A rational number is any number that can be written as one integer divided by another, and adding them fills the gaps on the line.
A rational number has the form p over q, where p and q are integers and q is not zero. Rational comes from ratio, not from being reasonable.
Which numbers are already rational
Every integer is rational, because 7 is 7 over 1. Every terminating decimal is, because 0.75 is 75 over 100. The new set does not replace the old ones; it contains them.
Two thirds, four sixths and 0.666 recurring are one single point on the number line wearing different costumes.
Putting a rational number on the line
The bottom number tells you how many equal pieces to cut each unit into; the top number tells you how many to count.
Give them the same bottom number and compare the tops, or turn both into decimals. Five sevenths is 50 over 70; seven tenths is 49 over 70.
Rational numbers as decimals
Either the division finishes, giving a terminating decimal, or a block of digits repeats for ever. There is no third possibility.
Dividing by 7, the only remainders are 0 to 6. Within seven steps some remainder must repeat, and from then the digits repeat too.
A calculator shows one third as 0.3333333, which is a rounded copy. Multiply it by 3 and you get 0.9999999 rather than 1.
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