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The conjugate of an irrational number is a non-zero number such that the product of the two numbers is rational.
In high school mathematics, you are usually required to know only the conjugates of surds of the form a + b*sqrt(x).
The conjugate is a - b*sqrt(x)
[-a + b*sqrt(x) is also a conjugate - all you need to do is to switch the sign of one of the two terms.]
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What is the conjugate of square root of 13?
If the square root of a number is irrational, it is its own conjugate. sqrt(13)*sqrt(13) = 13 and you no longer have an irrational!
How do you rationalise the denominator?
It depends on what the denominator was to start with: a surd or irrational or a complex number. You need to find the conjugate and multiply the numerator by this conjugate as well as the denominator by the conjugate. Since multiplication is by [conjugate over conjugate], which equals 1, the value is not affected. If a and b are rational numbers, then conjugate of sqrt(b) = sqrt(b) conjugate of a + sqrt(b) = a - sqrt(b), and conjugate of a + ib = a - ib where i is the imaginary square root of -1.
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.
Is an irrational number plus an irrational number rational?
No. The sum of an irrational number and any other [real] number is irrational.
What is a negative irrational number?
A negative irrational number can be thought of as an irrational number multiplied by -1, or an irrational number with a minus sign in front of it.
