Rational number and closure in multiplication
- Thread starter hi ia
- Start date
HallsofIvy
Elite Member
- Joined
- Jan 27, 2012
- Messages
- 7,763
WHAT??? 0 certainly IS a rational number! Where did you get the idea that it wasn't?
Dr.Peterson
Elite Member
- Joined
- Nov 12, 2017
- Messages
- 16,722
First, surely you didn't mean "closed under rational number", but "closed under multiplication".
Second, as you've already been told, 0 IS a rational number, as it can be written as 0/1. What definition are you using?
Third, if 0 were not a rational number, this would not show that the set is not closed, because you would be multiplying a number that is not rational! Closure involves starting with two members of the set under discussion.
So, what are you thinking?
Steven G
Elite Member
- Joined
- Dec 30, 2014
- Messages
- 14,583
I need to repeat what has been said.
Even if 0 is not a rational number then we can not conclude that the set is not closed. To show that the set is not closed we must start with two rational numbers, multiply them and not get a rational number. What are you trying to say and why is 0 not a rational number?
Even if 0 is not a rational number then we can not conclude that the set is not closed. To show that the set is not closed we must start with two rational numbers, multiply them and not get a rational number. What are you trying to say and why is 0 not a rational number?
I shall make a bet. He heard a definition of natural numbers starting with 1. Then he was told a definition of the rational numbers as all numbers that can be expressed as an integer divided by a positive integer. But what he heard was a whole number divided by a whole number. It is one of several reasons that I prefer the definition of the natural numbers as starting with zero.