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The proof is based on proving that sqt(2) is irrational, by the method of reductio ad absurdum.
We start by assuming that sqrt(2) is rational.
That means that it can be expressed in the form p/q where p and q are co-prime integers.
Thus sqrt(2) = p/q.
This can be simplified to 2*q^2 = p^2
Now 2 divides the left hand side (LHS) so it must divide the right hand side (RHS).
That is, 2 must divide p^2 and since 2 is a prime, 2 must divide p.
That is p = 2*r for some integer r.
Then substituting for p gives,
2*q^2 = (2*r)^2 = 4*r^2
Dividing both sides by 2 gives q^2 = 2*r^2.
But now 2 divides the RHS so it must divide the LHS.
That is, 2 must divide q^2 and since 2 is a prime, 2 must divide q.
But then we have 2 dividing p as well as q which contradicts the requirement that p and q are co-prime.
The contradiction implies that sqrt(2) cannot be rational.
Finally, sqrt(8) = 2*sqrt(2) and the product of a rational and an irrational is irrational. Therefore sqrt(8) is irrational.
Alternatively, you could go through 2 more cycles of the above process of 2 divides LHS/RHS and so must divide the other side, etc.
What else can I help you with?
A rational number whose square root is a whole number?
A perfect square.
What is A number whose square roots are integers or quotients of integers?
They are squares of rational numbers. there is no particular name for them.
What is a rational whose cube root is a whole number?
How about 27 whose cube root is 3 which is a rational whole number.
Is the number 0.112123123412345. a rational or irrational number?
It's rational. It can be written as the quotient of two numbers whose HCF is one.
What will happen to a whole number when it's multiplied by a fraction that is greater than 1?
The product will be a rational number whose absolute value is bigger than the absolute value of the whole number.The product will be a rational number whose absolute value is bigger than the absolute value of the whole number.The product will be a rational number whose absolute value is bigger than the absolute value of the whole number.The product will be a rational number whose absolute value is bigger than the absolute value of the whole number.
A rational number whose square root is a whole number?
A perfect square.
What is a number whose square root is a rational number?
It is another rational number whose numerator and denominator (in the ratio's simplest form) are perfect squares.
A short note on square root of rational number?
Square root of a rational number may either be rational or irrational. For example 1/4 is a rational number whose square root is 1/2. Similarly, 4 is 4/1 which is rational and the square root is 2 which of course is also rational. However, 1/2 and 2 are rational, but their square roots are irrational. We can say the square root of a rational number is always a real number. We can also say the rational numbers whose square roots are also rational are perfect squares or fractions involving perfect squares.
Is the square root over 3 rational number?
It depends on the number whose square root is being taken.sqrt(4)/3 is rational but sqrt(5)/3 is not.
Is the square root of a15 a rational number?
No, because 15 is not a perfect square. The closest perfect square is 16, whose square root is 4.
What is A number whose square roots are integers or quotients of integers?
They are squares of rational numbers. there is no particular name for them.
Is the square root of 2 imaginary?
No, but it is irrational, because there is no rational number whose square is two. Imaginary numbers are the square roots of negative numbers.
If you take the square root of an irrational number.Will it be irrational too?
Certainly. Otherwise, there would be a rational number whose square was an irrational number; that is not possible. To show this, let p/q be any rational number, where p and q are integers. Then, the square of p/q is (p^2)/(q^2). Since p^2 and q^2 must both be integers, their quotient is, by definition, a rational number. Thus, the square of every rational number is itself rational.
Are there two irrational number whose sum and production are rational?
Yes, for example: square root of 2, and the negative of the square root of 2.
Is -0.6 a perfect square?
No, -0.6 is not a perfect square. A perfect square is a number that can be expressed as the square of an integer or a rational number. Since the square of any real number is non-negative, there are no real numbers whose square equals -0.6.
What is a rational whose cube root is a whole number?
How about 27 whose cube root is 3 which is a rational whole number.
A number whose squares roots are integers or quotients of integers?
A number whose square roots are integers or quotients of integers is known as a rational number. Specifically, it can be expressed as the square of a rational number, meaning it can be written in the form ( \left( \frac{p}{q} \right)^2 ), where ( p ) and ( q ) are integers and ( q \neq 0 ). Examples of such numbers include perfect squares like 1, 4, and 9, as well as rational square roots like ( \frac{1}{4} ) or ( \frac{9}{16} ). In general, any rational number that can be expressed as a fraction of integers can also have rational square roots.
