WASHINGTON — The unemployment rate is suddenly sinking at the fastest pace in a half-century, falling to 9 percent from 9.8 percent in just two months — the most encouraging sign for the job market since the recession ended.

More than half a million people found work in January. A government survey found weak hiring by big companies. But more people appear to be working for themselves or finding jobs at small businesses.

The steepest two-month decline in unemployment since the Eisenhower administration is the latest sign that the economic recovery is picking up speed.

The service sector and manufacturing are growing again at pre-recession rates. The Dow Jones industrial average closed above 12,000 this week for the first time since mid-2008. And retail sales have reached a five-year high.

“It is not all rosy. But we seem to be headed in the right direction,” said economist Chris Rupkey at Bank of Tokyo-Mitsubishi. “The recovery is on track — warts and all.”

Yields on government bonds rose after the unemployment report came out, a sign that bond traders think the job market is improving and will lift the economy after a year and a half of only modest growth.

An unemployment rate of 9 percent remains very high by historical standards. But the swift decline in the rate could also lift confidence at a time when businesses and individuals are already spending more money, fueling more hiring and still-more spending.

Unemployment has not been this low since April 2009.

“It’s the thinking, ‘I survived so far and I’ll make it through,”’ said economist Nigel Gault of IHS Global Insight. “’I can buy the things I postponed buying over the past several years because I’m not worried about my job.”’

It could also encourage people who had given up to look for jobs again, which might push the unemployment rate up temporarily. People out of work aren’t counted as unemployed unless they’re looking for a job. Typically during a tight job market, some of the unemployed become discouraged and stop looking.

Still, economists think the unemployment rate will fall below 9 percent by year’s end — a far brighter outlook than they had a few months ago.

Those with jobs are making a little more, too. Average hourly earnings rose 8 cents to $22.86 in January. Over a 40-hour workweek, the increase works out to $3.20, a couple of cups of coffee.

Wages have risen nearly 2 percent over the past year, faster than the rate of inflation, which means people have more spending power.

The Labor Department survey of company payrolls showed a net gain of 36,000 jobs in January. That’s scarcely one-fourth the number needed to keep pace with population growth.

The government uses a separate survey of households to calculate the unemployment rate. It calls 60,000 households and asks people if they’re working or looking for a job.

This survey includes some people not counted in the payroll survey: the self-employed, farm workers and domestic help. It also includes those who work at small companies.

By contrast, in the payroll survey, about 140,000 businesses and government agencies send forms to the Labor Department showing how many people are on the payroll and how many hours they worked. The payroll survey can be slower than the household survey to recognize startup companies.

The number of people who described themselves as self-employed rose by 165,000 to 9.7 million in January, the report said. That was the most since May.

In addition, some economists said, the unusually snowy winter might have suppressed hiring at businesses in January. Some construction companies shut down, for instance, and transportation companies cut jobs for couriers and messengers. Neil Dutta, an economist at Bank of America Merrill Lynch, said the figure would probably rebound in February.

“The thumbprints of the weather were all over this report,” he said. “We know the job market is recovering.”

In another bright spot, the manufacturing sector added 49,000 jobs, the most since August 1998. Retailers added 28,000, the most in a year. Manufacturing output has grown for more than a year, and retail sales for January posted their biggest gain since 2005.

“The worst of the recession is behind us,” said Mike Dougherty, president of D&S Manufacturing in Black River Falls, Wis., which makes parts for manufacturers of farm machines and railroad equipment.

He expects demand for his products to rise this year and plans to add about 10 people to his 150-person work force.

The last time the unemployment rate fell so far so fast was in 1958, when it dropped to 6.2 percent from 7.1 percent in two months. At the time, the economy was bouncing back from an eight-month recession.

The rate is falling now in part because some people without jobs have stopped looking. The number who have given up looking rose to 2.8 million last month, from 2.6 million in December.

About 1 million of those workers said they were discouraged. The others stopped looking because they returned to school or for other reasons.

There are still nearly 14 million people unemployed in the United States. That’s about twice as in December 2007, when the recession began.

The government also said fewer jobs were created last year than first thought — a net 909,000, down from an estimated 1.1 million. The economy lost about 8 million jobs total in the two years before that.

In the past three months, the economy generated an average of 83,000 net jobs per month, according to the survey of business payrolls. It takes 125,000 jobs a month to keep up with population growth.

The weakness in the government payroll survey was widespread. Restaurants and hotels cut 2,200 jobs, governments 14,000, temporary help agencies 11,000 and financial services companies 10,000.

Health care, one of the few steady job generators through the downturn, added 11,000 jobs, the fewest in almost two years. Financial services lost 10,000 jobs.

The number of people who are employed part-time but would rather be working full-time fell to 8.4 million from 8.9 million in December. Combined with the 13.9 million unemployed and people who have given up looking for work, roughly 25 million people were “underemployed” last month. They amounted to 16.1 of the labor force, down from 16.7 percent in November.

While hiring has yet to pick up in force, businesses have cut back on layoffs. If you still have a job, you are less likely to lose it today than you have been at any point in the past 20 years.

As for those laid off during the recession, some are finding that striking out on their own is a better way.

Bobby Vasquez lost his job as a communications specialist for the city of Houston in April 2009. With financial help from friends and business associates, he launched a website about Deer Park, the Houston suburb where he lives. He doesn’t make as much money as before. But he says he is happier.

“Because of the way the economy worked out, I’m actually achieving a dream 10 or 20 years earlier than I thought I would,” he said.