%path = "maths/numbers/combination with operations" %kind = chindnum["texts"] %level = 9

If we combine with the counting number or natural numbers a reversible process like adding and its counter process subtracting, then we have introduced the integers.

\[\mathbb{Z} = \mathbb{N}\times\{+,-\}\]

Repeat the adding and we have a new operation: the multiplication.

Now let’s combine this new operation with the integers. The counter process is division. Now we have introduced the rational numbers.

\[\begin{split}\mathbb{Q} = \mathbb{Z}\times\{+,-\}\\ \mathbb{Q} = \mathbb{N}\times\{+,-\}\times\{\cdot,\div\}\end{split}\]

Repeat the multiplication and we have the new operation “to the power”.

Now not any more that analogously, but basically yes, rationals combined with power operation extend to a new set of numbers the algebraic numbers.

Numbers so far contain processes, so the make up an algorithm. If we allow infinite algorithms, we extend further by including the irrational numbers (algebraic and transcendental) and thus extend to the real numbers.

Note

Finite and infinte algorithms: \(\sqrt{2}\) is infinite if expressed with the basic operations. But if we do not evaluate \(\sqrt{2}\) but use it symbolically only then this is a finite algorithm.