are interpreted as multivalued.  

       If z is different from zero, z’ it verifies that |z’| = |z| and that Arg(z’) =  – Arg (z)

Properties of the module, the conjugate and the argument of a complex number

Some properties of the conjugate and the module of a complex number are: 

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Polar form and trigonometric form

  If ρ is equal to the module of nonzero complex number z and θ is an argument of z, then (ρ , θ) are the polar coordinates of the point z. The complex number z in polar form is written: 

   The conversión of polar coordinates into Cartesian and vice versa is done by the expressions:  

x = ρcosθ ,  y = ρsenθ , por lo que z = x + iy = ρ (cosθ + isenθ)

   The trigonometric form of this complex number is: z = ρ (cosθ + isenθ), which is valid even in the case that z=0, because then ρ=0, so it is verified for every θ.

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