Are all real numbers complex
The complex numbers
Let it be the field of real numbers. On the crowd
so on the Cartesian product of the set of real numbers with themselves, are by(1)
an addition (1) and a multiplication (2) are defined for all and from.
(1) is the component-wise addition of real numbers, i. H. is the direct product of with itself. Hence, is like an abelian group with the neutral element.
Obviously, the multiplication (2) is commutative and is a neutral element of.
The following applies to every three elements from
which shows the associativity of multiplication. So is a commutative monoid.
Also are because of
and the commutativity of multiplication satisfies both distributive laws, so that there is a commutative ring with one element.
After all, the element in exists for each element different from one another, since both squares and in the denominator cannot simultaneously vanish. But this element is due
the inverse to in the monoid. Thus the group of units of and is a commutative body, the Body of complex numbers.
The mapping with is obviously an injective homomorphism that maps onto the partial body of. So you can identify the body of real numbers with and understand it as part of the body of complex numbers. Because of
After this identification, you can write every complex number briefly as. One introduces the symbol for the number and calls it the imaginary unit. If you leave out the multiplication symbol *, you can write any complex number in the form with uniquely determined real numbers and. This is called the Real part and the Imaginary part of .
Complex numbers, the imaginary part of which vanishes, are exactly the real numbers, and different complex numbers, whose real part vanishes, are also called purely imaginary. Because of the fact that the square of such numbers is a really negative real number, which is impossible for real numbers and justifies their name. In particular, then(3)
If there is any complex number, the number is called the to conjugate complex number and the amount of . Thus a complex number agrees with its conjugate if and only if it is real, and because of this the (complex) absolute value of a real number is exactly its usual real value. Just as for real numbers, the absolute value for complex numbers has the properties of a norm, i.e. H. it holds for all complex numbers and(4)
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