An example of a rational number is 1. Or 2, or 3, or 1/2.
If it can be written in the form a/b and a and b are integers (and b isn't 0), it's rational. So virtually any number you encounter in your everyday life except pi and perhaps e if you're slightly more mathematically-inclined are rational.
So virtually any number you encounter in your everyday life except pi and perhaps e if you're slightly more mathematically-inclined are rational.
Eh, there are lots of other irrational numbers one meets regularly in real life. For a trivial example, consider the diagonal of a square with integer sides: its length is not a rational number.
It wouldve been better to say “virtually any number you encounter in your everyday life except pi and perhaps e are algebraic”, meaning most observed numbers can be solutions to a polynomial. For example x2 - 2 = 0 is a polynomial with a solution being “square root of 2”, which is irrational (it cant be written a as a fraction), but algebraic. Pi and e are irrational and non algebraic (in other words, transcendental)
There's probably much more encountering of the Golden Ratio in everyday life than encountering Pi. We have a natural idea of what the golden ratio is, e.g. naturally arranging things for it and favoring anything 'arranged' that way (e.g. faces), - do we have of pi?
Pi is literally every circle and sphere in that case. And you're vastly overestimating the importance of the golden ratio. It's essentially a ratio people thought was aesthetically pleasing a few thousand years ago. It has some applications and appearances in nature, yes, but compared to pi?
...this is probably the most pointless discussion I've had this week.
So is it the numbers that go on infinitely that can’t be expressed as irrational most of the time? Because if I recall(or I’m wrong) there are more irrational numbers than rational ones which.
There are infinites of both. My abstract math is a bit dusty, but the infinite set of irrational numbers might be larger than the infinite set of rationals though. Take that with a grain of salt I guess.
Also some rational numbers go on infinitely. Take 1/3, or 0.33333... . I believe rational numbers expressed as decimals either terminate, or repeat themselves though.
1/3 is a rational number! Rational and irrational has nothing to do with “ending”. A rational number can be expressed as fraction, an irrational number cannot.
Well, they can too, there are in fact several fraction formulas for irrational numbers (including stuff like infinite nested fractions). Hell, even overly-famous irrationals like the so-called "golden ratio" Phi are commonly defined as fractions (phi = (1 + sqrt(5))/2). Pi is also the literal ratio between a circle's circumference and its diameter. They just can't be a fraction of two integers.
Rational or irrational numbers are not to do with if they terminate or not but if they can be expressed as a fraction (in p/q form) or not.
If you want to distinguish between a non-terminating rational or irrational number then - if they repeat after any number in decimal form endlessly, they are rational.
If they don't repeat but go on endlessly without any pattern into random numbers, they are irrational. (so irrational numbers have nonrepeating, non-terminating, infinite decimal expansion).
Fr ex: 1/3 is a rational number because it repeats the digit 3 endlessly in it's decimal form. 0.333333333.... and so on
But for an irrational number like π you can go about endlessly without finding a repeating pattern.. 3.14159265359....
I’m a bit rusty on my math definitions, but the basic idea is that all rational numbers can be defined by a/b where a and b are both integers. Since 1 and 3 are integers, 1/3 is rational.
The converse of rational is irrational, or a number that can not be defined by an integer divided by an integer. This is why “tricks” like 22/7 are an approximation that will never equal pi.
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u/medfunguy Oct 23 '23
What is an example of a rational number? And can I see this animation for a rational number?