Numbers

By | 2019-12-02
Formal Definition

A Field is a set which elements are called Scalars. 2 scalars a,b can be combined in 2 ways, One is called their sum and is denoted by a+b . Another is called their product and is denoted  by ab (with composition symbol implied). Both compositions are associative and commutative and both have neutral elements called zero and one and denoted by 0 \mbox{ and } 1 respectively. Any element a has an additive inverse denoted by -a and, if non-zero, a multiplicative inverse denoted by \frac {1} {a}. In short, addition is a commutative group and multiplication restricted to non-zero elements is also a commutative group. Multiplication is distributive relative to addition. Term additive group is a common abbreviation for a commutative group which uses +,-,0 symbols as above.

Example: Wealth

Assume, for simplicity, that we live in a Kingdom which currency is a single denomination golden coin. Wealth in this case is just the count of coins in one’s possession. An example of addition would be combination of wealth by marriage of 2 families or by a merger of 2 corporations. The combined wealth will be the same if the attribution of  wealth between the 2 parties is reversed. In other words the composition is commutative. A penniless bride contributes nothing – has 0 wealth. If a marriage results in 0-wealth it means that one’s family debt was negative of the other’s family wealth. Assume that a bank promises to double your money in a year. We can treat the resulting wealth as a composition of 2 amounts:

  • amount with which the bank promises to replace every coin in your account (2 in this case)
  • initial wealth.

This is how we define multiplication. Scalars can be thought of as elements of a set of all potential instances of wealth. You can never go broke by acquiring additional funds. It is an experimental fact that cannot be derived by logical reasoning from the fundamental rules of a field. It is in fact theoretically possible for a field to be finite. Mathematical theory that correctly describes wealth is that of an infinite field.

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