Show that Q, the set of all rational numbers, is dense along the number line by showing that given any two rational line by showing that given any two rational numbers and with , there exists a rational number x such that .
Along the number line, the set of all rational numbers is dense by showing that, for any two rational numbers , there exists a rational number between two rational numbers .
The set of rational number is defined as follows.
The set rational number can also be written as .
Let be any two rational number on the number line such that .
Now show that there exists a rational number between
The midpoint between any two numbers on the number line is its average.
That means the midpoint for the numbers on the number line will be their average.
That means and it lies exactly middle of the numbers on the number line.
The number is obviously a rational number and it can be shown as follows.
Let where .
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