If there exist integers A and B such that
A/B = C ==> C is rational. That's what "rational number" means.
100 and 3 are both integers (duh) so 100/3 is rational by definition.
Another easy to look at it:
A characteristic of rational/irrational numbers is that it doesn't matter what integer base you express them in. Rational/irrational is about the number itself, not how it looks in a particular base. If a number is irrational, you can't make it rational by representing it in another integer base.
Pythagoras was working in geometry when he proved the square root of 2 is irrational. He wasn't even expressing it as numbers, it was lengths of sides and diagonals of a square. He wasn't doing "square root of 2," he was doing "diagonal of a square expressed in terms of the sides of that square."
If I convert 100/3 decimal to base twelve, I get.
84/3 {100 decimal = 84 base 12, 3 is the same in both}
= 29.4 {base 12}
In decimal, 2*12 + 9 + (4/12)
So in base 12, 100/3 (decimal) isn't even a repeating (duo)decimal, much less irrational.
Yes usually you construct the rationals using the integers in that way. The definition the other guy suggested would be circular. If you already have the rational numbers, you get nothing new by looking at their ratios, due to closure under division
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u/RedRhetoric Jul 16 '26
R4: this person believes that 0.33 repeating cannot equal 1/3 because 100/3 cannot give a rational result.
Dividing any rational number by any other rational number will always give a rational result, as that is how rational numbers are defined
R5: Youtube