SAN FRANCISCO, Oct. 16 — Silicon Valley's math is getting fuzzy again. Internet companies with funny names, little revenue and few customers are commanding high prices. And investors, having seemingly forgotten the pain of the first dot-com bust, are displaying symptoms of the disorder known as irrational exuberance.

Consider Facebook, the popular but financially unproven social network, which is reportedly being valued by investors at up to $15 billion. That is nearly half the value of Yahoo, a company with 38 times the number of employees and, based on estimates of Facebook's income, 32 times the revenue.

Google, which recently surged past $600 a share, is now worth more than I.B.M., a company with eight times the revenue. More broadly, Internet start-ups are drawing investment based on their ability to build an audience, not bring in revenue — the very alchemy that many say led to the inflation and bursting of the dot-com bubble.

The surge in the perceived value of some start-ups has even surprised some entrepreneurs who are benefiting from it. A year ago, Yahoo invested in Right Media, a New York-based company developing an online advertising network. Yahoo's investment valued the firm at $200 million. Six months later, when Yahoo acquired Right Media outright, the purchase price had swelled to $850 million.

What changed? According to Right Media's chief technology officer, Brian O'Kelley, very little, except that Yahoo's rivals, Microsoft and Google, were writing billion-dollar checks to buy online advertising networks, and Yahoo thought it needed to pay any price to keep up.

"I have to say I giggled," Mr. O'Kelley, 30, said of the deal that earned him millions. He has since left Right Media and is starting another company. "There is no way we quadrupled the value of the company in six months."

The trend is described as a return to madness (by skeptics) or as a rational approach to unlimited opportunities presented by the Internet (by true believers). Greed, fear and a desperate rush to pick the next big winner are all adding fuel to the fire that is Silicon Valley's resurgence.

"There's definitely a lot of betting going on, and it's not rational," said Tim O'Reilly, a technology conference promoter and book publisher.

Mr. O'Reilly is credited with coining the phrase "Web 2.0," which refers to a new generation of Web sites that encourage users to contribute material. His Web 2.0 conference, which begins Wednesday in San Francisco, has become a nexus for the optimism around the latest set of society-changing online tools. But that has not stopped Mr. O'Reilly from worrying that the industry is minting too many copycat companies, half-baked business plans and overpriced buyouts.

When the bubble inevitably pops, he said, "there are going to be a lot of people out of work again." Putting a value on start-ups has always been a mix of science and speculation. But as in the first dot-com boom and the recent surge in housing, seasoned financial professionals are seeming to indulge in some strange instinct to turn away from the science and lean instead on the speculation.

This time around, people indulging in that optimistic thinking are not mom-and-pop investors or day traders but venture capitalists whose coffers are overflowing with money from university endowments and hedge funds. Many of those financial professionals say that this time, everything is different.

More than 1.3 billion people around the world use the Internet, many with speedy broadband connections and a willingness to immerse themselves in digital culture. The flood of advertising dollars to the Web has become an indomitable trend and a proven way for these start-ups to make money, while the revenue models of the dot-coms of yesteryear were often little more than sleight of hand.

"The environmental factors are much different than they were eight years ago," said Roelof Botha, a partner at Sequoia Capital and an early backer of YouTube. "The cost of doing business has declined dramatically, and traditional media companies have also woken up to the opportunities of the Web.

"That does open up the aperture for a different outcome this time," he said. Some trace the start of the new bubble to eBay's $3.1 billion acquisition of the Internet telephone start-up Skype in 2005. EBay's chief executive, Meg Whitman, reportedly outbid Google for the company. This month, eBay conceded it had grossly overpaid for Skype by about $1.43 billion, and announced that Niklas Zennstrom, a Skype co-founder, had left the company.

Google's acquisition of YouTube last year for $1.65 billion, under similarly competitive bidding, might have accelerated the transition to loftier values. Google executives and many analysts argued that YouTube was well worth the price tag if it became the next entertainment juggernaut.

It still might. More than 205 million people visit YouTube each month, according to the research firm comScore. Still, Citigroup estimated that YouTube would bring in $135 million in revenue next year. At that rate, YouTube would have to grow considerably to account for just 5 percent of Google's annual revenue of nearly $12 billion.

"We are almost going back to year 2000 types of errors," said Aaron Kessler, an Internet analyst at Piper Jaffray. Internet companies "are buying users instead of revenue and profitability," he said.

The Skype and YouTube windfalls helped to give the newest batch of Internet entrepreneurs dreams of improbable wealth. They also brought back practices that had seemingly been discredited during the first boom. For example, in the first dot-com gold rush, Internet companies did not have to make money to acquire serious investments dollars. Now that once again is true.

Twitter, a company in San Francisco that lets users alert friends to what they are doing at any given moment over their mobile phones, recently raised an undisclosed amount of financing. Its co-founder and creative director, Biz Stone, says that the company was not currently focused on making money and that no one in the company was even working on how to do so.

"At the moment, we're focused on growing our network and our user experience," he said. "When you have a lot of traffic, there's always a clear business model."

That is not necessarily illogical in the current climate. A European competitor, Jaiku, which is similarly devoid of a mature business model, was acquired last week by Google for an undisclosed sum. With the competitive logic that prevails at the major Internet companies, the deal might have further raised Twitter's appeal to Google's rivals.

The high value placed on many start-ups and minimal requirements for financial performance are raising expectations of other entrepreneurs. Sharon Wienbar, managing director of Scale Ventures Partners, an investment firm, cited the $100 million valuation that investors gave to the Internet genealogy site Geni.com, founded last year in Los Angeles by a veteran of PayPal.

"Now every entrepreneur thinks he should get that," Ms. Wienbar said. "I have a feeling a lot of entrepreneurs are secretly meeting for beers on the Peninsula, saying, ‘Hey, look what I got.'"

Mr. O'Kelley, formerly of Right Media, said other entrepreneurs had begun to think that the financing game is best played by avoiding actual revenues — since that only limits the imagination of investors. "It's a screwed-up incentive structure, just like you had in the first bubble," he said.

Another company benefiting from the exuberance is Ning, which allows users to create their own MySpace-style ad-supported social networks. It was recently valued by investors at more than $200 million, mainly because its main backer and founder, Marc Andreessen, has a successful history with the Internet hits Netscape and Opsware.

Mr. Andreessen argues on his blog that there is no bubble and that the high prices represent a rational desire to stake a claim in the potentially huge markets of the future. But he acknowledges that a seemingly inexhaustible flood of capital into Silicon Valley is helping to power the boom. Venture capitalists are flush with cash from institutional investors, eager for Internet-style returns on their money.

"The upward valuations pressure is the result of decisions being made by people wearing suits in cities like New York and Boston who would never ever meet with start-ups," Mr. Andreessen said in an interview. "If that ever goes away, it will have consequences. But it doesn't look like they will change their minds."