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A number is rational if it can be written as the quotient of two integers.
This is why the set of rational numbers is denoted \(\mathbb Q\) for quotient.
Please explain @nbouscal .
other than the number itself over 1
example...2 is rational because i can express it as 4/2
For any rational number there is a \(p\) and \(q\) with \(p,q\in\mathbb Z\) such that the rational number can be written as \(\dfrac p q \)
Please mention differences between rational and irrational numbers also.
Irrational numbers are simply numbers that are not rational.
pi cannot be expressed as a quotient even though 22/7 is near therefore it is irrational
Some more examples ....@lgbasallote
The simplest proof of an irrational number is the one for \(\sqrt2\). http://www.homeschoolmath.net/teaching/proof_square_root_2_irrational.php shows the standard proof, due to the Greeks.
ALL integers are rational numbers
the integers are -1, -2, -3, 0, 1, 2, 3, etc
Decimal numbers that terminate are rational, because they can be written as a fraction. For example, 2.231 can just be written as 2231/1000. This is the case for any decimal that terminates.
What about 1/3...................?@nbouscal @lgbasallote @ParthKohli
some decimals that dont terminate can also be expressed as fractions...for example 0.333333 can be expressed as 1/3
1/3 is a quotient of two integers, so it is a rational number.
that is rational because 1 amd 3 are integers
do you get it now?
If you want to get even more fun you can look at the transcendental numbers, they are even more crazy than the irrational numbers. Irrational numbers that are not transcendental are called algebraic, they can be found as the roots of polynomials. There are all sorts of fun proofs to be read in this area of mathematics.
lol why make it confusing :p
Of coarse @lgbasallote is right.
the kid doesnt understand rational and irrational and you're suggesting "fun" :p
Also worth noting is that the union of the rational and irrational numbers forms the real numbers, \(\mathbb R\), which is the set that you usually work in. You can also go further to \(\mathbb C\), the complex numbers. And then even further than that :)
I'm giving him an idea of the cool stuff ahead, it doesn't just stop at rational and irrational :P
sometimes it's nicer to live in fantasies for some time
1/3 has recurring digits but is still expressed as a quotient of two integers, so it is rational.
i'd rather take the blue pill than the red