According to Hoffman, jazz can be loosely defined as a musical genre features the use of two primary musical elements: a swing beat and improvisation. According to Cyberessays.com's article on Jazz and Classical Music, some notable elements of jazz music include imprecise notation, interaction between band members (as opposed to the use of a conductor, as seen in classical music), and, again, the use of improvisation.
Cyberessays.com rightfully notes that jazz is a specifically American form of music. In the early 20th century, Americans began to produce jazz music, mostly in the Southern states. The first movement of jazz was known as "Dixieland Jazz," and its primary musician was the famous Louis Armstrong. After World War I, jazz became more popular, and the Swing movement emerged. With the growth of the recording industry, jazz was all the rage. In the 1940s and 1950s, the genre took a more sophisticated turn; the music became more complex, as heard, for example, in the work of Charlie Parker.
Most jazz musicians learned music by ear and were not formally educated in music theory, as noted at Cyberessays.com. This resulted in a great divide between jazz and classical music. Most classical composers were educated in music theory from a very young age. To play classical music, musicians were required to have an extensive knowledge of theory and technique. Jazz, on the other hand, as acknowledged by Hoffman, is primarily a folk music, originating with common, often uneducated people. Nevertheless, it is also true that jazz owes a great deal of its legacy to the classical tradition. The great Duke Ellington, for example, was a lover of classical music, as noted at Blogiversity.org.
This is not to say, however, that jazz had no influence on classical music or that the two are irreconcilable. In America, classical composers were dramatically affected by the rise of jazz. NPR.org's online outline of a "Milestones of the Millenium" show featuring the influence of jazz on classical music gives several examples of such musicians. Perhaps the most famous is George Gershwin and his "Rhapsody in Blue." In fact, as noted at Blogiversity.org, the premiere of this piece was played by Paul Whiteman, a jazz musician. Aaron Copland's "Music From the Theater," among other pieces, is unmistakably influenced by jazz music, as he acknowledged.
The influence of jazz stretched across the ocean. In 1923, Darius Milhaud was one of the first composers to use jazz elements in a classical piece, with his ballet, "The Creation of the World." Igor Stravinsky and Leonard Bernstein also used jazz elements regularly in their compositions, such as Stravinsky's "Ebony Concerto" and Bernstein's "Prelude, Fugue and Riff."
Jazz continues to influence classical music. As noted by Hoffman, several hybrids of jazz and classical have come into existence since the emergence of jazz, often called the "jazz crossover" movement.