Question

Ms. Fox asked her class is the sum of 4.2 and square root of 2 rational or irrational? Patrick answered that the sum would be irrational. State whether Patrick is correct or incorrect. Justify your reasoning.

Answer 1

The sum `4.2 + sqrt2` is irrational; it inherits the never-repeating decimal expansion property of `sqrt 2`.

Explanation:

An irrational number is a number that cannot be expressed as a ratio of two integers. If a number is irrational, then its decimal expansion goes on forever without a pattern, and vice versa.

We already know that `sqrt 2` is irrational. Its decimal expansion begins:

`sqrt 2 = 1.414213562373095...`

The number `4.2` is rational; it can be expressed as `42/10.` When we add 4.2 to the decimal expansion of `sqrt 2`, we get:

`sqrt 2 + 4.2 = color(white)+ 1.414213562373095...`
`color(white)(sqrt 2) color(white)+ color(white)(4.2 = )+ 4.2`
`color(white)(sqrt 2) color(white)+ color(white)(4.2 = )bar(color(white)(+)5.614213562373095...)`

It is easily seen that this sum also does not terminate nor have a repeating pattern, so it is also irrational.

In general, the sum of a rational number and an irrational number will always be irrational; the argument is similar to above.

Answer 2

`color(blue)("correct")`

Explanation:

If we start by saying the sum is rational: All rational numbers can be written as the quotient of two integers `a/bcolor(white)(88)` `b!=0`

`4.2=21/5`

`21/5+sqrt(2)=a/b`

`sqrt(2)=a/b-21/5`

`sqrt(2)=(5a-21b)/(5b)`

The product of two integers is an integer:

The difference of two integers is an integer:

So:

`5a-21b` is an integer.

`5b` is an integer.

Hence:

`(5a-21b)/(5b)` is rational.

But we know that `sqrt(2)` is irrational, so this is a contradiction from our assumption that the sum was rational, therefore the sum of an irrational number and a rational number is always irrational.