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.
A perfect square.
They are squares of rational numbers. there is no particular name for them.
How about 27 whose cube root is 3 which is a rational whole number.
It's rational. It can be written as the quotient of two numbers whose HCF is one.
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 perfect square.
It is another rational number whose numerator and denominator (in the ratio's simplest form) are perfect squares.
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.
It depends on the number whose square root is being taken.sqrt(4)/3 is rational but sqrt(5)/3 is not.
No, because 15 is not a perfect square. The closest perfect square is 16, whose square root is 4.
They are squares of rational numbers. there is no particular name for them.
No, but it is irrational, because there is no rational number whose square is two. Imaginary numbers are the square roots of negative numbers.
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.
Yes, for example: square root of 2, and the negative of the square root of 2.
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.
How about 27 whose cube root is 3 which is a rational whole number.
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.