Which number produces an irrational number when added to 1/4

Any irrational number, e. g.

√

2

Step-by-step explanation:

x

+

1

4

is irrational if and only if

x

is irrational.

Equivalently,

x

+

1

4

is rational if and only if

x

is rational.

To prove this we can proceed as follows:

First suppose that

x

+

1

4

is rational.

Then there are some integers

p

,

q

, with

q

>

0

such that:

x

+

1

4

=

p

q

Subtracting

1

4

from both sides, this becomes:

x

=

p

q

-

1

4

=

4

p

-

q

4

q

which is rational.

Conversely, if

x

is rational, then there are integers

m

,

n

with

n

>

0

such that

x

=

m

n

and we find:

x

+

1

4

=

m

n

+

1

4

=

4

m

+

n

4

n

which is also rational.