Question

The difference of a rational and irrational number is always

Answer 1

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Answer 2

The difference of a rational and irrational number is always irrational

Definition of irrational number:

Any real number that cannot be expressed as the quotient of two integers. For example, there is no number among integers and fractions that equals the square root of 2.

Rational and irrational numbers:

Any number that can be expressed as an integer split by another integer is a rational number.

Take integers $A, B, C, D$

$\frac{A}{B}+X=\frac{C}{D}$

Now we'll show that $X$ has to be rational.

$\begin{aligned}&X=\frac{C}{D}-\frac{A}{B} \\&X=\frac{C B-A D}{B D}\end{aligned}$

• The top and bottom of the fraction are both integers since an integer added, subtracted, or multiplied with another integer is always an integer. As a result, $X$ has a rational value.
• If you're subtracting, just pretend $X$ is negative; the math is the same.
• As a result, if you add or subtract a rational from an unknown and get a rational result, the unknown is rational.
• Irrational is a rational plus or minus an irrational.

Then the difference of a rational and irrational number is always irrational