is every rational number is a real number?If no explain
Answers
Answer:
Yes, every rational number is a real number. Let's take a look at the definitions of each of these types of numbers.
Step-by-step explanation:
The answer can be yes or no, depending how you define rationals and reals.
The normal set-theoretic approach is to axiomatize the reals and then define the rationals as a subset of the reals. In this case YES every rational is (trivially) a real number.
A different (category-theoretic) approach is to axiomatize the rationals and reals independently. In this case NO, none of the rational numbers are real numbers — they live in separate realms.
But in the latter case, you can map each rational number to a corresponding real number, and also map each operation on rational numbers into a corresponding operation on real numbers, with the important additional property that the map commutes.
Another way to describe this situation is to imagine that you can pick subsets of the real numbers and real operations in such a way that those subsets looks like a perfect copy of the rational numbers and operations. I.e., anything you can do using the rational numbers and operations can be mapped into something perfectly analogous using the real numbers and operations, and it doesn’t matter whether you make the jump from rationals to reals first with the arguments or last with the results.
In practice, the set theoretic approach is usually assumed, but it has a major conceptual problem: where do you start? If integers are rationals and rationals are reals and reals are complex numbers and complex numbers are quaternions and quaternions are …, while rationals are also quotient sets over pairs of integers, then you need to set up a huge amount of complicated but ignored structure just to define the integer subset and you need to wade through complicated circularities. The category-theoretic approach lets you define things much more simply, with the complication that you need to set up and deal with the maps from the simpler theories into the more complicated ones.
From a computer scientific approach, another advantage of the category-theoretic formulation is that you can use different implementation strategies for each kind of number (integer, rational, real, etc.), as long as you also have the appropriate coercion functions to move back and forth. For the set-theoretic approach, think of the problems you would encounter implementing integers as rationals at the same time that rationals are implemented as pairs of integers.
yes
every rational number is a real number
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