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Show that the sum of rational no with an irrational no is always irrational?
Wiki User
∙ 9y ago
Suppose x is a rational number and y is an irrational number.
Let x + y = z, and assume that z is a rational number.
The set of rational number is a group.
This implies that since x is rational, -x is rational [invertibility].
Then, since z and -x are rational, z - x must be rational [closure].
But z - x = y which implies that y is rational.
That contradicts the fact that y is an irrational number. The contradiction implies that the assumption [that z is rational] is incorrect.
Thus, the sum of a rational number x and an irrational number y cannot be rational.
Wiki User
∙ 9y ago
Wiki User
∙ 9y agoSuppose x is a rational number and y is an irrational number.
Let x + y = z, and assume that z is a rational number.
The set of rational number is a group.
This implies that since x is rational, -x is rational [invertibility].
Then, since z and -x are rational, z - x must be rational [closure].
But z - x = y which implies that y is rational.
That contradicts the fact that y is an irrational number. The contradiction implies that the assumption [that z is rational] is incorrect.
Thus, the sum of a rational number x and an irrational number y cannot be rational.
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The sum of two irrational is always irrational?
No. In fact, the sum of conjugate irrational numbers is always rational.For example, 2 + sqrt(3) and 2 - sqrt(3) are both irrational, but their sum is 4, which is rational.
Explain why the sum of a rational number and an irrational number is an irrational number?
Let R1 = rational number Let X = irrational number Assume R1 + X = (some rational number) We add -R1 to both sides, and we get: -R1 + x = (some irrational number) + (-R1), thus X = (SIR) + (-R1), which implies that X, an irrational number, is the sum of two rational numbers, which is a contradiction. Thus, the sum of a rational number and an irrational number is always irrational. (Proof by contradiction)
Is it true that sum of a rational number and irrational number is irrational?
Yes
Why an irrational number plus an irrational number equal a rational?
That simply isn't true. The sum of two irrational numbers CAN BE rational, but it can also be irrational. As an example, the square root of 2 plus the square root of 2 is irrational.
If you add a rational and irrational number what is the sum?
an irrational number PROOF : Let x be any rational number and y be any irrational number. let us assume that their sum is rational which is ( z ) x + y = z if x is a rational number then ( -x ) will also be a rational number. Therefore, x + y + (-x) = a rational number this implies that y is also rational BUT HERE IS THE CONTRADICTION as we assumed y an irrational number. Hence, our assumption is wrong. This states that x + y is not rational. HENCE PROVEDit will always be irrational.
Is the sum of a rational and irrational number rational or irrational?
It is always irrational.
Is the sum of two rational numbers is it rational or irrational?
Such a sum is always rational.
The sum of a rational number and an irrational number?
Such a sum is always irrational.
What is an irrational plus two rational numbers?
Since the sum of two rational numbers is rational, the answer will be the same as for the sum of an irrational and a single rational number. It is always irrational.
Is the sum of an irrational number and rational always irrational?
Yes.
Are the sum of two rational numbers rational or irrational?
They are always rational.
Is the sum of a rational number and an irrational number always irrational?
Yes, always.
Is the sum of a rational and irrational number always irrational?
Yes, that is so.
The sum of rational numbers and an irrational number?
It is always an irrational number.
What is the classification of the sum of a rational number and an irrational number?
It is always an irrational number.
Is the sum of a rational and irrational number never an irrational number?
Wrong. It is always an irrational number.
The sum of two irrational is always irrational?
No. In fact, the sum of conjugate irrational numbers is always rational.For example, 2 + sqrt(3) and 2 - sqrt(3) are both irrational, but their sum is 4, which is rational.
