Which axiom is used to prove that the product of two rational numbers is rational?
Holy cow, what is an axiom?
An axiom is a fundamental truth or principle that serves as a basis for a system of reasoning. In mathematics, axioms are the building blocks from which all other mathematical principles and theorems are derived. They are self-evident and do not require any further proof.
To prove that the product of two rational numbers is rational, we need to use the axiom of closure of rational numbers under multiplication. This axiom states that the product of any two rational numbers is also a rational number.
To understand this axiom better, let's first define what rational numbers are. Rational numbers are numbers that can be expressed as the quotient or ratio of two integers, where the denominator is not zero. For example, 3/4, -2/5, and 1 are all rational numbers.
Now, to prove that the product of two rational numbers is rational, we can use a direct proof. Let's say we have two rational numbers, a/b and c/d, where a/b and c/d are in the form of fractions, and b and d are non-zero integers.
We can express their product as (a/b) * (c/d) = (a * c) / (b * d).
Here, a * c is the product of the numerators, and b * d is the product of the denominators. Since multiplying two integers always results in another integer, both (a * c) and (b * d) will be integers.
Moreover, since b and d are non-zero integers, their product (b * d) is also non-zero. Therefore, we have shown that the product of two rational numbers, (a/b) * (c/d), can be expressed as the ratio of two integers, (a * c) / (b * d), where the denominator is non-zero. Hence, the product is a rational number.
In summary, the axiom of closure of rational numbers under multiplication is used to prove that the product of two rational numbers is rational.