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Why 3 is not a perfect square for any rational number plus proof?
Wiki User
∙ 13y ago
Assume 3 is a perfect square for a rational number, a/b where a and b are integers in lowest terms.
Now means 3=a2 /b2 which implies the square root of 3=a/b but the square root of 3 is irrational so this is not possible.
But we need one more part to show that.
Suppose
3 = p2/q2
where p and q are integers and p/q is in lowest terms. So
3 q2 = p2 as we did above with a and b.
This means p^2 is divisible by 3. That means that p must be as well, so p^2 is
divisible by nine.
So
q2 = p2/3
and q^2 is divisible by three.
But that means that p and q are both divisible by three, so they weren't
in lowest terms, which is a contradiction because we said they were.
Wiki User
∙ 13y ago
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Is the square root of 14 rational or irrational number?
Search for the proof for the irrationality of the square root of 2. The same reasoning applies to any positive integer that is not a perfect square. In summary, the square root of any positive integer is either a whole number, or - as in this case - it is irrational.
Why is the square root of 33 irrational?
Most high school algebra books show a proof (by contradiction) that the square root of 2 is irrational. The same proof can easily be adapted to the square root of any positive integer, that is not a perfect square. You can find the proof (for the square root of 2) on the Wikipedia article on "irrational number", near the beginning of the page (under "History").
Why is 3 square root of 2 is irrational?
sqrt(2) is irrational. 3 is rational. The product of an irrational and a non-zero rational is irrational. A more fundamental proof would follow the lines of the proof that sqrt(2) is irrational.
How is the sum of a rational and irrational number irrational?
This can easily be proved by contradiction. Without loss of generality, I will take specific numbers as an example. The proof can easily be extended to any rational + irrational number. Assumption: 1 plus the square root of 2 is rational. (It is a well-known fact that the square root of 2 is irrational. No need to prove it here; you can use any other irrational number will do.) This rational sum can be written as p / q, where "p" and "q" are whole numbers (this is basically the definition of a "rational number"). Then, the square root of 2, which is equal to the sum minus 1, is: p / q - 1 = p / q - q / q = (p - q) / q Since the difference of two whole numbers is a whole number, this makes the square root of 2 rational, which doesn't make sense.
Is there a proof that irrational numbers can be derived from rationals numbers?
I am not quite sure what you mean with "derive" - what sort of derivation you will accept. If you take the square root of an integer, unless the integer happens to be a perfect square, you get an irrational number. And yes, there is proof of that. The can be found in most high school algebra books.
Is the square root of 32 an irrational or a rational number?
The square root of a positive integer can ONLY be:* Either an integer, * Or an irrational number. (The proof of this is basically the same as the proof, in high school algebra books, that the square root of 2 is irrational.) Since in this case 32 is not the square of an integer, it therefore follows that its square root is an irrational number.
Is the square root of 14 rational or irrational number?
Search for the proof for the irrationality of the square root of 2. The same reasoning applies to any positive integer that is not a perfect square. In summary, the square root of any positive integer is either a whole number, or - as in this case - it is irrational.
Is the square root of 3 rational?
Root of '3' is NOT rational. It is an IRRATIONAL Number. To 9 d.p. it is sqrt(3) = 1.732050808.... NB THE square roots of prime numbers are irrational , just like 'pi = 3.141592....'. NNB A irrational number, put casually, is a number were the decimals go to inifinty and there is no regular order in the number2.
P is irrationalHow can you prove square root of p is irrational?
By an indirect proof. Assuming the square root is rational, it can be written as a fraction a/b, with integer numerator and denominator (this is basically the definition of "rational"). If you square this, you get a2/b2, which is rational. Hence, the assumption that the square root is rational is false.
Proof that square of any rational number is rational?
Consider a rational number, p.p is rational so p = x/y where x and y are integers.x is an integer so x*x is an integer, and y is an integer so y*y is an integer.So p2 = (x/y)2 = x2/y2 is a ratio of two integers and so is rational.
Why is the square root of 33 irrational?
Most high school algebra books show a proof (by contradiction) that the square root of 2 is irrational. The same proof can easily be adapted to the square root of any positive integer, that is not a perfect square. You can find the proof (for the square root of 2) on the Wikipedia article on "irrational number", near the beginning of the page (under "History").
Why is 3 square root of 2 is irrational?
sqrt(2) is irrational. 3 is rational. The product of an irrational and a non-zero rational is irrational. A more fundamental proof would follow the lines of the proof that sqrt(2) is irrational.
How is the sum of a rational and irrational number irrational?
This can easily be proved by contradiction. Without loss of generality, I will take specific numbers as an example. The proof can easily be extended to any rational + irrational number. Assumption: 1 plus the square root of 2 is rational. (It is a well-known fact that the square root of 2 is irrational. No need to prove it here; you can use any other irrational number will do.) This rational sum can be written as p / q, where "p" and "q" are whole numbers (this is basically the definition of a "rational number"). Then, the square root of 2, which is equal to the sum minus 1, is: p / q - 1 = p / q - q / q = (p - q) / q Since the difference of two whole numbers is a whole number, this makes the square root of 2 rational, which doesn't make sense.
Can it be demonstrated that there is a difference between the number of rational numbers and the number of irrational numbers?
Yes. Google Cauchy's proof.
Why is the square root of 10 irrational?
The argument why the square root of 2 is irrational can be found in most high school algebra books. You can also find this proof, and several other proofs, that the square root of 2 is irrational, in the Wikipedia article "Square root of 2".The same argument can be applied to the square root of any natural number that is not a perfect square.
Is there a proof that irrational numbers can be derived from rationals numbers?
I am not quite sure what you mean with "derive" - what sort of derivation you will accept. If you take the square root of an integer, unless the integer happens to be a perfect square, you get an irrational number. And yes, there is proof of that. The can be found in most high school algebra books.
Is the 2 square root of 5 rational or irrational why?
It is irrational. The proof depends on the proof that sqrt(5) is irrational. However, judging by this question, I suggest that you are not yet ready for that proof. So, assume that sqrt(5) is irrational. Any multiple of an irrrational number by a non-zero rational isirrational. For suppose 2*sqrt(5) were rational that is 2*sqrt(5) = p/q for some integers p and q, where q is nonzero. then dividing both isdes by 2 gives sqrt(5) = p/(2q) where p and 2q are both integers and 2q in non-zero. But that implies that sqrt(5) is rational! That is a contradiction so 2*sqrt(5) cannot be rational.
