Since not, we are not going to speak about this type bodies, but about the mathematical Body concept, that is to say, of algebraic structures. In particular, we are going to center on the numerical bodies.
In principle, a body is an algebraic structure on a set of elements (commonly called things) in that we have defined 2 types of operations: the sum (+) and the product (*). In principle, these operations do not have porqué to be what we usually understand for sum and product, but if we center on the set of the numbers, we can think that there are nustras well-known basic arithmetical operations.
But skylight, it is not enough to only have the operations, but these must to fulfill a series of properties. We are going to fix ideas. A set (X, +, *) (that is to say, a set and his 2 operations) it is a body when the following properties are fulfilled:

With these properties, and we are already going to center in the Numbers the smallest body that can be, with the sum and the habitual product (and this is tremendously important), it is the Body of the Rational Numbers.
In the following level, the body most used by all is: the Body of the Real Numbers. We even can spread a little more and come to the Body of the Complex Numbers. Really the latter set, that of the complexes, is an extension of the real numbers so that the RADICAL operation (to take root of any index) has sense. Basically, this body arises of setting off of (R 2, +), that is to say, the plane with the habitual sum of vectors, and there is invented an operation, called a product (*), that fulfills all the properties of body. East operation acts of the following form: (to, b) * (c, d) = (ac-bd, ad+bc) And if we write this operation between complex numbers in his habitual form, it proves that (a+bi) * (c+di) =ac-bd + (ad+bc) i
to include like authorized number the imaginary unit i = -1 and to work with him, as if of the second coordinate it was talking each other.
But: might we extend, of any form, the concept of complex number any dimension more? Really yes one can. In fact, we can speak about the Cuaterniones, which are to the Complexes, what these to the real ones. But the problem that arises is that this set does not have body structure. And he lacks her, not for his own definition, but because it is impossible that it has it, since somewhere near 1863, some Weierstrass proved an interesting fact he is the acquaintance like Final Theorem of the Arithmetic, who affirms that for n≥3 it is impossible to provide the additive group R n with a product operation (*) so that (R n, +, *) it has body structure. In other words, that for much that we get into debt, we are not going to be able to find a definition of the product that it extends to the product of complex numbers and that fulfills all the previously described properties of Body.
To conclude, we are going to turn a little back, there where he was speaking of

the smallest body that can be…

If we forget of the operations sum and standard product, the smallest body that it is possible to construct with numbers one usually calls Z2 and it is formed by 2 neutral elements: {0,1}. The operations it adds up and product they are defined as it continues:

• SUM:
• 0+0=0
• 0+1=1+0=1
• 1+1=0
• PRODUCT
• 0*0=0
• 0*1=1*0=0
• 1*1=1

Basically the operations standard are, but inside the Modular Arithmetic, where, in this particular case, 0=2=4=6 =…, 1=3=5=7 =…
Anyway, that I believe that we have already got tired of speaking about bodies. If you have not liked the content of this article, at least I wait that you should have enjoyed the image incial. Certainly: have you seen the girl who exists after the spiral?.