In Fahrenheit 451, Part 2, "The Sieve and the Sand," what is the importance of the dentifrice commercial?

Assuming that "importance" refers to literary
importance—or the importance of the scene to literary elements and development—rather
than referring to social criticism importance, then the literary
importance of the Denham's Dentifrice commercial is that it quite
intensely reveals the violent inner struggle Montag is going through. He is trying to
extricate himself from one false society and embed himself in a true society because he
has learned "of a time when books were legal and people did not live in fear" ( href="http://www.spaceagecity.com/bradbury/bio.htm">Jepsen and Johnston,
spaceagecity.com).

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"Shut up, shut up, shut up!" [His] was a plea, a
cry so terrible that Montag found himself on his feet, ... this man with the insane,
gorged face, the gibbering, dry mouth, the flapping book in his
fist.

Montag has been reading
his stolen books to Mildred, whose only response is, "Books aren't people. You read and
I look around, but there isn't anybody!" when an electronic dog comes sniffing at their
front door, exhaling "the smell of blue electricity blowing under the locked door."
Montag—the fireman—knows full well what the sniffing dog means. Beatty knows Montag has
stolen and expects the return of the book ("If I pick a substitute and Beatty does know
which book I stole, he'll guess we've an entire library here!") that very night. He is,
as he says himself, "numb" ("I'm numb, he thought") as he slams the house door and goes
to board the subway. He has decided to go to Faber and ask to have a duplicate of the
stolen book made so he can safely—safely for himself and Mildred and safely for the
book—return the stolen book to Beatty.

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"There's only one thing to do," he said. "Some
time before tonight when I give the book to Beatty, I've got to have a duplicate
made."

Riding on the subway
amongst so many people, Montag is both scared of what he is doing and earnestly
determined to memorize a portion of the New Testament that he holds open (foolhardy
action) in his hands. The Denham's Dentifrice jingle has all the passengers tapping
their feet and quietly singing along with the jingly words. The jingle acts as a
literary counterpoint as it lauds "Denham's Dentifrice. Denham's. Spelled D-E-N-" while
Montag struggles to retain the sentence "Consider the lilies of the field. ... Consider
the lilies, the lilies, the lilies ...." The old, meaningless society fights against
Montag's mind, as detergent would against impurities, "Denham's dental detergent," until
he breaks down, shouting "Shut up, shut up, shut up!" to the "rhythm of Denham's
Dentifrice, Denham's Dandy Dental Detergent, Denham's Dentifrice Dentifrice Dentifrice,
one two, one two three,...."

Montag battles, against dire
consequences, for thought under the fear, strain, desperation and desire that compel him
forward to Knoll View (symbolic as a rise from which to gain a vantage place for seeing
the panorama). There he hopes to perpetuate the life of a book that will stand against
the totality of "'the family'" and the "White Clown" and keep alive the society that he
seeks to embrace, the free society where books were desired. Montag's struggle for one
society over the other is the literary importance of this
scene in "The Sieve and the Sand." Montag's efforts to memorize "Consider the lilies of
the field" are so much sand through a sieve: "the sand was boiling, the sieve was empty.
Seated there in the midst of July, without a sound, he felt the tears move down his
cheeks."

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"'Denham's. Spelled : D-E-N-' 
They
toil not, neither do they...
A fierce whisper of hot sand through empty
sieve. 
'Denham's does it!'
Consider the lilies, the lilies, the
lilies...
'Denham's dental detergent.'" 
 

readability="16.218">Regarding social criticism, the social importance of the scene
is significant also. After all, Bradbury wrote Fahrenheit 451 as a
social dystopia novel because the new technology of television and the resultant
escalations in advertising were disconcerting to Bradbury and cause for his negative
prognostics of the future. As stated by Biography.com, Bradbury expressed his " href="http://www.biography.com/people/ray-bradbury-9223240#literary-works-and-honors">distaste
for television" when he explained Fahrenheit 451 as a
novel about "how television drives away interest in
reading."