It's been more than a year now! Someday, I will be frequent here.
Anyways, it's been a while, I love studying maths - not advanced maths, but that high school level maths - everyone knows! I am at a stage where I feel that whatever I studied during my school was just a bunch of formulas and equations, without even questioning the basic facts. Here's one:
Why is (-1)(-1) = 1?
Why is it that way defined, can we prove. We just accepted the fact :D
Okay, that brings to this post's topic - Maths is indeed fun, but sometimes it questions our beliefs, intuitions and common sense! Let's consider the system of rational numbers - that we studied in schools - 1/2, 21/7,99/100 etc. On the number line, we can identify each of these numbers. If I am given an interval (or a line segment) [a,b], then I can easily show a rational number q such that a<=q<=b, however small the interval may be! [read that again]
This is the reason, why we say that the rational numbers are dense on the line. That implies there may infintely many of them, even if the interval is teeny-tiny! This seems intuitive. But hang-on, here is an interesting thing - Even though rational numbers are dense on the line, what if I show a point on the line that does not collide with any of the infinitely many rational numbers? Seems paradoxical right? Given an interval [1,2] with infinite rational numbers, there is a point $p$ on the interval that manages to avoid contact with any of the rational numbers!
Hold your breath, I chose $p=\sqrt{2}$. It lies in [1,2] which has infinitely many rational numbers but this point evades all of them! Confused! That's the reason no one believed such numbers exist - from Greeks to even neo-Mathematicians - that questioned their rational thoughts - hence they are irrational numbers!
Source: What is Mathematics?
Anyways, it's been a while, I love studying maths - not advanced maths, but that high school level maths - everyone knows! I am at a stage where I feel that whatever I studied during my school was just a bunch of formulas and equations, without even questioning the basic facts. Here's one:
Why is (-1)(-1) = 1?
Why is it that way defined, can we prove. We just accepted the fact :D
Okay, that brings to this post's topic - Maths is indeed fun, but sometimes it questions our beliefs, intuitions and common sense! Let's consider the system of rational numbers - that we studied in schools - 1/2, 21/7,99/100 etc. On the number line, we can identify each of these numbers. If I am given an interval (or a line segment) [a,b], then I can easily show a rational number q such that a<=q<=b, however small the interval may be! [read that again]
This is the reason, why we say that the rational numbers are dense on the line. That implies there may infintely many of them, even if the interval is teeny-tiny! This seems intuitive. But hang-on, here is an interesting thing - Even though rational numbers are dense on the line, what if I show a point on the line that does not collide with any of the infinitely many rational numbers? Seems paradoxical right? Given an interval [1,2] with infinite rational numbers, there is a point $p$ on the interval that manages to avoid contact with any of the rational numbers!
Hold your breath, I chose $p=\sqrt{2}$. It lies in [1,2] which has infinitely many rational numbers but this point evades all of them! Confused! That's the reason no one believed such numbers exist - from Greeks to even neo-Mathematicians - that questioned their rational thoughts - hence they are irrational numbers!
Source: What is Mathematics?
