Complex number is an element of a numerical set created to solve algebraic equations that involve the even root of a negative rational number.
Summary
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- 1 Bases and definition
- 2 Representations of complex numbers.
- 1 Punctual representation
- 2 Algebraic representation
- 3 Trigonometric representation
- 4 Exponential representation
- 3 Arithmetic of complex numbers
- 1 Addition
- 2 Subtraction
- 3 Multiplication
- 4 Division
- 4.1 Conjugate
- 4.2 Definition of the division operation between complex numbers
- 5 Empowerment
- 4 Relationship of the complexes with other numerical sets
- 5 See also
- 6 References
- 7 Sources
Bases and definition
The set of real numbers , despite their usefulness and universality, has the great deficiency that: not every algebraic equation of degree greater than 2, with integer coefficients, has a real root . The equations x 2n = m, where n natural and p real negative have no real roots.
A repeated and notable example is the quadratic equation x 2 + 1 = 0 , from which it is obtained that x 2 = -1 . But according to the rules of algebra, no positive or negative number squared can give -1, that is, there is no real number x that satisfies the equation of the previous example.
The aforementioned insufficiency has forced mathematicians to invent a number i , with the property that i 2 + 1 = 0 , the admission of this number within the great family of numbers has considerably simplified algebraic calculations.
Each complex number z can be defined as an ordered pair of reals (a, b) where a is the real part and b is the imaginary part.
Representations of complex numbers.
Complex numbers have various forms of representation. Namely:
- Punctual representation.
- Algebraic representation.
- Trigonometric representation.
- Exponential representation.
Punctual representation
The number z is represented as a point on the plane in Cartesian coordinates (x, y) , where x is the real part y and the imaginary component.
Note that other forms of point representation in the plane, such as polar coordinates, are not included in this form of strut representation of the complex number.
Algebraic representation
The complex number z is represented by an algebraic expression x + yi , where x is the real part ; ‘and , the imaginary part .
Trigonometric representation
The trigonometric representation of a complex number is based on the representation of a point by polar coordinates (a, b) where a is the length of the radius vector to the point in question and b the angle to the X axis .
Then the complex number z = x + yi can be represented as z = a cos (b) + a isen (b) where the representations are related as follows:
Exponential representation
The Act Euler for complex raises the association:
that allows to establish a link between polar representation and exponentiation; hence we can say that we are in the presence of a complex number normalized in its exponential form:
Where a and b are those defined in the previous section (it should be noted that b is an angle in radians ), while k is any integer due to the angular equivalence after each integer turn. If c = ln (a) is known , then we have an exponential representation of the complexp z as follows:
