Beauty of maths

The Beauty of MAtheMazic


1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321

1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 +10= 1111111111

9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8888
9876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888
987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888
98765432 x 9 + 0 = 888888888

Brilliant, isn't it?
And finally, take a look at this symmetry:

1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111=123456789 87654321

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Google fourth-quarter profit nearly triples

Google's fourth-quarter revenue rose 67 percent and profit nearly tripled on continued strength in its cash cow paid search business.

Handily beating analysts estimates, net earnings for the quarter ended December 31 were US$1.03 billion, or US$3.29 a share, including one-time items such as stock-based compensation, compared with US$372 million, or US$1.22 a share, during the same quarter in 2005. Excluding those items, earnings were US$997 million, or US$3.18 a share.

Total revenue for the quarter rose 67 percent to US$3.2 billion, compared to US$1.92 billion a year ago. Excluding traffic acquisition costs, or commission paid to content partners, revenue was US$2.23 billion.

Analysts polled by Thomson Financial were expecting Google to post earnings per share of US$2.92 excluding items, and revenue of US$2.19 billion, excluding traffic acquisition costs.

Paid search represents nearly all of Google's revenue. The company is expanding its advertising platform beyond the Web into radio and print.

"We paid over US$3 billion in 2006 to our partners and this is a figure that we expect is going to increase as we ramp up our video, radio and print programs," founder Sergey Brin said in a conference call with analysts.

Asked about whether the company plans to help companies advertise on television, Chief Executive Eric Schmidt said: "We have already said that we are experimenting with traditional television advertising...There are many reasons to believe that the targeting technology we have invented can apply well, that advertisers will pay much higher rates for ads that are targeted."

As for YouTube, Google suspects content creators will take advantage of the direct link to fans that the video-sharing site provides, said Schmidt. "We can connect the copyright owner with the user," he said. "We're pushing for a model where the people who produce the content get some revenue back." In addition, Google is working on audio and video fingerprinting technologies to protect copyright holders, he said.

When it comes to search, paid clicks on ads on Google's network and its publisher partners' Web sites grew more than 60 percent in the aggregate, according to Chief Financial Officer George Reyes.

While things look rosy for Google, its largest search rival posted more modest fourth quarter results. Last week, Yahoo's earnings were down more than 60 percent from a year ago.

Web traffic to Google grew 24 percent in the fourth quarter from the same period a year earlier and Google accounts for more than half of all Web searches, according to Nielsen/NetRatings. The company is expected to capture two-thirds of the search advertising revenue this year, according to eMarketer.

Google shares, which closed at US$501.50, dropped as much as 2 percent in after-hours trading after the company failed to make so-called "whisper numbers" on the street that were slightly higher than the estimates that they beat. The stock has risen nearly 18 percent over the past year, about the same amount that Yahoo's has fallen.