Assume = a/b with positive integers a und b. Now, for some natural number n define the functions f and F as follows. Strictly speaking, f and F should each have n as an index as they depend on n but this would render things unreadable; remember that n is always the same constant throughout this proof. Let f(x) = xn(a-bx)n/n! and let F(x) = f(x) + ... + (-1)jf(2j)(x) + ... + (-1)nf(2n)(x) where f(2j) denotes the 2j-th derivative of f. Then f and F have the following properties: f is a polynomial with coefficients that are integer, except for a factor of 1/n! f(x) = f(-x) 0

I'm assuming that you mean 'square root'. Yes, this sum is irrational. So are each of the two numbers alone. A simple proof can be done by writing x=square root 2 + square root 3 and then "squareing away" the square roots and then use the rational roots theorem. The sum or difference of two irrational number need not be irrational! Look at sqrt(2)- sqrt(2)=0 which is rational.

rational

is it

'Quarante' is the french word for forty.

peenutbudr

No because integers are whole numbers

The square roots of 36 are +6 & -6

Remember +6 x + 6 = (+)36

Also

-6 x -6 = (+)36 Double minus becomes positive, so square roots must show both plus and minus answers.

= +36

Yes, and here's the proof:

Let's start out with the basic inequality 4 < 7 < 9.

Now, we'll take the square root of this inequality:

2 < √7 < 3.

If you subtract all numbers by 2, you get:

0 < √7 - 2 < 1.

If √7 is rational, then it can be expressed as a fraction of two integers, m/n. This next part is the only remotely tricky part of this proof, so pay attention. We're going to assume that m/n is in its most reduced form; i.e., that the value for n is the smallest it can be and still be able to represent √7. Therefore, √7n must be an integer, and n must be the smallest multiple of √7 to make this true. If you don't understand this part, read it again, because this is the heart of the proof.

Now, we're going to multiply √7n by (√7 - 2). This gives 7n - 2√7n. Well, 7n is an integer, and, as we explained above, √7n is also an integer; therefore, 7n - 2√7n is an integer as well. We're going to rearrange this expression to (√7n - 2n)√7, and then set the term (√7n - 2n) equal to p, for simplicity. This gives us the expression √7p, which is equal to 7n - 2√7n, and is an integer.

Remember, from above, that 0 < √7 - 2 < 1.

If we multiply this inequality by n, we get 0 < √7n - 2n < n, or, from what we defined above, 0 < p < n. This means that p < n and thus √7p < √7n. We've already determined that both √7p and √7n are integers, but recall that we said n was the smallest multiple of √7 to yield an integer value. Thus, √7p < √7n is a contradiction; therefore √7 can't be rational, and so must be irrational.

Q.E.D.

They never repeat in any pattern.

If they never repeated, you could have at most 10 digits after the decimal point and therefore the decimal representation would be terminating.

It is rational because it can be expressed as a fraction which is 21/5

rational

'Zero' and 'one' are the only numbers that are written the same in any base.

A number is said to be rational if it can be expressed as a ratio of two [finite] integers [the second of which is non-zero].

Here, if pi is to be rational, both p and q must be integers. If p is an integer then q will be irrational and if q is an integer, p will be irrational. They can never both be rational and so the ratio definition cannot be satisfied.

No, it is rational.

No, it is rational.

No, it is rational.

Rational.

Yes

Yes, just look at decimals.

Note: integers are a subset of rational numbers.

Assuming that the notation "666..." represents the number with infinitely many 6s, the number is rational.

Because it can't be expressed as a fraction