Video Transcript
Is every noninteger number a
rational number?
In this question, we need to
determine if every noninteger number is necessarily a rational number.
We can start by recalling that
integers are the whole numbers. So they are the positive whole
numbers, the negative whole numbers, and zero. The nonintegers will include any
number that is not a whole number. We can also recall that the
rational numbers is the set of quotients of integers where the denominator is
nonzero. We can then answer this question by
recalling that the square root of two is not an integer, and it is also not a
rational number. So, the answer is no, not every
noninteger number is a rational number.
Although this is enough to answer
this question, we can prove that this is the case by showing that the square root of
two cannot be written as the quotient of integers. To do this, let’s start by assuming
that root two is a rational number. So, it can be written as 𝑎 over
𝑏, where 𝑎 and 𝑏 are integers and 𝑏 is nonzero. We can assume that the greatest
common divisor of 𝑎 and 𝑏 is one, since we can cancel any common factors in the
numerator and denominator.
We can now square both sides of the
equation to get that two is equal to 𝑎 squared over 𝑏 squared. If we multiply through by 𝑏
squared, we can note that the left-hand side of the equation is even since it is the
product of an even number and an integer. This means that the right-hand side
of the equation must also be even; this can only be true if 𝑎 is even. Let’s say that 𝑎 is two times some
integer 𝑐. If we substitute 𝑎 equals two 𝑐
into the equation, we obtain two 𝑏 squared equals two 𝑐 all squared. We can rewrite two 𝑐 all squared
as four 𝑐 squared and then divide both sides of the equation through by two. This gives us that 𝑏 squared
equals two 𝑐 squared.
We can now use the same reasoning
we did before. We see that two 𝑐 squared is
even. So, 𝑏 squared must also be even,
meaning that 𝑏 itself is even. However, we already showed that 𝑎
and 𝑏 have a greatest common divisor of one. And we also know that neither 𝑎 or
𝑏 can be zero, so this is not possible. Hence, our original assumption that
the square root of two is rational must be incorrect. So, the square root of two is an
example of a noninteger that is not a rational number.
