no an irrational number is one that cannot be expressed by a fraction of 2 integers. It has nothing to do with how you happen to write the integers whether it be 45 or 0x2D or 101101 they are all the same number. There are some numbers like one third 0.3333 that are infinite in length in base 10 but not in base 3 where one third is just 0.1 but either way 1/3 is not an irrational number because it can be expressed as the fraction of 2 integers
It's not only math that comes into play here. I know what an irrational number is, but all the definitions are meant for Arabic base-10 number systems. I can easily make up my own number system and say the character "3" now stands for pi (base 10). If everyone used my number system "3" would not be irrational because it's just "3" and "1" would be pi/3 (base 10) and "2" is pi/2(base 10) and "4" is pi*2(base 10) and so on. So my number system would start out like this if it were base 10: pi/3, pi/2, pi, pi*2, pi*3... but it would look like 1,2,3,4... and those would all be rational.
The definition of a rational number heavily relies on the representation of the number, and if the standards of that representation is changed the term does not apply.
I'm going to have to disagree with you on that. You can't redefine numbers like that. If you did you would be changing the definition of "1" which, by definition, gives the same number when multiplied by itself. one is a whole number. You can change bases but you can never change the definition of an integer.
I'm not changing the definition of 1 (base 10)
in my system there is a solid 1(base 10), it would just be represented a different way.
The actual character "1" is just a symbol. I can make it mean whatever I want. However most people decided it should mean 1(base 10) or 0001(base 2) and so on. It would be no different than saying what I have in the picture. If everyone agreed to use those symbols to mean those values(base 10) then it would be good. It's like making your own language and what you'd be saying is that grammar rules from a similar version still apply, which they wouldn't.
It's all based on conventions. We found whole numbers to be more useful when counting and we has 10 fingers so we used base 10 with whole numbers.
It has nothing to do with the base of the number system (essentially the format for how numbers are written). If we assume that 1 is a pretty good fundamental unit for math, all numbers are just multiples of 1. There is no ratio of multiples of 1 that is equal to pi, regardless of if you are in base-10, binary, whatever. It is just fundamentally separate.
No, it is irrational no matter what because 3.14... isn't actually high enough for base 10 to matter. More than that though, pi is a ratio between the diameter and circumference of a circle, all circles; it has no units nor is it unique to any counting system. Geometric ratios are about as fundamental as things get.
Oh right, the decimals going the other way also follow it, silly me. Didn't really think about the first statement much because I knew the second half to be true. So here's another reason then: Quirky decimals caused by your choice of base always have repeating decimals. 1/7 for example just repeats .142857 over and over. An irrational number in one base is irrational in all bases.
P.S. here's another comment for you to thumb down Mr. internet tough guy.
Could be base 4, 5, 6....whatever and it wouldn't matter. The only way it would be rational is if you made the circumference of a circle a unit, and then the diameter would be irrational, in order to keep the ratio the same.
"It" would not be rational even in that case, if by "it" you mean the ratio of a circle's circumference to its diameter, which is what pi is and always will be. Units are irrelevant.
What i meant is that if you were to make your counting system base-pi for some reason, though i expressed it poorly. Units wasn't really the right word, but I think most people refer to radians as a unit, and it is almost exactly this. The lack of units on radians is why using them always seems to confuse students who actually check their units at the end; you don't need to cancel out radians.
yeah, but if helps as a visual for people who might be visual learners/ not great at understanding math... I would argue that for someone first learning the concept of pi, this would be a great tool
I was the top student in my math class and they never taught us that. When I asked, teachers would only say "because that's the way it is, now shut up and solve the problem". I now study a no-math-related major.
C = pi * d
C is circumference, or the length around the circle. This is useful for a bunch of things.
d is diameter, or the length from one end of the circle to the opposite.
pi is simply the exact number you need to multiply d by to get C. It was found through centuries of guessing (22/7), then through more exact methods once the enlightenment came around.
I took geometry and then algebra 2 at my old highschool (which was a joke) and we used pi, but no one ever explained that it was anything more than "3.14159261...."
English is not my native language, so here goes nothing:
pi is (as the image shows) the length of the outside of a circle with a diameter of 1
pi is a number, therefore you can say: your dick is as long as pi, but not nearly as tasteful. (horribly lame pun, I know).
In case I wasn't clear on something. if you take the diameter of a circle (max. length between two sides) and multiply it by pi, you get the length of it's outside. I hope I could make myself clear enough to teach you.