Question

Identify the number that does not belong with the other three. Explain your reasoning. 50.1 repeating 1, negative 50 over 2, negative 50.1, square root 50

Answer 1

Let's quickly go over rational and irrational numbers
A rational number is a number which can be expressed in the form of
\[\frac{p}{q}\]
where
\[q \neq 0\]
and
\[p,q \in \mathbb{Z}\]
that is p and q are integers
irrational numbers r just opposite
Let's look at first number,
\[50.111...\]
Let
\[S=50.111....\]
therefore
\[10S=501.111...\]
Multiplying by 10 will take 1 out from the decimal, but still an infinitely amount of 1's in
subtracting 2nd equation by first we get
\[10S-S=(501.111...-50.111...)=(501-50)+(0.111...-0.111...)\]
We have simply taken the decimal parts apart
it's the same as writing
\[2.34=2+0.34\]
Now we have
\[9S=451\]\[S=\frac{451}{9}\]
Which is of the form
\[\frac{p}{q}, p,q \in \mathbb{Z}, q \neq 0\]
If you do the division of 451 by 9 in a calculator you will indeed get your number back again
Thus 50.111..... is a RATIONAL number, in general all numbers with terminating decimals(decimals that end after some numbers)
eg. \[2.45\]\[55.2\]
and numbers with non terminating and repeating decimals(decimals that keep on repeating to infinity)
\[2.474747... \]
\[5.77777...\]
are RATIONAL
But if a number has non terminating and NON repeating decimals, it is IRRATIONAL
eg. infamous pi
\[\pi = 3.14159...\]
You will never find a pattern that is repeating in the decimal expansion of pi
Note that 22/7 is an approximation of pi, pi does not equal to 22/7
\[\pi \approx \frac{22}{7}\]
another eg.
\[\sqrt{2}=1.4142....\]
In general square root of a number which is not a prefect square is IRRATIONAL
By perfect square I mean a number who has a simple and integer square root
\[\sqrt{4}=2\]
So 4 is a perfect square, it has an integer square root
but root 2 cannot be written as an integer so it is an irrational number
Your 2nd number
\[-\frac{50}{2}=-25\]
It is a rational number where the denominator is 1(an integer not equal to 0) and numerator is also integer(-25)
similarly your 3rd number has a terminating decimal, so it's a rational number
however what about
\[\sqrt{50}\]
rational or irrational??
The notation
\[p,q \in \mathbb{Z}, q \neq 0\]
Means p and q are integers and q is not equal to zero