# Irrational Numbers

## An irrational number is a number that can't be expressed by a fraction having integers in both its numerator and denominator. While their existence was once kept secret from the public for philosophical reasons, they are now well accepted, yet still surprisingly hard to prove on an individual basis. Please post all questions about irrational numbers, including the famous examples of π, e, and √2, into this category.

###### Asked in Math and Arithmetic, Proofs, Irrational Numbers

### Give a proof that the square root of 7 is an irrational number?

Proof by contradiction: suppose that root 7 (I'll write sqrt(7))
is a rational number, then we can write sqrt(7)=a/b where a and b
are integers in their lowest form (ie they are fully cancelled).
Then square both sides, you get 7=(a^2)/(b^2) rearranging gives
(a^2)=7(b^2). Now consider the prime factors of a and b. Their
squares have an even number of prime factors (eg. every prime
factor of a is there twice in a squared). So a^2 and b^2 have an
even number of prime factors. But 7(b^2) then has an odd number of
prime factors. But a^2 can't have an odd and an even number of
prime factors by unique factorisation. Contradiction X So root 7 is
irrational.

###### Asked in Math and Arithmetic, Algebra, Numbers , Irrational Numbers

### What are the subsets of integers?

There are an infinite number of subsets.
The empty set,
Even numbers
Odd numbers
Multiples of 3
Multiples of 4, etc
Integers from 1 to 31 (days of the month)
Integers from 0 to 23 (hour display on some digital clocks)
{1}, {1,4,456,-5}, (1,2,3,4,5,6,7,3492}
and so on, and on, and on.
There are an infinite number of possible subsets.
Odd numbers, even numbers, the age (in whole years) of each
member of my family, my friends' telephone numbers, prime numbers,
integers between 70 and 83 are all examples of subsets.

###### Asked in Math and Arithmetic, Numbers , Irrational Numbers

### Is the square root of 50 a irrational number?

7.071678118.......
rounded to
7.1
*****
To a whole lot of places, it's actually
7.0710678118654752440084436210485...
But no finite number of places can be enough to establish its
irrationality.
Certainly, √50 is indeed irrational. Here is how we can see that
it is so:
First of all, we note that √50 = 5√2.
To show that the above is irrational, we need to show two
things:
(1) that √2 is irrational; and
(2) that the product of a integer and an irrational is
irrational.
We show (1), above, by reductio ad absurdum: We assume
the opposite of that which we wish to prove and show that it
entails a contradiction.
Suppose that √2 is rational. Then it must equal the quotient
p/q, where p and q are two integers that are relatively prime. In
particular, not both, p and q may be even numbers.
(√2 might also equal some p'/q', where p' and q' have a common
divisor greater than 1; but, in that case, the fraction could be
reduced to p/q. Thus, we can assume, without loss of generality,
that p and q are relatively prime; that is, that p/q is
irreducible.)
Before continuing our proof, we shall need to establish the
following: The square of every even integer is an even integer; and
the square of every odd integer is an odd integer.
Let n be even; thus, n = 2k, where k is some integer. Then, n2 =
4k2; thus, n2 is divisible by 4 and, therefore, it is also
divisible by 2.
Let m be odd; thus, m = 2j + 1.
Then m2 = (2j + 1)2 = 4j2 + 4j + 1 = 2(2j2 + 2j) + 1, which is
evidently odd.
We may conclude, from the above, that, if a perfect square is an
even integer, then its square root must also be even. Now, we may
continue the main proof:
From √2 = p/q, which we assumed above, p and q being relatively
prime, we deduce the following:
p2 = 2q2; therefore, p2 is even; therefore, p is even.
Because p is even, p = 2r, for some integer r. Then, p2 = 4r2 =
2q2; thus, q2 = 2r2, and q is similarly shown to be even. P and q,
therefore can not be relatively prime.
But, wait! Our assumption was that p and q are relatively prime
and can not both be even; and, so, we have the contradiction we
sought.
Therefore, (1) is proved, and there exist no integers p and q,
such that p/q = √2, and p/q is an irreducible fraction. In other
words, √2 must, after all, be irrational.
Now, we proceed to show (2), that the product of an integer and
an irrational is irrational; it will suffice to do this by means of
the example in hand: namely, that, if √2 is irrational, then √50 =
5√2 must also be irrational.
First, we know that the sum of two integers is also an integer;
likewise, by induction, the product of two integers is an integer.
From this, we can show that the product of two rationals is also a
rational:
Let the two rationals be a/b and c/d, where a, b, c, and d are
all integers. Then, their product is ac/bd, and, because ac and bd
are integers, ac/bd must, by definition, be rational.
Multiplying both sides by 1/5, we obtain (1/5)(√50) = √2.
Now, 1/5 is rational; then, if √50 is rational, then √2, being
the product of two rationals, must also be rational. But we have
shown, in (1) above, that √2 is not rational.
We conclude that √50 can not be rational, either; therefore, it
is irrational, which is what we set out to prove.

###### Asked in Math and Arithmetic, Mathematical Finance, Numbers , Irrational Numbers

### Who is the founder of irrational numbers?

The history of this is confused. The first irrational number
found was almost certainly root 2. The man who proved this was
probably Hippasus.
Hippasus died by drowning, and some stories say that Pythagoras
drowned him for proving the existence of an irational number which
he thought was impossible. Others say that Pythagoras
ordered the drowning of Hippasus for revealing the existence of an
irrational number which should be kept fron public knowledge.

