Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.4

4. Prove or disprove each of the following statements.
a) The sum of a rational and an irrational number is irrational.
b) The sum of two irrational numbers is irrational.
c) The product of a rational number and an irrational number is irrational.
d) The product of two irrational numbers is irrational.

Solution:

(a) The sum of rational and irrational number is irrational. We prove the statement by contradiction. Let us assume that there is a rational number x and an irrational number y such that (x+y) is rational.
Now we define rational numbers as  $x=\frac{a}{b}$…..(1)

For some integers a,b with $b\neq0$ And $x+y=\frac{c}{d}$…..(2)

For some integers c,d with $d\neq0$

we substitute (1) in (2) and get

$\frac{a}{b}+y=\frac{c}{d}$

and so $y=\frac{c}{d}-\frac{a}{b}$ $y=\frac{(bc-ad)}{bd}$

Now (bc-ad) and bd are both integers since products and differences of integers are integers and $bd\neq0$.
Hence, y is a ratio of integers (bc-ad) and bd with $bd\neq0$. So, by the definition of Rational, y is rational. This contradicts the fact that y is irrational. Hence our assumption is wrong and the statement is true.

(b) The sum of two irrational numbers may not necessarily be irrational.
Let us take two irrational numbers $\pi$ and $-\pi$.$\pi$ is irrational as it cannot be expressed as a ratio of two integers.

Summing these irrational numbers gives us $\pi+(-\pi)=\pi-\pi=0$

The number 0 is a rational number. Hence, our statement is justified.

(c) The product of a rational and an irrational number may not necessarily be irrational.

Let us take 0 as a rational number and $\pi$ as an irrational number. We know that any number multiplied by 0 gives 0, that is,
$\pi.0=0$
Hence, our statement is justified.

(d) The product of two irrational numbers may not necessarily be irrational.

Let us take two irrational numbers $\pi$ and $\frac{1}{\pi}$.

Now, $\pi.\frac{1}{\pi}=\frac{\pi}{\pi}=1$

But 1 is rational number. Hence our statement is justified.

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