From the Story
"Platform nine and three-quarters is the train platform from which students board the Hogwarts Express, the scarlet steam engine that brings students to and from Hogwarts. It can be accessed by walking straight through the apparently solid barrier between platforms nine and ten."
J. K. Rowling's Thoughts
"In choosing the number of the concealed platform that would take young witches and wizards to boarding school, I decided that it would have to be a number between those of the Muggle platforms - therefore, it was clearly a fraction. This raised the interesting question of how many other fractional platforms lay between the whole-numbered platforms at King's Cross, and I concluded that were probably quite a few. Although these are never mentioned in the book, I like to think that it is possible to take a version of the Orient Express off to wizard-only villages in continental Europe (try platform seven and a half), and that other platforms may be opened on an as-required-basis, for instance for large, one-off events such as Celestina Warbeck concerts (see your ticket for details).
"The number nine and three-quarters presented itself without much conscious thought, and I liked it so much that I took it at once. It is the 'three-quarters' that makes it, of course."