###### Asked in Math and Arithmetic, Numbers , Irrational Numbers

### What is the difference between rational numbers and irrational numbers?

An irrational number is a number that can't be expressed by a
fraction having integers in both its numerator and denominator. A
rational number can be.
A rational number is defined to be a number that can be expressed
as the ratio of two integers. An irrational number is any real
number that is not rational. A rational number is a number that can
be expressed as a fraction. An irrational number is one that can
not.
Some examples of rational numbers would be 5, 1.234, 5/3, or
-3
Some examples of irrational numbers would be π, the square root
of 2, the golden ratio, or the square root of 3.
A rational number is a number that either has a finite end or a
repeating end, such as .35 or 1/9 (which is .1111111
repeating).
An irrational number has an infinite set of numbers after the
decimal that never repeat, such a the square root of 2 or pi.
A rational number is one that can be expressed as a ratio of two
integers, x and y (y not 0). An irrational number is one that
cannot be expressed in such a form.
In terms of decimal numbers, a rational number has a decimal
representation that is terminating or [infinitely] recurring. The
decimal representation for an irrational is neither terminating nor
recurring. (Recurring decimals are also known as repeating
decimals.)
A rational number is a number that can be expressed as a fraction.
An irrational number is one that can not.
Some examples of rational numbers would be 5, 1.234, 5/3, or
-3
Some examples of irrational numbers would be π, the square root
of 2, the golden ratio, or the square root of 3.
An irrational number is a number that can't be expressed by a
fraction having integers in both its numerator and denominator. A
rational number can be.
A rational number can be represented by a ratio of whole numbers.
An irrational number cannot. There are many more irrational numbers
than there are rational numbers
Rational numbers are numbers that can be written as a fraction.
Irrational numbers cannot be expressed as a fraction.
A rational number can be expressed as a fraction, with integers in
the numerator and the denominator. An irrational number can't be
expressed in that way. Examples of irrational numbers are most
square roots, cubic roots, etc., the number pi, and the number
e.
A rational number can always be written as a fraction
with whole numbers on the top and bottom.
An irrational number can't.
A rational number can always be written as a fraction with whole
numbers on top and bottom.
An irrational number can't.
Any number that you can completely write down, with
digits and a decimal point
or a fraction bar if you need them, is a rational number.
A rational number can be expressed as a fraction whereas an
irrational can not be expressed as a fraction.
Just look at the definition of a rational number. A rational number
is one that can be expressed as a fraction, with integers (whole
numbers) in the numerator and the denominator. Those numbers that
can't be expressed that way - for example, the square root of 2 -
are said to be irrational.
A rational number is any number that can be written as a ratio or
fraction. If the decimal representation is finite orhas a
repeating set of decimals, the number is rational.
Irrational numbers cannot be reached by any finite use of the
operators +,-, / and *. These numbers include square roots of
non-square numbers, e.g.√2
.
Irrational numbers have decimal representations that never end
or repeat.
Transcendental numbers are different again - they are
irrational, but cannot be expressed even with square roots or other
'integer exponentiation'. They are the numbers in between the
numbers between the numbers between the integers. Famous examples
include
e or pi (π).
By definition: a rational number can be expressed as a ratio of two
integers, the second of which is not zero. An irrational cannot be
so expressed.
One consequence is that a rational number can be expressed as a
terminating or infinitely recurring decimal whereas an irrational
cannot.
This consequence is valid whatever INTEGER base you happen to
select: decimal, binary, octal, hexadecimal or any other - although
for non-decimal bases, you will have the "binary point" or "octal
point" in place of the decimal point and so on.
A rational number can be expressed as a fraction whereas an
irrational number can't be expressed as a fraction
Rational numbers can be expressed as a ratio of two integers, x/y,
where y is not 0. Conventionally, y is taken to be greater than 0
but that is not an essential element of the definition. An
irrational number is one for which such a pair of integers does not
exist.
Rational numbers can be expressed as one integer over another
integer (a "ratio" of the two integers) whereas irrational numbers
cannot.
Also, the decimal representation of
a rational number will either:
terminate (eg 31/250 = 0.124); or
go on forever repeating a sequence of digits at the end (eg
41/330 = 0.1242424... [the 24 repeats]);
whereas an irrational number will not terminate, nor will there
be a repeating sequence of digits at the end (eg π = 3.14159265....
[no sequence repeats]).
Rational numbers are numbers that keeps on going non-stop, for
example pie. Irrational numbers end. Its as simple as that!
Improved Answer:-
Rational numbers can be expressed as fractions whereas
irrational numbers can't be expressed as fractions.
a rational number can be expressed as a fraction in the form a/b
(ie as a fraction).
a irrational number cannot be expressed as a fraction (e.g. pi,
square root of 2 etc)
Rational numbers can be represented as fractions.
That is to say, if we can write the number as a/b where a and b
are any two integers and b is not zero. If we cannot do this, then
the number is irrational.
For example, .5 is a rational number because we can write it as
5/10=1/2
The square root of 2 is irrational because there do not exist
integers a and b such
that square root of 2 equals a/b.
Rational numbers can be expressed as fractions whereas irrational
numbers can't be expressed as fractions.